Solucion ejercicios beer

COSMOS: Complete Online Solutions Manual Organization System
Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P. Beer, E. Russell Johnston, Jr.,
Elliot R. Eisenberg, William E. Clausen, David Mazurek, Phillip J. Cornwell
© 2007 The McGraw-Hill Companies.
Chapter 8, Solution 1.
FBD Block B:
Tension in cord is equal to lb25=AW from FBD’s of block A and
pulley.
0: cos30 0,y BF N WΣ = − ° = cos30BW= °N
(a) For smallest ,BW slip impends up the incline, and
0.35 cos30s BF N Wµ= = °
0:xFΣ = 25 lb sin30 0BF W− + ° =
( )0.35cos30 sin30 25 lbBW° + ° =
min 31.1 lbBW =
(b) For largest ,BW slip impends down the incline, and
0.35 cos30s BF N Wµ= − = − °
0: sin30 25 lb 0x s BF F WΣ = + ° − =
( )sin30 0.35cos30 25 lbBW° − ° =
lb0.127max =BW

FBD Block B:
Tension in cord is equal to 40 lbAW = from FBD’s of block A and
pulley.
(a) ( )0: 52 lb cos25 0,yF NΣ = − ° = 47.128 lb=N
( )max 0.35 47.128 lb 16.495 lbsF Nµ= = =
( )eq0: 40 lb 52 lb sin 25 0xF FΣ = − + ° =
So, for equilibrium, lb024.18eq =F
Since eq max,F F> the block must slip (up since F > 0)
∴There is no equilibrium
(b) With slip, ( )0.25 47.128 lbkF Nµ= =
11.78 lb=F 35°

FBD Block:
Tension in cord is equal to 40 N,P = from FBD of pulley.
( )( )2
10 kg 9.81 m/s 98.1 N= =W
( ) ( ) 020sinN4020cosN1.98:0 =°+°−=Σ NFy
78.503 NN =
( )( )max 0.30 78.503 N 23.551 NsF Nµ= = =
For equilibrium: ( ) ( )0: 40 N cos20 98.1 N sin 20 0xF FΣ = ° − ° − =
eq max4.0355 N , Equilibrium existsF F= < ∴
eqF F= 4.04 N=F 20°

Tension in cord is equal to 62.5 N,P = from FBD of pulley.
( )( )2
10 kg 9.81 m/s 98.1 N= =W
( ) ( )0: 98.1 N cos20 62.5 N sin15 0yF NΣ = − ° + ° =
76.008 NN =
( )( )max 0.30 76.008 N 22.802 NsF Nµ= = =
For equilibrium: ( ) ( )0: 62.5 N cos15 98.1 N sin 20 0xF FΣ = ° − ° − =
eq max26.818 N so no equilibrium,F F= >
and block slides up the incline
( )( )slip 0.25 76.008 N 19.00 NxF Nµ= = =
19.00 N=F 20°

Tension in cord is equal to P from FBD of pulley.
( )( )2
10 kg 9.81 m/s 98.1 N= =W
( )0: 98.1 N cos20 sin 25 0yF N PΣ = − ° + ° = (1)
( )0: cos25 98.1 N sin 20 0xF P FΣ = ° − ° + = (2)
For impending slip down the incline, 0.3 NsF Nµ= = and solving
(1) and (2), 7.56 NDP =
For impending slip up the incline, 0.3 NsF Nµ= − = − and solving
(1) and (2), 59.2 NUP =
so, for equilibrium 7.56 N 59.2 NP≤ ≤

FBD Block:
( )( )2
20 kg 9.81 m/s 196.2 N= =W
For minθ motion will impend up the incline, so F is downward and
sF Nµ=
( ) ( )0: 220 N sin 196.2 N cos35 0yF N θΣ = − − ° =
( )0.3 220 sin 196.2 cos35 NsF Nµ θ= = + ° (1)
( ) ( )0: 220 N cos 196.2 N sin35 0xF FθΣ = − − ° = (2)
( ) ( ) ( )1 2 : 0.3 220 sin 196.2cos Nθ θ+ +
( ) ( )220 cos N 196.2sin35 Nθ= − °
or 220cos 66sin 160.751θ θ− =
Solving numerically: 28.9θ = °

FBD Block:
For minP motion will impend down the incline, and the reaction force R
will make the angle
( )1 1
tan tan 0.35 19.2900s sφ µ− −
= = = °
with the normal, as shown.
Note, for minimum P, P must be ⊥ to R, i.e. sβ φ= (angle between
P and x equals angle between R and normal).
(b) 19.29β = °
then ( ) ( )160 N cos 40P β= + °
( )160 N cos59.29 81.71 N= ° =
(a) min 81.7 NP =

FBD block (impending motion
downward)
( )1 1
tan tan 0.25 14.036s sφ µ− −
= = = °
(a) Note: For minimum P, ⊥P R
So ( )90 30 14.036 45.964β α= = ° − ° + ° = °
and ( ) ( ) ( )30 lb sin 30 lb sin 45.964 21.567 lbP α= = ° =
21.6 lbP =
(b) 46.0β = °

FBD Block:
For impending motion. ( )1 1
tan tan 0.40s sφ µ− −
= =
21.801sφ = °
Note 1,2 1,2 sβ θ φ= −
From force triangle:
s 1,2
10 lb 15 lb
sin sinφ β
=
( )1
1,2
33.85415 lb
sin sin 21.801
10 lb 146.146
β − ° 
= ° =  
°  
So 1,2 1,2
55.655
167.947
sθ β φ
°
= + = 
°
So (a) equilibrium for 0 55.7θ≤ ≤ °
(b) equilibrium for 167.9 180θ° ≤ ≤ °

FBD A with pulley:
FBD E with pulley:
Tension in cord is T throughout from pulley FBD’s
0: 2 20 lb = 0,yF TΣ = − 10 lbT =
For max,θ motion impends to right, and
( )1 1
tan tan 0.35 19.2900s sφ µ− −
= = = °
From force triangle,
( )
( )
20 lb 10 lb
, 2sin sin
sin sin
s s
s s
φ θ φ
θ φ φ
= = −
−
( )1
sin 2sin19.2900 19.2900 60.64θ −
= ° + ° − °
max 60.6θ = °

FBD top block:
FBD bottom block:
FBD block:
10: 196.2 N 0yF NΣ = − =
1 196.2 N=N
(a) With cable in place, impending motion of bottom block requires
impending slip between blocks, so ( )1 1 0.4 196.2 NsF Nµ= =
1 78.48 N=F
20: 196.2 N 294.3 N 0yF NΣ = − − =
2 490.5 N=N
( )2 2 0.4 490.5 N 196.2 NsF Nµ= = =
0: 78.48 N 196.2 N 0xF PΣ = − + + =
275 N=P
(b) Without cable AB, top and bottom blocks will move together
0: 490.5 N 0, 490.5 NyF N NΣ = − = =
Impending slip: ( )0.40 490.5 N 196.2 NsF Nµ= = =
0: 196.2 N 0xF PΣ = − + =
196.2 N=P

FBD top block:
FBD bottom block:
FBD block:
Note that, since ( )1 1
tan tan 0.40 21.8 15 ,s sφ µ− −
= = = ° > ° no motion
will impend if 0,P = with or without cable AB.
(a) With cable, impending motion of bottom block requires impending
slip between blocks, so 1 sF Nµ=
1 10: cos15 0,yF N W′Σ = − ° = 1 1 cos15 189.515 NN W= ° =
( )1 1 1 10.40 cos15 0.38637sF N W Wµ= = ° =
1 75.806 N=F
0:xF ′Σ = 1 1 sin15 0T F W− − ° =
75.806 N 50.780 N 126.586 NT = + =
( )( )2
2 30 kg 9.81 m/s 294.3 NW = =
( ) ( )20 : 189.515 N cos 15 294.3 NyF NΣ = − ° −
( )75.806 N sin15 0+ ° =
2 457.74 N=N
( )( )2 2 0.40 457.74 N 183.096 NsF Nµ= = =
( ) ( )0: 189.515 N 75.806 N cos15xF PΣ = − + + °
126.586 N 183.096 N 0+ + =
361 N=P
(b) Without cable, blocks remain together
1 20: 0yF N W WΣ = − − = 196.2 N 294.3 NN = +
490.5 N=
( )( )0.40 490.5 N 196.2 NsF Nµ= = =
0: 196.2 N 0xF PΣ = − + = 196.2 N=P

FBD A:
FBD B:
Note that slip must impend at both surfaces simultaneously.
10: N sin 16 lb = 0yF T θΣ = + −
1 16 lb sinN T θ= −
Impending slip: ( )( )1 1 0.20 16 lb sinsF N Tµ θ= = −
( )1 3.2 lb 0.2 sinF T θ= − (1)
10: cos 0xF F T θΣ = − = (2)
2 1 2 10: 24 lb 0, 24 lbyF N N N NΣ = − − = = +
30 lb sinT θ= −
Impending slip: ( )( )2 2 0.20 30 lb sinsF N Tµ θ= = −
6 lb 0.2 sinT θ= −
1 20: 10 lb 0xF F FΣ = − − =
( ) ( ) ( )1 2 1 110 lb 0.2 24 lbs N N N Nµ  = + = + + 
1 110 lb 0.4 N 4.8 lb, 13 lbN= + =
Then ( )( )1 1 0.2 13 lb 2.6 lbsF Nµ= = =
Then ( )1 : sin 3.0 lbT θ =
( )2 : cos 2.6 lbT θ =
Dividing 13 3
tan , tan 49.1
2.6 2.6
θ θ −
= = = °
49.1θ = °

FBD’s:
A:
B:
Note: Slip must impend at both surfaces simultaneously.
1 10: 20 lb 0, 20 lbyF N NΣ = − = =
Impending slip: ( )( )1 1 0.25 20 lb 5 lbsF Nµ= = =
0: 5 lb 0, 5 lbxF T TΣ = − + = =
( ) ( )20: 20 lb + 40 lb cos 5 lb sin 0yF N θ θ′Σ = − − =
( ) ( )2 60 lb cos 5 lb sinN θ θ= −
Impending slip: ( )( )2 2 0.25 60cos 5sin lbsF Nµ θ θ= = −
( ) ( )20: 5 lb 5 lb cos 20 lb 40 lb sin 0xF F θ θ′Σ = − − − + + =
20cos 58.75sin 5 0θ θ− + − =
Solving numerically, 23.4θ = °

FBD:
For impending tip the floor reaction is at C.
( )( )2
40 kg 9.81 m/s 392.4 N= =W
For impending slip ( )1 1
tan tan 0.35s sφ φ µ− −
= = =
19.2900φ = °
0.8 m 0.4 m
tan , 1.14286 m
0.35
EG
EG
φ = = =
0.5 m 0.64286 mEF EG= − =
(a) 1 1 0.64286 m
tan tan 58.109
0.4 m 0.4 m
s
EF
α − −
= = = °
58.1sα = °
(b)
sin19.29 sin128.820
P W
=
°
( )( )392.4 N 0.424 166.379 NP = =
166.4 NP =
Once slipping begins, φ will reduce to 1
tan .k kφ µ−
=
Then maxα will increase.

First assume slip impends without tipping, so sF Nµ=
FBD
0: sin 40 0, sin 40yF N P W N W PΣ = + ° − = = − °
( )0.35 sin 40sF N W Pµ= = − °
0: cos40 0xF F PΣ = − ° =
( )0.35 cos40 0.35sin 40W P= ° + °
0.35317sP W= (1)
Next assume tip impends without slipping, R acts at C.
( ) ( ) ( )0: 0.8 m sin 40 0.5 m cos40 0.4 m 0AM P P WΣ = ° + ° − =
0.4458t sP W P= > from (1)
( )( )2
max 0.35317 40 kg 9.81 m/ssP P∴ = =
138.584 N=
(a) max 138.6 NP =
(b) Slip is impending

FBD Cylinder:
For maximum M, motion impends at both A and B
;A A A B B BF N F Nµ µ= =
0: 0x A B A B B BF N F N F NµΣ = − = = =
A A A A B BF N Nµ µ µ= =
( )0: 0 1y B A B A BF N F W N Wµ µΣ = + − = + =
or
1
1
B
A B
N W
µ µ
=
+
and
1
B
B B B
A B
F N W
µ
µ
µ µ
= =
+
1
A B
A A B B
A B
F N W
µ µ
µ µ
µ µ
= =
+
( )
1
0: 0
1
A
C A B B
A B
M M r F F M Wr
µ
µ
µ µ
+
Σ = − + = =
+
(a) For 0 and 0.36A Bµ µ= =
0.360M Wr=
(b) For 0.30 and 0.36A Bµ µ= =
0.422M Wr=

FBD’s:
(a) FBD Drum:
10
0: ft 50 lb ft 0
12
DM F
 
Σ = − ⋅ = 
 
60 lbF =
Impending slip:
60 lb
150 lb
0.40s
F
N
µ
= = =
FBD arm:
( ) ( ) ( )0: 6 in. 6 in. 18 in. 0AM C F NΣ = + − =
( )60 lb + 3 150 lb 390 lbC = − =
cw 390 lbC =
(b) Reversing the 50 lb ft⋅ couple reverses the direction of F, but the magnitudes of F and N are not changed.
Then, using the FBD arm: ( ) ( ) ( )0: 6 in. 6 in. 18 in. 0AM C F NΣ = − − =
( )60 lb 3 150 lb 510 lbC = + =
ccw 510 lbC =

FBD’s: For slipping, 0.30 NkF Nµ= =
(a) For cw rotation of drum, the friction force F is as shown.
From FBD arm:
( )( ) ( ) ( )0: 6 in. 600 lb 6 in. 18 in. 0AM F NΣ = + − =
600 lb + 3 0
0.30
F
F − =
600
lb
9
F =
Moment about ( )10 in. 666.67 lb in.D F= = ⋅
cw = 55.6 lb ft⋅M
(b) For ccw rotation of drum, the friction force F is reversed
( )( ) ( ) ( )0: 6 in. 600 lb 6 in. 18 in. 0AM F NΣ = − − =
600 lb 3 0
0.30
F
F− − =
600
lb
11
F =
Moment about
10 600
ft lb 45.45 lb ft
12 11
D
  
= = ⋅  
  
ccw 45.5 lb ft= ⋅M

FBD: (a) ( )0: 0,CM r F T T FΣ = − = =
Impending slip: sF Nµ= or
s s
F T
N
µ µ
= =
( )0: cos 25 sin 25 0xF F T WθΣ = + ° + − ° =
( )1 cos 25 sin 25T Wθ + ° + = °  (1)
( )0: cos25 sin 25 0yF N W T θΣ = − ° + ° + =
( )
1
sin 25 cos25
0.35
T Wθ
 
+ ° + = ° 
 
(2)
Dividing (1) by (2):
( )
( )
1 cos 25
tan 25
1
sin 25
0.35
θ
θ
+ ° +
= °
+ ° +
Solving numerically, 25 42.53θ° + = °
17.53θ = °
(b) From (1) ( )1 cos42.53 sin 25T W+ ° = °
0.252T W=

FBD ladder:
Note: slope of ladder
4.5 m 12
,
1.875 m 5
= = so ( )
13
4.5 m 4.875
12
AC = =
6.5 m,L = so
4.875 m 3 1
,
6.5 m 4 2
AC L AD L= = =
and
1
4
DC BD L= =
For impending slip: ,A s A C s CF N F Nµ µ= =
Also 1 12
tan 15 52.380
5
θ −  
= − ° = ° 
 
0: sin15 cos sin 0x A C CF F W F Nθ θΣ = − ° + − =
10 10
sin15 cos sin
39 39
A sF W W Wµ θ θ= ° − +
( )0.46192 0.15652 s Wµ= −
0: cos15 sin cos 0y A C CF N W F Nθ θΣ = − ° + + =
10 10
cos15 sin cos
39 39
A sN W W Wµ θ θ= ° − −
( )0.80941 0.20310 s Wµ= −
But 2
: 0.46192 0.15652 0.80941 0.20310A A s s sF Nµ µ µ µ= − = −
2
4.7559 2.2743s sµ µ− +
0.539,sµ = 4.2166
min 0.539sµ =

FBD ladder:
Slip impends at both A and B, ,A s AF Nµ= B s BF Nµ=
0: 0,x A B B A s AF F N N F NµΣ = − = = =
0: 0,y A B A BF N W F N F WΣ = − + = + =
A s BN N Wµ+ =
( )2
1A sN Wµ+ =
( )
5 5
0: 6 m m m 0
4 2
O B AM N W N
   
Σ = + − =   
   
( )25 5
6 1 0
4 2
s A A s AN N Nµ µ+ + − =
2 24
1 0
5
s sµ µ+ − =
2.4 2.6sµ = − ± min 0.200sµ =

FBD rod:
(a) Geometry: cos cos tan
2 2
L L
BE DEθ θ β
 
= =  
 
cos
sin
2 tan s
L
EF L DF
θ
θ
φ
= =
So
1 cos
cos tan sin
2 2 tan s
L
L
θ
θ β θ
φ
 
+ = 
 
or
1 1 1
tan 2tan 2.5
tan 0.4s s
β θ
φ µ
+ = = = = (1)
Also, sin sinL L Lθ β+ =
or sin sin 1θ β+ = (2)
Solving Eqs. (1) and (2) numerically 1 14.62 66.85θ β= ° = °
2 248.20 14.75θ β= ° = °
Therefore, 4.62 and 48.2θ θ= ° = °
(b) Now 1 1
tan tan 0.4 21.801s sφ µ− −
= = = °
and
( )sin sin 90s s
T W
φ β φ
=
+ −
or
( )
sin
sin 90
s
s
T W
φ
β φ
=
+ −
For 4.62 0.526T Wθ = ° =
48.2 0.374T Wθ = ° =

FBD:
Assume the weight of the slender rod is negligible compared to P.
First consider impending slip upward at B. The friction forces will be
directed as shown and , ,B C s B CF Nµ=
( )0: sin 0
sin
B C
a
M L P Nθ
θ
 
Σ = − = 
 
2
sinC
L
N P
a
θ=
0: sin cos 0x C C BF N F Nθ θΣ = + − =
( )sin cosC s BN Nθ µ θ+ =
so ( )2
sin sin cosB s
L
N P
a
θ θ µ θ= +
0: cos sin 0y C C BF P N F Fθ θΣ = − + − − =
cos sinC s C s BP N N Nθ µ θ µ= − −
so ( ) ( )2 2
sin cos sin sin sin coss s s
L L
P P P
a a
θ θ µ θ µ θ θ µ θ= − − + (1)
Using 35θ = ° and 0.20, solve for 13.63.s
L
a
µ = =
To consider impending slip downward at B, the friction forces will be
reversed. This can be accomplished by substituting 0.20 insµ = −
equation (1). Then solve for 3.46.
L
a
=
Thus, equilibrium is maintained for 3.46 13.63
L
a
≤ ≤

FBD ABC:
( ) ( )0: 0.045 m + 0.30 m sin30 400 N sin30CM    Σ = ° °   
( ) ( )0.030 m + 0.30 m cos30 400 N cos30   + ° °   
( ) ( )
12 5
0.03 m 0.045 m 0
13 13
BD BDF F
   
− − =   
   
3097.64 NBDF =
FBD Blade:
( )
25
0: 3097.6 N 0 1191.4
65
xF N NΣ = − = =
( )0.20 1191.4 N 238.3 NsF Nµ= = =
( )
60
0: 3097.6 N 0
65
yF P FΣ = + − =
2859.3 238.3 2621.0 NP = − =
Force by blade 2620 N=P

FBD CD:
Note: The plate is a 3-force member, and for minimum ,sµ slip
impends at C and D, so the reactions there are at angle sφ from the
normal.
From the FBD, OCG 20 sφ= ° +
and ODG 20 sφ= ° −
Then ( ) ( )0.5 in. tan 20 sOG φ= ° +
and ( )
1.2 in.
0.5 in. tan 20
sin70
sOG φ
 
= + ° − 
° 
Equating, ( ) ( )tan 20 3.5540tan 20s sφ φ° + = ° −
Solving numerically, 10.5652sφ = °
( )tan tan 10.5652s sµ φ= = °
0.1865sµ =

FBD pin A: From FBD Whole the force at 750 lbA =
( )
4
0: 0,
5
x AB AB AB ABF F F F F′ ′Σ = − = =
3
0: 750 lb 2 0, 625 lb
5
y AB ABF F FΣ = − = =
FBD Casting:
0: 0,x D D D DF N N N N N′ ′Σ = − = = =
Impending slip , or D
D D D D
s
F
F F N Nµ
µ
′= = =
0: 2 750 lb 0, 375 lby D DF F FΣ = − = =
375 lb
D
s
N
µ
=
FBD ABCD:
( ) ( ) ( ) ( )
4
0: 12 in. 6 in. 42.75 in. 625 lb 0
5
CM N FΣ = − − =
( ) ( )( ) ( ) ( )
375 lb 4
12 in. 6 in. 375 lb 42.75 in. 625 lb 0
5sµ
= + =
0.1900sµ =

From FBD Whole, and neglecting weight of clamp compared to 550 lb plate, = −P W Since AB is a
two-force member, B is vertical and .B W=
FBD BCD:
( ) ( )0: 1.85 in. 2.3 in. cos40CM W DΣ = − °
( )0.3 in. sin 40 0, 0.94642D D W− ° = =
FBD EG:
( ) ( ) ( )0: 0.9 in. 1.3 in. 1.3 in. cos40 0E G G DM N F NΣ = − − ° =
Impending slip: G s GF Nµ=
Solving: ( )0.9 1.3 0.94250s GN Wµ− = (1)
FBD Plate:
By symmetry ,G G G G s GN N F F Nµ′ ′= = =
0: 2 0, ,
2 2
y G G G
s
W W
F F W F N
µ
Σ = − = = =
Substitute in (1): ( )0.9 1.3 0.94250
2
s
s
W
Wµ
µ
− =
Solving, 0.283,sµ = sm 0.283µ =

FBD table + child:
( )2
18 kg 9.81m/s 176.58 NCW = =
( )2
16 kg 9.81m/s 156.96 NTW = =
(a) Impending tipping about , 0, andF FE N F= =
( )( ) ( )( ) ( ) ( )0: 0.05 m 176.58 N 0.4 m 156.96 N 0.5 m cos 0.7 m sin 0EM P Pθ θΣ = − + − =
33cos 46.2sin 53.955θ θ− =
Solving numerically 36.3 and 72.6θ θ= − ° = − °
Therefore 72.6 36.3θ− ° ≤ ≤ − °
Impending tipping about F is not possible
(b) For impending slip: 0.2 0.2E s E E F s F FF N N F N Nµ µ= = = =
( ) ( )0: cos 0 or 0.2 66 N cosx E F E FF F F P N Nθ θΣ = + − = + =
0: 176.58 N 156.96 N sin 0y E FF N N P θΣ = + − − − =
( )66sin 333.54 NE FN N θ+ = +
So 330cos 66sin 333.54θ θ= +
Solving numerically, 3.66 and 18.96θ θ= − ° = − °
Therefore, 18.96 3.66θ− ° ≤ ≤ − °

Geometry of four-bar:
Considering the geometry when 0,α =
( ) ( )
1/ 2
2 2
60 mm 52 mm 36 mm 22 mm 58.549 mmCDL  = − + + =  
In general, ( ) ( )52 mm 36 mm sin 60 mm 58.549 mm sinα β− = −
so 1 36sin 8
sin
58.549
α
β − + 
=  
 
(a) FBD ACE: 0 7.8533 ,α β= = ° note that the links at E and K are prevented from pivoting
downward by the small blocks
0: sin 0,
sin 7.8533
E
y CD E CD
F
F F F FβΣ = − = =
°
( ) ( ) ( )0: 60 mm cos7.8533 32 mm 212 mm 0
sin 7.8533
E
A E E
F
M F N
 
Σ = ° − − = 
° 
Impending slip on pad ,E
E
s
F
N
µ
= so
212
435.00 32 0E
s
F
µ
 
− − = 
 
0.526sµ =
(b) 30 , 26.364α β= ° = °
3
0: cos26.364 0
2
x AB CD EF F F NΣ = − + ° − =
1
0: sin 26.364 0
2
y AB CD EF F F FΣ = − + ° − =
Eliminating ( ), 0.89599 0.76916 0AB CD E EF F N F− − + =
Impending slip ,E s EF Nµ= so ( )0.126834 1E s EF Nµ= −
( )0: 60 mm cos26.364A CDM FΣ = °
( ) ( )212 mm 32 mm 0E s EN Nµ− − =
( )53.759 212 32 0CD s EF Nµ= − =
212 32 53.759
1 0.12634
s
s
µ
µ
−
=
−
0.277sµ =

FBD ABD:
FBD Pipe:
FBD DF:
( ) ( )0: 15 mm 110 mm 0D A AΜ N FΣ = − =
Impending slip: A SA AF Nµ=
So 15 110 0SAµ− = 0.136364SAµ =
0.1364SAµ =
0: 0,x A xF F DΣ = − = x A SA AD F Nµ= =
60 mmr =
0: 0,y C A C AF N N N NΣ = − = =
( ) ( ) ( )0: 550 mm 15 mm 500 mm 0F C C xM F N DΣ = − − =
Impending slip: C SC C SC AF N Nµ µ= =
So, 550 15 500 0SC A A SA AN N Nµ µ− − =
( )550 15 500 0.136364SCµ = +
0.1512SCµ =

FBD Plate:
Assume reactions as shown, at ends of sleeves,
For impending slip ,A s A B s BF N F Nµ µ= =
0: sin 0x s A s BF P N Nθ µ µΣ = − − =
2.5 sinA BN N P θ+ =
0: cos 0, cosy A B A BF N N P N N Pθ θΣ = − − = − =
Solving: ( ) ( )2.5sin cos , 2.5sin cos
2 2
A B
P P
N Nθ θ θ θ= + = − (1)
( ) ( ) ( )0: 23.5 in. sin 16 in. 1 in. 0B A AM P N FθΣ = − + =
( ) ( ) ( )23.5 in. sin 16 in. 0.4 1 in. 2.5sin cos 0
2
P
P θ θ θ − − + =  (2)
4sin 7.8cos 0, 62.9θ θ θ− = = °
For 62.9 ,θ > ° the panel will be self locking, ∴motion for 62.9 .θ ≤ °
As θ decreases, BN will reverse direction at 2.5sin cos 0,θ θ− =
(see equ. 1) or at 21.8 .θ = ° So for 21.8θ ≤ °
( )0 : sin 0x s A BF P N Nθ µΣ = − + =
2.5 sinA BN N P θ+ =
0: cos 0, cosy A B A BF N N P N N Pθ θΣ = + − = + =
2.5sin cos , 21.8θ θ θ∴ = = °
So impending motion for 21.8 62.9θ° ≤ ≤ °

FBD Plate:
Assuming reactions as shown, at ends of sleeves,
For impending slip ,A s A B s BF N F Fµ µ= =
( )0: sin 0x s A BF P N Nθ µΣ = − + =
2.5 sinA BN N P θ+ = (1)
0: cos 0, cosy A B A BF N N P N N Pθ θΣ = − − = − = (2)
Solving: ( ) ( )2.5sin cos , 2.5sin cos
2 2
A B
P P
N Nθ θ θ θ= + = −
Note that, for 21.8 ,θ < ° BN becomes negative, so we must change equ. 2 to
cos ,A BN N P θ+ = ( 2′ )
but equ. (1) does not change. Solving (1) and ( 2′ ) gives cos 2.5 sin ,P Pθ θ=
or 21.8 ,θ = ° so the lower limit for impending slip is 21.8 .θ = °
For 21.8 ,θ ≥ ° the forces are as shown, and
( ) ( ) ( )0: 23.5 in. sin cos 1 in. 16 in. 0B A AM P xP F Nθ θΣ = + + − =
( ) ( ) ( ) ( )23.5 in. sin cos 0.4 1 in. 16 in. 2.5sin cos 0
2
P
P x Pθ θ θ θ + + − + = 
or ( )4sin 7.8 in. cos 0, tan 1.950
4 in.
x
xθ θ θ − − = = − 
(a) For 4 in.,x = tan 1.950,θ = 43.5 .θ = ° For 43.5θ > ° self locking
∴ impending motion for 21.8 43.5θ° ≤ ≤ °
(b) As x increases from 4 in., the upper bound for θ decreases, becoming
( )21.8 tan 0.4000θ° = when ( )( )4 in. 1.950 0.400 6.2 in.x = − =
Thus max 6.20 in.x =
at which θ must equal 21.8 .°

FBD Collar:
Impending motion down:
Impending motion up:
Stretch of spring
cos
a
x AB a a
θ
= − = −
( )( )
1
1.5 kN/m 0.5 m 1
cos cos
s
a
F kx k a
θ θ
   
= = − = −   
   
( )
1
0.75 kN 1
cosθ
 
= − 
 
0: cos 0x sF N F θΣ = − =
( )( )cos 0.75 kN 1 cossN F θ θ= = −
Impending slip: ( )( )( )0.4 0.75 kN 1 cossF Nµ θ= = −
( )( )0.3 kN 1 cosθ= −
+ down, – up
0: sin 0y sF F F WθΣ = ± − =
( )( ) ( )( )0.75 kN tan sin 0.3 kN 1 cos 0Wθ θ θ− ± − − =
or ( ) ( ) ( )[ ]0.3 kN 2.5 tan sin 1 cosW θ θ θ= − ± −
with 30 :θ = ° ( )up 0.01782 kN OKW =
( )down 0.0982 kN OKW =
Equilibrium if 17.82 N 98.2 NW≤ ≤

Geometry:
FBD B:
( ) ( )1 m 0.5 m cos tan 0.5 m sinα θ α − = 
( )tan 2 cos sinθ α α− =
30 60θ α= ° → = °
then ( )
3
1 m cos30 m
2
ABL = ° =
( )0
kN 3 1
1.5 m m
m 2 2
s ABF k L L
  
= − = −     
( )0.75 3 1 kN 549.04 NsF = − =
0: sin60 549.04 N 0xF F WΣ = + ° − =
549.04 N
2
W
F = −
3
0: cos60 0,
2
yF N W N WΣ = − ° = =
For impending slip upward, F is as shown and ,sF Nµ= so
3
549.04 N 0.40 ,
2 2
W
W− = min 648.61 NW =
For impending slip downward, F is reversed, or ,sF Nµ= − so
max
3
549.04 N 0.40 , 3575 N
2 2
W
W W− = − =
( )2
9.81 m/s
W
m = so 66.1 kg 364 kgm≤ ≤

FBD Collar:
FBD AB:
Note: BC is a two-force member, and for max,M slip will impend to the
right.
0: cos 0, cosy BC BCF F N N Fθ θΣ = − = =
Impending slip: coss s BCF N Fµ µ θ= =
0: sin 0x BCF F F PθΣ = − − =
( )sin cosBC sF Pθ µ θ− =
( )0: 2 cos 0A ABM M l F θΣ = − =
2 cos
sin coss
P
M l θ
θ µ θ
=
−
max
2
tan s
Pl
M
θ µ
=
−
max
max
For tan ,
self locking
For tan , 0
s
s
M
M
µ θ
µ θ
= = ∞

> < 

Geometry:
FBD AB:
1 2 4 2cos 60
2
L L L
L
θ −
+ −
= = °
For min
a
L
slip will impend to right and reactions will be at
( )1 1
tan tan 0.35 19.2900s sφ µ− −
= = = ° from normal.
Note: AB is a three-force member
( ) ( ) ( )tan 60 tan 60s sCD a L aφ φ= + = − ° −
( ) ( ) ( )tan 79.29 tan 40.71a L a° = − °
6.1449 1
L
a
= −
0.13996
a
L
=
min 0.1400
a
L
=

FBD A:
FBD B:
FBD A:
FBD B:
Note: Rod is a two force member. For impending slip the reactions are at
angle
( )1 1
tan tan 0.40 21.801s sφ µ− −
= = = °
Consider first impending slip to right
9 lb
3.8572 lb
tan66.801
ABF = =
( ) ( )0: 3.8522 lb sin30 6 lb cos30 0y BF NΣ = − ° − ° =
( )7.1223 lb, 0.40 7.1223 lbB B s BN F Nµ= = =
2.8489 lbBF =
( ) ( )0: 2.8489 lb 3.8572 lb cos30 6 lb sin30 0xF PΣ = − + ° − ° − =
min 2.508 lbP = −
Next consider impending slip to left
( )9 lb tan66.801 21.000 lbABF = ° =
( ) ( )0: 21 lb sin30 6 lb cos30 0, 15.6959 lby B BF N NΣ = − ° − ° = =
( )0.4 15.6959 lb 6.2784 lbB s BF Nµ= = =
( ) ( )0: 6.2784 lb 21 lb cos30 6 lb sin30 0Fx PΣ = + ° − ° − =
max 21.465 lbP =
equilibrium for 2.51 lb 21.5 lbP− ≤ ≤

FBD AB:
( )2 2
0: 8 in 4 in 0A AM N MΣ = + − =
( )( )12 lb ft 12 in./ft
16.100 lb
8.9443 in.
N
⋅
= =
Impending motion: ( )0.3 16.100 lb 4.83 lbsF Nµ= = =
Note: For max MC, need F in direction shown; see FBD BC.
FBD BC + collar:
( ) ( ) ( )
1 2 2
0: 17 in. 8 in. 13 in. 0
5 5 5
C CM M N N FΣ = − − − =
or ( ) ( ) ( )
17 in. 16 in. 26 in.
16.100 lb 16.100 lb 4.830 lb 293.77 lb in.
5 5 5
CM = + + = ⋅
( )max
24.5 lb ftC = ⋅M

FBD yoke:
FBD wheel and slider:
0: 0, 8 lbxF P N N PΣ = − = = =
For impending slip, ( )125 8 lbsF Nµ= =
2 lbF =
For max,M F on yoke is down as shown
For min,M F on yoke is up.
(a) For maxM the 2 lb force is up as shown.
( ) ( ) ( ) ( )0: 3 in. sin 65 8 lb 3 in. cos65 2 lb 0B BM M    Σ = − ° − ° =   
max 24.3 lb in.B = ⋅M
(b) For minM the 2 lb force is reversed, and
( ) ( ) ( ) ( )0: 3in. sin65 8 lb 3 in. cos65 2 lb 0B BM M    Σ = − ° + ° =   
min 19.22 lb in.B = ⋅M

FBD Rod:
FBD Cylinder:
( ) ( )( )10: 20 in. 12.5 in. 12 lb 0AM NΣ = − =
1 7.5 lb.N =
2 20: 7.5 lb 36 lb 0, 43.5 lbyF N NΣ = − − = =
since 1 2µ µ= and 1 2,N N< slip will impend at top of cylinder first, so
1 1sF Nµ= .
( )1 0.35 7.5 lb 2.625 lbF = =
( ) ( )( )0: 4.25 in. 12.5 in. 2.625 lb 0, 7.7206 lbDM P PΣ = − = =
max 7.72 lbP =
To check slip analysis above, 20: 36 lb 7.5 lb 0yF NΣ = − − =
2 43.5 lbN =
( )2max 2 0.35 43.5 lb 15.225 lbsF Nµ= = =
1 2 20: 0, 7.72 lb 2.625 lb 0xF P F F FΣ = − − = − − =
2 max5.095 lb ,F F= < OK

FBD pulley:
FBD block A:
( )( )2
2.4 kg 9.81 m/s 23.544 NAW = =
FBD block C:
Note that ( )1 1
tan tan 0.5 26.565 30 ,SA SAφ µ− −
= = = ° < ° Cable is needed
to keep A from sliding downward.
0: 2 0, , 2
2
B
y B B
W
F T W T W TΣ = − = = = (1)
(a) For minimum ,BW there will be impending slip of block A
downward, and A SA AF Nµ= as shown.
0: cos30 0, cos30A A A AFy N W N W′Σ = − ° = = °
23.544 Ncos30 20.390 N= ° =
( )( )0.50 20.390 N 10.195 NAF = =
0: sin30 0x A AF T W F′Σ = − ° + =
( )23.544 N sin30 10.195 N 1.577 NT = ° − =
From (1)
( )2
3.154 N
2 3.154 N, 0.322 kg,
9.81 m/s
B BW T m= = = =
min 322 gBm =
0: 0, 58.86 Ny C C CF N W NΣ = − = =
( )max 0.30 58.86 N 17.658 NC SC CF Nµ= = =
Since max1.577 N ,CT F= < block B doesn’t slip and above answer for
minBm is correct.

(b) For maxBm assume impending slip of block C to left, maxCF F=
max0: 0, 17.658 Nx C C CF T F T F FΣ = − + = = = =
From (1) 2
35.316 N
2 35.316 N, 3.6 kg
9.81 m/s
B
B B
W
W T m
g
= = = = =
From FBD block A,
0: sin30 0, sin30x A A A AF T W F F W TΣ = − ° + = = ° −
( ) max23.544 N sin30 17.658 N 5.886, 10.195 NA AF F= ° − = − =
Since max,A AF F< A does not slip max 3.6 kgBM =

FBD A:
FBD B and C:
FBD B:
For impending motion A must start up and C down the incline. Since the
normal force between A and B is less than that between B and C, and the
friction coefficients are the same, maxF will be reached first between A
and B, and B and C will stay together.
( )1 10: 4 lb cos30 0, 2 3 lbyF N NΣ = − ° = =
Impending slip: 1 1 2 3 lbs sF Nµ µ= =
( )0: 4 lb sin30 2 3 lb 0x sF T µΣ = − ° − =
( )2 1 3 lbsT µ= + (1)
( )20: 2 3 lb 3 lb 8 lb cos30 0yF NΣ = − − + ° =
2
15
3 lb
2
N =
Impending slip: 2 2
15
3 lb
2
s sF Nµ µ= =
( )
15
0: 2 3 3 lb 3 8 lbsin30 0
2
µ
  
Σ = + + − + ° =    
x sF T
11 19
3 lb
2 2
sT µ
 
= − 
 
(2)
Equating (1) and (2): ( )4 1 3 lb 11 19 3s sµ µ+ = −
min23 3 7, 0.1757s sµ µ= =
To check slip reasoning above:
( )3 3
7
0: 2 3 lb 3 lb cos30 0, 3 lb
2
yF N NΣ = − − ° = =
3max 3
7
3
2
s sF Nµ µ= =
( ) 30: 3 lb sin30 2 3 lb 0x sF FµΣ = − ° + − =
( )3
3
2 3 0.1757 lb lb 0.891 lb
2
F = − = −
3 3max,F F< OK

FBD rod:
( )
3 in.
0: 4.5 in. cos 0
cos
A BM N Wθ
θ
 Σ = − = 
or ( )2
1.5cosBN Wθ=
Impending motion: ( )2
1.5 cosB s B sF N Wµ µ θ= =
( )2
0.3cos Wθ=
0: sin cos 0x A B BF N N Fθ θΣ = − + =
or ( ) ( )2
1.5cos sin 0.2cosAN Wθ θ θ= −
Impending motion: A s AF Nµ=
( ) ( )2
0.3cos sin 0.2cosWθ θ θ= −
0: cos sin 0y A B BF F N F Wθ θΣ = + + − =
or ( )3 2
1 1.5cos 0.3cos sinAF W θ θ θ= − −
Equating FA’s
( )2 3 2
0.3cos sin 0.2cos 1 1.5cos 0.3cos sinθ θ θ θ θ θ− = − −
2 3
0.6cos sin 1.44cos 1θ θ θ+ =
Solving numerically 35.8θ = °

FBD pin A:
FBD B:
FBD C:
12 3
0: 0
13 5
x AB ACF F FΣ = − =
5 4
0: 0
13 5
y AB ACF F F PΣ = + − =
Solving:
13 20
,
21 21
AB ACF P F P= =
12 13 12
0: 0,
13 21 21
x B BF N P N PΣ = − ⋅ = =
For minP slip of B impends down, so
11
20
B s B BF N Nµ= =
min
11 12 5 13
0: 18 lb 0, 236.25 lb
20 21 13 21
yF P P PΣ = ⋅ − ⋅ − = =
(For 236.25 lb,P < A will slip down)
4 20 16
0: 80 lb 0, 80 lb
5 21 21
y C CF N P N PΣ = − − ⋅ = = +
For maxP slip of C impends to right, C s CF Nµ=
or
11 16 44
80 lb 44 lb
20 21 105
CF P P
 
= + = + 
 
3 20
0: 0,
5 21
x CF P FΣ = ⋅ − =
12 44
44 lb
21 105
P P= +
max 288.75 lbP =
∴ equilibrium 236 289P≤ ≤

( )1 1
tan tan 0.4 21.801 ,s sφ µ− −
= = = ° slip impends at wedge/block wedge/wedge and block/incline
FBD Block:
2 530 lb
sin 41.801 sin 46.398
R
=
° °
2 487.84 lbR =
FBD Wedge:
487.84 lb
sin51.602 sin 60.199
P
=
° °
441 lbP =

( )1 1
tan tan 0.40 21.801 ,s sφ µ− −
= = = ° and slip impends at wedge/lower block, wedge/wedge, and upper
block/incline interfaces.
FBD Upper block and wedge:
2 530 lb
sin 41.801 sin38.398
R
=
° °
2 568.76 lbR =
FBD Lower wedge:
568.76 lb
sin51.602 sin68.199
P
=
° °
480 lbP =

( )( )2
18 kg 9.81 m/s 176.58 NDW = =
( )( )3.5 kN/m 0.1 m 0.35 kN = 350 NsF kx= = =
( )1 1
tan tan 0.25 14.0362s sφ µ− −
= = = °
FBD Lever:
( )( ) ( )( )0: 0.3 m 350 N 0.4 m 176.58 NCMΣ = −
( )0.525 m cos4.0362AR− °
( )0.05 m sin 4.0362 0AR+ ° =
66.070 NAR =
( )0: 66.07 N sin 4.0362 0,x xF CΣ = ° + = 4.65 NxC = −
( )0: 66.07 N cos4.0362 350 N 176.58 N = 0yFΣ = ° − −
FBD Wedge:
66.070 N
sin18.072 sin75.964
P
=
° °
21.1 lbP =
(a) 21.1 lbP =
(b) 4.65 Nx =C
461 Ny =C

( )( )2
18 kg 9.81 m/s 176.58 NDW = =
( )( )3.5 kN/m 0.1 m 0.35 kN = 350 NsF kx= = =
( )1 1
tan tan 0.25 14.0362s sφ µ− −
= = = °
FBD Lever:
( )( ) ( )( )0: 0.3 m 350 N 0.4 m 176.58 NCMΣ = −
( )0.525 m cos24.036AR− °
( )0.05 m sin 24.036 0AR− ° =
68.758 NAR =
( )0: 68.758 N sin 24.036 0,x xF CΣ = − ° = 28.0 NxC =
( )0: 350 N 176.58 N + 68.758 N cos24.036 0y yF CΣ = − − ° =
464 NyC =
FBD Wedge:
68.758 N
sin38.072 sin75.964
P
=
° °
(a) 43.7 NP =
(b) 28.0 Nx =C
464 Ny =C

For steel/steel contact, ( )1 1
1 1
tan tan 0.3 16.6992s sφ µ− −
= = = °
For steel/concrete interface, ( )2 2
1 1
tan tan 0.6 30.964s sφ µ− −
= = = °
FBD Plate CD:
0: 90 kN 0,yF NΣ = − = 90 kNF =
Impending slip: ( )1
0.3 90 kN 27 kNsF Nµ= = =
0: 0,xF F QΣ = − = 27 kNQ F= =
FBD Top wedge assuming impending slip between wedges:
0: cos26.699 90 kN = 0,y wF RΣ = ° − 100.74 kNwR =
( )0: 27 kN 100.74 kN sin 26.699 0xF PΣ = − − ° =
72.265 kN,P = (a) 72.3 kN=P
(b) 27.0 kN=Q
To check above assumption; note that bottom wedge is a two-force member so the reaction of the floor on that
wedge is Rw, at 26.699° from the vertical. This is less than 2
30.964 ,sφ = ° so the bottom wedge doesn’t slip
on the concrete.

For steel/steel contact, ( )1 1
1 1
tan tan 0.30 16.6992s sφ µ− −
= = = °
For steel/concrete contact, ( )2 2
1 1
tan tan 0.60 30.964s sφ µ− −
= = = °
FBD Plate CD and top wedge:
90 kN tan 26.6992 = 45.264 kNQ = °
90 kN
100.741 kN
cos26.6992
wR = =
°
FBD Bottom wedge: slip impends at both surfaces
100.714 kN
sin57.663 sin59.036
P
=
° °
(a) 99.3 kN=P
(b) 45.3 kN=Q

FBD Wedge:
FBD Block C:
( )1 1
tan tan 0.4 21.801s sφ µ− −
= = = °
By symmetry B CR R=
( )0: 2 sin 29.801 0,y CF R PΣ = ° − = 0.9940 CP R=
175 lb
,
sin 41.801 sin18.397 lb
CR
=
°
367.3 lbP =
(a) 367 lbP =
b) Note: That increasing friction between B and the incline will mean that block B will not slip, but the above
calculations will not change.
(b) 367 lbP =

FBD Block C: ( )1 1
tan tan 0.4 21.8014s sφ µ− −
= = = °
0: 0x ACx CFxF R RΣ = − =
0: 175 lb = 0y CFy ACyF R RΣ = − −
so
175 lbCFy ACy
CFx ACx ACx
R R
R R R
− =
( ) ( )cot 20 cot 32.2 0φ° + − ° >
12.2 21.8sφ φ< ° < = °
so block C does not slip (or impend)
FBD Block B:
(a) ( )1 1
tan tan 0.4 21.8014B Bφ µ− −
= = = °
175 lb
,
sin 41.8014 sin 46.3972
BR
=
° °
161.083 lbBR =
(b) ( )1 1
tan tan 0.6 30.9638B Bφ µ− −
= = = °
175 lb
,
sin50.9638 sin37.2330
BR
=
° °
224.65 lbBR =
FBD Wedge:
,
sin59.6028 sin52.1986
BP R
=
° °
1.09163 BP R=
(a) 161.083 lb,BR = 175.8 lbP =
(b) 224.65 lb,BR = 245 lbP =

Since vertical forces are equal and ground wood,s sµ µ> assume no impending motion of board. Then there
will be impending slip at all wood/wood contacts, ( )1 1
tan tan 0.35 19.2900s sφ µ− −
= = = °
FBD Top wedge:
1
8 kN
8.4758 kN
cos19.29
R = =
°
1
sin52.710 cos56.580
R P
=
° °
8.892 kNP =
To check assumption, consider
FBD wedges + board:
( )1 1 8 kN = 0.35 8 kN 2.8 kNF µ= =
0: 8 kN 0,y GF NΣ = − = 8 kNGN =
( )( )max 0.6 8 kN 4.8 kNG G GF Nµ= = =
10: 0,x GF F FΣ = − = 1 2.8 kNGF F= =
max,G GF F< OK
8.89 kNP∴ =

Assume no impending motion of board on ground. Then there will be impending slip at all wood/wood
interfaces.
FBD Top wedge:
Wedge is a two-force member so 2 1= −R R
and ( )1 1
2 2tan 2tan 0.35s sθ φ µ− −
= = =
38.6θ = °
To check assumption, consider
FBD wedges + board:
( )1 18 kN = 0.35 8 kN 2.8 kNF µ= =
0: 8 kN 0,y GF NΣ = − = 8 kNGN =
( )( )max 0.6 8 kN 4.8 kNG G GF Nµ= = =
10: 0,x GF F FΣ = − = 1 2.8 kNGF F= =
max,G GF F< OK
8.89 kNP∴ =

FBD Cylinder:
Slip impends at B
( )1
tan 0.35 19.2900SCφ −
= = °
( )
4
0: cos 12 19.29 0
3
A C
r
M r R W
π
Σ = ° + ° − =
0.49665,CR = 124.163 lbW =
FBD Wedge:
( )1 1
tan tan 0.50 26.565SF SFφ µ− −
= = = °
124.163 lb
sin58.855 sin63.435
P
=
° °
117.5 lbP =

FBD tip of screwdriver:
( )1 1
tan tan 0.12 6.8428s sφ µ− −
= = = °
by symmetry 1 2R R=
( )10: 2 sin 6.8428 8 3.5 N 0yF RΣ = ° + ° − =
1 2 6.8315 NR R= =
If P is removed quickly, the vertical components of R1 and R2 vanish, leaving the horizontal components
( )1 2 6.8315 N cos14.8428H H= = °
6.6035 N=
Side forces = 6.60 N
This is only instantaneous, since 8 ,sφ° > so the screwdriver will be forced out.

As the plates are moved, the angle θ will decrease.
(a) 1 1
tan tan 0.2 11.31 .s sφ µ− −
= = = ° As θ decreases, the minimum angle at the contact approaches
12.5 11.31 ,sφ° > = ° so the wedge will slide up and out from the slot.
(b) 1 1
tan tan 0.3 16.70 .s sφ µ− −
= = = ° As θ decreases, the angle at one contact reaches 16.7 .° (At this
time the angle at the other contact is 25 16.7 8.3 )sφ° − ° = ° < The wedge binds in the slot.

FBD Wedge:
( )1 1
tan tan 0.35 19.2900s sφ µ− −
= = = °
by symmetry 1 2R R=
10: 2 sin 22.29 60 lb 0yF RΣ = ° − =
2 79.094 lbR =
When P is removed, the vertical component of R1 and R2 will vanish, leaving the horizontal components
( )1 2 79.094 lb cos22.29H H= = °
73.184 lb=
Final forces 1 2 73.2 lbH H= =
Since these are at ( )3 sφ° < from the normal, the wedge is self-locking and will remain in place.

FBD Cylinder:
note 3
tan30
r
d r= =
FBD Wedge:
( )( )2
80 kg 9.81 m/s 784.8 NW = =
0: 0,G A B A BM F F F FΣ = − = = (1)
0: 0,
3
D B A A B
W
M dN dN rW N NΣ = − + = = + (2)
so max max,A B A BN N F F> >
∴ slip impends first at B. 0.25B s B BF N Nµ= =
( ) ( ) ( )( )0: cos30 sin30 1 sin30 0.25 0A B BM r N r W r NΣ = ° − ° − + ° =
1.01828 799.15 NBN W= =
0.25 199.786 NB BF N= =
From (2) above,
784.8 N
799.15 N + 1252.25 N
3
AN = =
From (1), 199.786 NA BF F= =
( )0: 1252.25 N cos10 199.786 N sin10 0y CF NΣ = − ° + ° =
1198.53 NCN =
Impending slip ( )0.25 1198.53 N 299.63 NC s CF Nµ= = =
( )0: 299.63 N 199.786 N cos10xF PΣ = − − °
( )1252.25 N sin10 0− ° =
714 N=P 20.0°

FBD Cylinder:
( )( )2
80 kg 9.81 m/s 784.8 NW = =
For impending slip at B, 0.30B sB B BF N Nµ= =
( ) ( )( )0: cos30 1 sin30 0.30A B BM r N r NΣ = ° − + °
sin30 0r W− ° =
1.20185 943.21 NBN W= =
0.30 0.36055B BF N W= =
( )0: 0, 0.36055G A B A BM r F F F F WΣ = − = = =
0: sin30 cos30 0x A A BF N F NΣ = ° + ° − =
( )0.36055 cos30 1.20185
sin30
A
W W
N
− ° +
=
°
1.77920AN W=
For minimum ,Aµ slip impends at A, so
min
0.36055
0.2026
1.77920
A
A
A
F W
N W
µ = = =
min 0.203Aµ =

FBD plank + wedge:
( ) ( )( )( )0: 8 ft 1.5 ft 48 lb/ft 3 ftA BM NΣ = −
( ) ( )( )
1
2 ft 48 lb/ft 3 ft
2
−
( )( )
5 1
3 ft 96 lb/ft 5 ft 0
3 2
  
− + =  
  
185 lbBN =
( )
48 96
0: 185 lb lb/ft 3ft
2
y WF N
+ 
Σ = + −  
 
( )( )
1
96 lb/ft 5 ft 0
2
+ =
271 lbWN =
Since ,W BN N> and all sµ are equal, assume slip impends at B and between wedge and floor, and not at A.
Then ( )0.45 271 lb 121.95 lbW s WF Nµ= = =
( )0.45 185 lb 83.25 lbB s BF Nµ= = =
0: 121.95 lb 83.25 lb 0, 205.20 lbxF P PΣ = − − = =
Check Wedge for assumption
0: 271 lb cos 0y AF R θΣ = − =
0: 205.2 lb 121.95 lb sin 0x AF R θΣ = − − =
so tan 0.3072 tan9sθ µ
83.25
= = < + °
271
so no slip here
∴ (a) 205 lb=P
(b) impending slip at B

FBD plank + wedge:
( ) ( )( )( )0: 8 ft 1.5 ft 48 lb/ft 3 ftA BM NΣ = −
( ) ( )( )
1
2 ft 48 lb/ft 3 ft
2
−
( )( )
5
3 ft 96 lb/ft 5 ft 0
3
  
− + =  
  
185 lbWN =
( )
48 96
0: 185 lb lb/ft 3ft
2
y AF N
+ 
Σ = + −  
 
( )( )
1
96 lb/ft 5 ft 0
2
− =
271 lbAN =
Since ,A WN N> and all sµ are equal, assume impending slip at top and bottom of wedge and not at A. Then
( )0.45 185 NW s WF Nµ= =
83.25 lbWF =
FBD Wedge:
( )1 1
tan tan 0.45 24.228s sφ µ− −
= = = °
( )0: 185 lb cos 24.228 9 0y BF RΣ = − ° + ° =
221.16 lbBR =
( )0: 221.16 lb sin33.228 83.25 lb 0xF PΣ = ° + − =
204.44 lbP =
Check assumption using plank/wedge FBD
0: 0, 204.44 lb 83.25 lb 121.19 lbx A W AF F F P FΣ = + − = = − =
( )max 0.45 271 lb 121.95 lbA s AF Nµ= = =
max,A AF F< OK
∴ (a) 204 lb=P
(b) no impending slip at A

( )( ) ( )( )2 2
10 kg 9.81 m/s 98.1 N, 50 kg 9.81 m/s 490.5 NA BW W= = = =
Slip must impend at all surfaces simultaneously, sF Nµ=
FBD I: A + B
0: 150 N 98.1 N 490.5 N 0, 738.6 Ny B BF N NΣ = − − − = =
impending slip: ( )738.6 NB s B sF Nµ µ= =
( )0: 0, 738.6 Nx A B A sF N F N µΣ = − = =
FBD II: A ( ) ( )0: 738.6 N sin 20 150 N + 98.1 N cos20 0y AB sF F µ′  Σ = + ° − ° = 
( )233.14 252.62 NAB sF µ = − 
( ) ( )0: 738.6 N cos20 150 N + 98.1 N sin 20 0x s ABF Nµ′  Σ = ° − ° − = 
( )84.855 694.06 NAB sN µ = + 
233.14 252.62
84.855 694.06
AB s
s
AB s
F
N
µ
µ
µ
−
= =
+
2
0.48623 0.33591 0s sµ µ= − =
0.24312 0.62850sµ = − ±
Positive root 0.385sµ =

( )( ) ( )( )2 2
10 kg 9.81 m/s 98.1 N, 50 kg 9.81 m/s 490.5 NA BW W= = = =
Slip impends at all surfaces simultaneously
FBD I: A + B
0: 0,x A B A B s BF N F N F NµΣ = − = = = (1)
( )0: 150 N + 98.1 N + 490.5 N 0y A BF F NΣ = − + =
738.6 Ns A BN Nµ + = (2)
Solving (1) and (2) 2 2
738.6 N 738.6
, N
1 1
s
B B
s s
N F
µ
µ µ
= =
+ +
FBD II: B ( )0: 490.5 N cos70 cos70 sin70 0x AB B BF N N F′Σ = + ° − ° − ° =
( ) ( )2
738.6 N
cos70 sin 70 490.5 N cos70
1
AB s
s
N µ
µ
= ° + ° − °
+
(1)
( )0: 490.5 N sin 70 sin70 cos70 0y AB B BF F N F′Σ = − − ° + ° − ° =
( ) ( )2
738.6 N
sin 70 cos70 490.5 N sin70 0
1
AB s
s
F µ
µ
= ° − ° − ° =
+
Setting ,AB s ABF Nµ=
3 2
6.8847 2.0116 1.38970 0s s sµ µ µ− − + =
Solving numerically, 0.586, 0.332, 7.14sµ = −
Physically meaningful solution: 0.332sµ =

FBD jack handle:
See Section 8.6
0: 0CM aP rQΣ = − = or
r
P Q
a
=
FBD block on incline:
(a) Raising load
( )tan sQ W θ φ= +
( )tan s
r
P W
a
θ φ= +
continued

PROBLEM 8.66 CONTINUED
(b) Lowering load if screw is self-locking (i.e.: if sφ θ> )
( )tan sQ W φ θ= −
( )tan s
r
P W
a
φ θ= −
(c) Holding load is screw is not self-locking ( )i.e: if sφ θ<
( )tan sQ W θ φ= −
( )tan s
r
P W
a
θ φ= −

FBD large gear:
( )0: 12 in. 7.2 kip in. 0, 0.600 kipsCM W WΣ = − ⋅ = =
600 lb=
Block on incline:
( )
1 0.375 in.
tan 2.2785
2 1.5 in.
θ
π
−
= = °
1 1
tan tan 0.12 6.8428s sφ µ− −
= = = °
( )tan sQ W θ φ= +
( )600 lb tan9.1213 96.333 lb= ° =
FBD worm gear:
1.5 in.r =
( )( )0: 1.5 in. 96.333 lb 0BM MΣ = − =
144.5 lb in.M = ⋅

FBD large gear:
( )0: 12 in. 7.2 kip in. 0CM WΣ = − ⋅ =
0.600 kips 600 lbW = =
Block on incline:
( )
1 0.375 in.
tan 2.2785
2 1.5 in.
θ
π
−
= = °
1 1
tan tan 0.12 6.8428s sφ µ− −
= = = °
( )tan sQ W φ θ= −
( )600 lb tan 4.5643 47.898 lb= ° =
FBD worm gear:
1.5 in.r =
( )( )0: 1.5 in. 47.898 lb 0BM MΣ = − =
71.8 lb in.M = ⋅

Block/incline analysis:
1 0.125 in.
tan 2.4238
2.9531 in.
θ −
= = °
( )1
tan 0.35 19.2900sφ −
= = °
( )47250tan 21.714 18.816 lbQ = ° =
( )
0.94
Couple in. 18.516 lb 8844 lb in.
2 2
d
Q
 
= = = ⋅ 
 
Couple 7.37 lb ft= ⋅

FBD joint D:
By symmetry: AD CDF F=
0: 2 sin 25 4 kN 0y ADF FΣ = ° − =
4.7324 kNAD CDF F= =
FBD joint A:
By symmetry: AE ADF F=
( )0: 2 4.7324 kN cos25 0x ACF FΣ = − ° =
8.5780 kNACF =
Block and incline A:
( )
1 2 mm
tan 4.8518
7.5 mm
θ
π
−
= = °
1 1
tan tan 0.15 8.5308s sφ µ− −
= = = °

( ) ( )8.578 kN tan 13.3826Q = °
2.0408 kN=
Couple at A: AM rQ=
( )
7.5
mm 2.0408 kN
2
 
=  
 
7.653 N m= ⋅
By symmetry: Couple at C: 7.653 N mCM = ⋅
( )Total couple 2 7.653 N mM = ⋅ 15.31 N mM = ⋅

FBD joint D:
By symmetry: AD CDF F=
0: 2 sin 25 4 kN 0y ADF FΣ = ° − =
4.7324 kNAD CDF F= =
FBD joint A:
By symmetry: AE ADF F=
( )0: 2 4.7324 kN cos25 0x ACF FΣ = − ° =
8.5780 kNACF =
Block and incline at A:

( )
1 2 mm
tan 4.8518
7.5 mm
θ
π
−
= = °
1 1
tan tan 0.15s sφ µ− −
= =
8.5308sφ = °
3.679sφ θ− = °
( )8.5780 kN tan3.679Q = °
0.55156 kNQ =
Couple at : AA M Qr=
( )
7.5 mm
0.55156 kN
2
 
=  
 
2.0683 N m= ⋅
By symmetry: Couple at : 2.0683 N mCC M = ⋅
( )Total couple 2 2.0683 N mM = ⋅ 4.14 N mM = ⋅

FBD lower jaw:
By symmetry 540 NB =
0: 540 N 540 N 0, 1080 NyF A AΣ = − + − = =
(a) since A > B when finished, adjust A first when there will be no force
Block/incline at B:
(b) 1 4 mm
tan 6.0566
12 mm
θ
π
−
= = °
( )1 1
tan tan 0.35 19.2900s sφ µ− −
= = = °
( )540 N tan 25.3466 255.80 NQ = ° =
( )( )Couple 6 mm 255.80 N 1535 N mmrQ= = = ⋅
1.535 N mM = ⋅

FBD lower jaw:
By symmetry 540 NB =
0: 540 N 540 N 0, 1080 NyF A AΣ = − + − = =
since A > B, A should be adjusted first when no force is required.
If instead, B is adjusted first,
Block/incline at A:
1 4 mm
tan 6.0566
12 mm
θ
π
−
= = °
( )1 1
tan tan 0.35 19.2900s sφ µ− −
= = = °
( )1080 N tan 25.3466 511.59 NQ = ° =
( )( )Couple 6 mm 511.59 N 3069.5 N mmrQ= = = ⋅
3.07 N mM = ⋅
Note that this is twice that required if A is adjusted first.

Block/incline:
1 0.25 in.
tan 2.4302
1.875 in.
θ
π
−
= = °
( )1 1
tan tan 0.10 5.7106s sφ µ− −
= = = °
( ) ( )1000 lb tan 8.1408 143.048 lbQ = ° =
( )( )Couple 0.9375 in. 143.048 lb 134.108 lb in.rQ= = = ⋅
134.1 lb in.M = ⋅

FBD Bucket:
( )1
sin sin tanf s sr r rφ µ−
= =
( ) ( )1
0.18 m sin tan 0.30 0.05172 m−
= =
( ) ( )0: 1.6 m + 0.05172 m 0.05172 m 0AM T WΣ = − =
0.031314T W=
( )
kN
0.031314 50 Mg 9.81
Mg
 
=  
 
15.360 kN=
15.36 kNT = !
NOTE FOR PROBLEMS 8.75–8.89
Note to instructors: In this manual, the simplification sin ( )1
tan µ µ−
≈ is NOT used in the solution of journal
bearing and axle friction problems. While this approximation may be valid for very small values of ,µ there
is little if any reason to use it, and the error may be significant. For example, in Problems 8.76–8.79,
0.50,sµ = and the error made by using the approximation is about 11.8%.

FBD Windlass:
( )1
sin sin tanf b s b sr r rφ µ−
= =
( ) ( )1
1.5 in. sin tan 0.5 0.67082 in.−
= =
( ) ( )0: 8 0.67082 in. 5 0.67082 in. 160 lb 0AM P   Σ = − − + =   
123.797 lbP =
123.8 lbP =
tan µ µ−

FBD Windlass:
( )1
= =
( ) ( )1
1.5 in. sin tan 0.5 0.67082 in.−
= =
( ) ( ) ( )0: 8 0.67082 in. 5 0.67082 in. 160 lb 0AM P   Σ = + − + =   
104.6 lbP =
tan µ µ−

FBD Windlass:
( )1
= =
( ) ( )1
1.5 in. sin tan 0.50 0.67082 in.−
= =
( ) ( ) ( )0: 8 0.67082 in. 5 0.67082 in. 160 lb 0AM P   Σ = + − − =   
79.9 lbP =
tan µ µ−

FBD Windlass:
( )1
= =
( ) ( )1
1.5 in. sin tan 0.50 0.67082 in.−
= =
( ) ( ) ( )0: 8 0.67082 in. 5 0.67082 in. 160 lb 0AM P   Σ = − − − =   
94.5 lbP =
tan µ µ−

(a) FBD lever (Impending CW
rotation):
(b) FBD lever (Impending CCW
rotation):
( )( ) ( )( )0: 0.2 m 75 N 0.12 m 130 N 0C f fM r rΣ = + − − =
0.0029268 m 2.9268 mmfr = =
sin
f
s
s
r
r
φ =
*
1 1 2.9268 mm
tan tan sin tan sin
18 mm
f
s s
s
r
r
µ φ − −   
= = =   
  
0.34389=
0.344sµ =
( )( )0: 0.20 m 0.0029268 m 75 NDMΣ = −
( )0.12 m 0.0029268 m 0P− + =
120.2 NP =
tan µ µ−

Pulley FBD’s:
Left:
Right:
30 mmpr =
( )
*
1
axle axlesin sin tanf k kr r rφ µ−
= =
( ) ( )1
5 mm sin tan 0.2−
=
0.98058 mm=
Left:
( )( )0: 600 lb 2 0C p f p ABM r r r TΣ = − − =
or
( )
( )
30 mm 0.98058 mm
600 N 290.19 N
2 30 mm
ABT
−
= =
290 NABT =
0: 290.19 N 600 N 0y CDF TΣ = − + =
or 309.81 NCDT = 310 NCDT =
Right:
( ) ( )0: 0G p f CD p f EFM r r T r r TΣ = + − − =
or ( )
30 mm 0.98058 mm
309.81 N 330.75 N
30 mm 0.98058 mm
EFT
+
= =
−
331 NEFT =
tan µ µ−

Pulley FBDs:
Left:
Right:
30 mmpr =
( )
*
1
axle axlesin sin tanf k kr r rφ µ−
= =
( ) ( )1
5 mm sin tan 0.2−
=
0.98058 mm=
( )( )0: 600 N 2 0C p f p ABM r r r TΣ = + − =
or
( )
( )
30 mm 0.98058 mm
600 N 309.81 N
2 30 mm
ABT
+
= =
310 NABT =
0: 600 N 0y AB CDF T TΣ = − + =
or 600 N 309.81 N 290.19 NCDT = − =
290 NCDT =
( ) ( )0: 0H p f CD p f EFM r r T r r TΣ = − − + =
or ( )
30 mm 0.98058 mm
290.19 N
30 mm 0.98058 mm
EFT
−
=
+
272 NEFT =
tan µ µ−
bearing and axle friction problems. While this approximation may be valid for very small values of ,µ there is
little if any reason to use it, and the error may be significant. For example, in Problems 8.76–8.79, 0.50,sµ =
and the error made by using the approximation is about 11.8%.

FBD link AB:
Note: That AB is a two-force member. For impending motion, the pin
forces are tangent to the friction circles.
1
sin
25 in.
fr
θ −
=
where ( )
*
1
sin sin tanf p s p sr r rφ µ−
= =
( ) ( )1
1.5 in. sin tan 0.2 0.29417 in.−
= =
Then 1 0.29417 in.
sin 1.3485
12.5 in.
θ −
= = °
(b) 1.349θ = °
vert horizcos sinR R R Rθ θ= =
( )horiz vert tan 50 kips tan1.3485 1.177 kipsR R θ= = ° =
(a) horiz 1.177 kipsR =
tan µ µ−

FBD gate:
( )2
1 66 kg 9.81m/s 647.46 NW = =
( )2
2 24 kg 9.81m/s 235.44 NW = =
( )1
sin sin tanf s s s sr r rφ µ−
= =
( ) ( )1
0.012 m sin tan 0.2 0.0023534 m−
= =
( ) ( ) ( )1 20: 0.6 m 0.15 m 1.8 m 0C f f fM r W r P r WΣ = − + − − + =
( )( ) ( )( )
( )
1.80235 m 235.44 N 0.59765 m 647.46 N
0.14765 m
P
−
=
253.2 N=
253 NP = !
tan µ µ−

It is convenient to replace the ( )66 kg g and( )24 kg g weights with a single combined weight of
( )( )2
90 kg 9.81m/s 882.9 N,= located at a distance
( )( ) ( )( )1.8 m 24 kg 0.6 m 66 kg
0.04 m
90 kg
x
−
= = to the
right of B.
( ) ( ) ( )
*
1 1
sin sin tan 0.012 m sin tan 0.2f s s s sr r rφ µ− −
= = =
0.0023534 m=
FBD pulley + gate:
1 0.04 m 0.15
tan 14.931 0.15524 m
0.15 m cos
OBα
α
−
= = ° = =
1 1 0.0023534 m
sin sin 0.8686 then 15.800
0.15524 m
fr
OB
β θ α β− −
= = = ° = + = °
tan 249.8 NP W θ= =
250 NP = !
tan µ µ−

FBD gate:
( )2
1 66 kg 9.81m/s 647.46 NW = =
( )2
2 24 kg 9.81m/s 235.44 NW = =
( )
*
1
sin sin tanf s s s sr r rφ µ−
= =
( ) ( )1
0.012 m sin tan 0.2 0.0023534 m−
= =
( ) ( ) ( )1 20: 0.6 m 0.15 m 1.8 m 0C f f fM r W r P r WΣ = + + + − − =
( )( ) ( )( )1.79765 m 235.44 N 0.60235 m 647.46 N
0.15235 m
P
−
=
218.19 N=
218 NP = !
tan µ µ−

It is convenient to replace the ( )66 kg g and( )24 kg g weights with a single weight of
( )( )90 kg 9.81 N/kg 882.9 N,= located at a distance
( )( ) ( )( )1.8 m 24 kg 0.15 m 66 kg
0.04 m
90 kg
x
−
= = to the
right of B.
FBD pulley + gate:
( ) ( ) ( )
*
1 1
sin sin tan 0.012 m sin tan 0.2f s s s sr r rφ µ− −
= = =
0.0023534 mfr =
1 0.04 m 0.15 m
tan 14.931 0.15524 m
0.15 m cos
OBα
α
−
= = ° = =
1 1 0.0023534 m
sin sin 0.8686 then 14.062
0.15524 m
fr
OB
β θ α β− −
= = = ° = − = °
tan 221.1 NP W θ= =
221 NP = !
tan µ µ−

FBD Each wheel:
( )1
f axle axler r sin r sin tanφ µ−
= =
0: sin 0
4
x
P
F R θΣ = − =
0: cos 0
4
y
W
F R θΣ = − =
tan or tan
P
P W
W
θ θ∴ = =
but ( )1axle
sin sin tan
f
w w
r r
r r
θ µ−
= =
(a) For impending motion, use 0.12sµ =
( )10.5 in.
sin sin tan 0.12
5 in.
θ −
= 0.68267θ = °
( ) ( )tan 500 lb tan 0.68267P W θ= = °
5.96 lbP =
(b) For constant speed, use 0.08kµ =
( )11
sin sin tan 0.08
10
θ −
= 0.45691θ = °
( ) ( )500 lb tan 0.45691P = °
3.99 lbP =
tan µ µ−

FBD Each wheel:
For equilibrium (constant speed) the two forces R and
2
W
must be equal
and opposite, tangent to the friction circle, so
( )1
sin where tan slope
f
w
r
r
θ θ −
= =
( )
( )1
1
sin tan
sin tan 0.03
B k
w
r
r
µ−
−
=
( )
( )
( )
1
1
sin tan 0.12
12.5 mm 49.666 mm
sin tan 0.03
wr
−
−
= =
99.3 mmwd =
tan µ µ−

FBD
( )0: 8 in. 0,OM Q MΣ = − =
8 in.
M
Q =
but, from equ. 8.9,
( )( )
2 2 7 in.
0.60 10.1 lb
3 3 2
kM WRµ
 
= =  
 
14.14 lb=
so,
14.14
,
8
Q = 1.768 lbQ =

Eqn. 8.8 gives
3 3 3 3
2 1 2 1
2 2 2 2
2 1 2 1
2 1
3 3
s s
R R D D
M P P
R R D D
µ µ
− −
= =
− −
so ( )( )( )( ) ( )
( ) ( )
3 3
2
2 2
0.030 m 0.024 m1
0.15 80 kg 9.81 m/s
3 0.030 m 0.024 m
M
−
=
−
1.596 N mM = ⋅

Let the normal force on A∆ be ,N∆ and
N k
A r
∆
=
∆
As in the text ,F N M r Fµ∆ = ∆ ∆ = ∆
The total normal force
2
0 00
lim
R
A
k
P N rdr d
r
π
θ
∆ →
 
= Σ∆ =  
 
∫ ∫
( )0
2 2 or
2
R P
P kdr kR k
R
π π
π
= = =∫
The total couple
2
worn 0 00
lim
R
A
k
M M r rdr d
r
π
µ θ
∆ →
 
= Σ∆ =  
 
∫ ∫
2 2
worn 0
2 2 2
2 2 2
R R P R
M k rdr k
R
πµ πµ πµ
π
= = =∫
or worn
1
2
M PRµ=
Now new
2
3
M PRµ= [Eq. (8.9)]
Thus
1
worn 2
2
new 3
3
75%
4
PRM
M PR
µ
µ
= = =

Let normal force on A∆ be ,N∆ and
N k
A r
∆
=
∆
As in the text ,F N M r Fµ∆ = ∆ ∆ = ∆
The total normal force P is
2
1
2
00
lim
R
RA
k
P N rdr d
r
π
θ
∆ →
 
= Σ∆ =  
 
∫ ∫
( )
( )
2
1
2 1
2 1
2 2 or
2
R
R
P
P kdr k R R k
R R
π π
π
= = − =
−
∫
The total couple is 2
1
2
worn 00
lim
R
RA
k
M M r rdr d
r
π
µ θ
∆ →
 
= Σ∆ =  
 
∫ ∫
( ) ( )
( )
( )
2
1
2 2
2 12 2
worn 2 1
2 1
2
2
R
R
P R R
M k rdr k R R
R R
πµ
πµ πµ
π
−
= = − =
−
∫
( )worn 2 1
1
2
M P R Rµ= +

Let normal force on A∆ be ,N∆ and ,
N
k
A
∆
=
∆
so
sin
r
N k A A r s sφ
θ
∆
∆ = ∆ ∆ = ∆ ∆ ∆ =
where φ is the azimuthal angle around the symmetry axis of rotation
sinyF N kr rθ φ∆ = ∆ = ∆ ∆
Total vertical force
0
lim y
A
P F
∆ →
= Σ∆
( )2 2
1 1
2
0
2
R R
R R
P krdr d k rdr
π
φ π= =∫ ∫ ∫
( )
( )
2 2
2 1 2 2
2 1
or
P
P k R R k
R R
π
π
= − =
−
Friction force F N k Aµ µ∆ = ∆ = ∆
Moment
sin
r
M r F r krµ φ
θ
∆
∆ = ∆ = ∆
Total couple 2
1
2 2
00
lim
sin
R
RA
k
M M r dr d
π µ
φ
θ∆ →
 
= Σ∆ =  
 
∫ ∫
( )
( )2
1
2 3 3
2 32 2
2 3
2
2
sin 3 sin
R
R
k P
M r dr R R
R R
µ πµ
π
θ θ π
= = −
−
∫
3 3
2 1
2 2
2 1
2
3 sin
P R R
M
R R
µ
θ
−
=
−

If normal force per unit area (pressure) of the center is OP , then as a function
of r, 1O
r
P P
R
 
= − 
 
2
0 0
1
R
N O
r
F W PdA P rdrd
R
π
θ
 
Σ = = = − 
 
∫ ∫ ∫
2 3 2
2
0
2
2 3 6
O O
R R R
W P d P
R
π
θ π
 
= − =  
 
∫
so 2
3
O
W
P
Rπ
=
For slipping, ( )kdF PdAµ=
2
0 0
Moment 1
R
k O
r
rdF P r rdrd
R
π
µ θ
 
= = − 
 
∫ ∫ ∫
3 4 3
2
0
2
3 4 12
k O k O
R R R
P d P
R
π
µ θ πµ
 
= − =  
 
∫
so
3
2
3 1
2
12 2
k k
W R
M WR
R
πµ µ
π
= =
( )0: 8 in. 0OM Q MΣ = − =
( )
( )( )
1 7 in.
0.6 10.1 lb
2 2
8 in. 8 in.
M
Q
 
 
 = =
1.326 lbQ =

FBD pipe:
1 0.025 in. 0.0625 in.
sin 1.00257
5 in.
θ − +
= = °
tanP W θ= for each pipe, so also for total
( ) ( )2000 lb tan 1.00257P = °
35.0 lbP =

FBD disk:
tan slope 0.02θ = =
( )( )tan 60 mm 0.02b r θ= =
1.200 mmb =

FBD wheel:
230 mmr =
1mmb =
1
sin
b
r
θ −
=
1
tan tan sin
b
P W W
r
θ − 
= =  
 
for each wheel, so for total
( )( )2 1 1
1000 kg 9.81m/s tan sin
230
P − 
=  
 
42.7 NP =

FBD wheel:
( )1
axle axlesin sin tan , orφ µ µ µ−
= =f s kr r r
sin tan
f
w
r b
r
θ θ
= +
For small , sin tan , so tan
f
w
r b
r
θ θ θ θ
+
0: cos 0
4
y
W
F R θΣ = − =
0: sin 0
4
x
P
F R θΣ = − + =
Solving: tan
P
W
θ =
so tan
f
w
r b
P W W
r
θ
+
= =
(a) For impending slip, use ( )10.5 in.
, sin tan 0.12 0.029786 in.
2
s frµ − 
= = 
 
so ( )
0.02986 in. + 0.25 in.
500 lb 55.96 lb
2.5 in.
P = =
56.0 lbP =
(b) For constant speed, use ( )10.5 in.
, sin tan 0.08 0.019936 in.
2
k frµ − 
= = 
 
so ( )
( )0.019936 0.25 in.
500 lb 53.99 lb
2.5 in.
P
+
= =
54.0 lbP =

FBD wheel:
For equilibrium (constant speed), R and
2
W
are equal and opposite
and tangent to the friction circle as shown
( ) ( ) ( )1 1
axle sin tan 12.5 mm sin tan 0.12f kr r µ− −
= =
1.48932 mmfr =
From diagram,
sin tan
f
w
r b
r
θ θ
= +
For small ,θ sin tan ,θ θ so
tan
f
w
r b
r
θ
+
tan slopeθ =
1.48932 mm 1.75 mm
107.977 mm
0.03
wr
+
= =
216 mmwd =

Two full turns of rope → 4 radβ π=
(a) 2 2
1 1
1
ln or lns s
T T
T T
µ β µ
β
= =
1 20 000 N
ln 0.329066
4 320 N
sµ
π
= =
0.329sµ =
(b) 2
1
1
ln
s
T
T
β
µ
=
1 80 000 N
ln
0.329066 320 N
=
16.799 rad=
2.67β = turns

FBD A:
( )( )2
10 kg 9.81 m/s 98.1 NAW = =
0: sin30 0,
2
A
x A A A
W
F T W TΣ = − ° = =
FBD B:
0: sin30 0,
2
B
x B B B
W
F W T T′Σ = ° − = =
(a) Motion of B impends up incline and 8 kgBm =
1 1
, ln lnsA A A
s
B B B
T T W
e
T T W
µ β
µ
β β
= = =
1 3 10 kg
ln ln
8 kgB
Am
mβ π
 
= =  
 
From hint, β is not dependent
on shape of support
0.21309sµ =
0.213sµ =
(b) For maximum ,Bm motion of B impend down incline
0.21309
3, 1.250sB
B A A
A
T
e T T e T
T
π
µ β
= = =
1.25B AW W∴ = and ( )1.25 1.25 10 kgB Am m= =
max 12.50 kgBm =

FBD A:
0: sin30 0,
2
A
x A A A
W
F T W TΣ = − ° = =
FBD B:
0: sin30 0,
2
B
x B B B
W
F W T T′Σ = ° − = =
For min,Bm motion of B impends up incline
And
0.50
3 1.68809A
B
T
e
T
π
= =
But 1.68809A A A
B B B
m W T
m W T
= = =
so min 5.9238 kgBm =
From hint, β is not dependent
on shape of C
For max,Bm motion of B impends down incline
so
0.50
3 1.68809sB B B
A A A
m W T
e e
m W T
β
π
µ
= = = = =
so max 16.881 kgBm =
For equilibrium 5.92 kg 16.88 kgBm≤ ≤

1.5 turns 3 radβ π= =
For impending motion of W up
( ) ( )s 0.15 3
1177.2 NP We e
πµ β
= =
4839.7 N=
For impending motion of W down
( ) ( )0.15 3
1177.2 NsP We e
πµ β −−
= =
286.3 N=
For equilibrium
286 N 4.84 kNP≤ ≤

Horizontal pipe: Vertical pipe
Contact angles
2
H
π
β = Contact angle Vβ π=
0.25sHµ = 0.2sVµ =
For P to impend downward,
( ) ( ) ( )2 2 2 2 100 lbsH sH sH sHsV sVP e Q e e R e e e
π π π πµ µ µ µµ π µ π       = = =       
       
( )
( ) ( ) 0.45
max 100 lb 100 lb 411.12 lbsH sV
P e e
π µ µ π+ = = =  
For 100 lb to impend downward, the ratios are reversed, so
0.45
min100 lb , 24.324 lbPe Pπ
= =
So, for equilibrium, 24.3 lb 411 lbP≤ ≤

Horizontal pipe Vertical pipe
Contact angles
2
H
π
β = Contact angle Vβ π=
0.30sHµ = ?sVµ =
For min,P the 100 lb force impends downward, and
( ) ( ) PeeeQeeRe sHsVssVsHsH











=




=




= 2222
lb100
ππ
π
ππ
µπµµπµµµ
( )
( ) ( )0.30 0.30
100 lb 20 lb , so 5sV sV
e e
π µ π µ+ + = =  
(a) For maxP the force P impends downward, and the ratios are reversed, so ( )max 5 100 lb 500 lbP = =
(b) ( )0.30 ln5sVπ µ+ =
1
ln5 0.30 0.21230sVµ
π
= − = 0.212sVµ =

FBD motor and mount:
Impending belt slip: cw rotation
0.40
2 1 1 13.5136sT T e T e Tµ β π
= = =
( )( ) ( ) ( )2 10: 12 in. 175 lb 7 in. 13 in. 0DM T TΣ = − − =
( )( ) 12100 lb 7 in. 3.5136 13 in. T = + 
1 2 155.858 lb, 3.5136 196.263 lbT T T= = =
FBD drum at B:
( )( )0: 3 in. 196.263 lb 55.858 lb 0B BM MΣ = − − =
421 lb in.BM = ⋅
3 in.r = (Compare to 857 lb in.⋅ using V-belt, Problem 8.130)

Impending belt slip: ccw rotation
0.40
1 2 2 23.5136sT T e T e Tµ β π
= = =
( )( ) ( ) ( )1 20: 12 in. 175 lb 13 in. 7 in. 0DM T TΣ = − − =
( )( ) 22100 lb 13 in. 3.5136 7 in. 0T = + = 
2 1 239.866 lb, 3.5136 140.072 lbT T T= = =
FBD drum at B:
( )( )0: 3 in. 140.072 lb 39.866 lb 0B BM MΣ = − − =
301 lb in.BM = ⋅

FBD lower portion of belt:
0: 48 N 0, 48 Ny D DF N NΣ = − = =
Slip on both platen and wood
( )0.10 48 N 4.8 ND kD DF Nµ= = =
( )48 NE kE E kEF Nµ µ= =
FBD Drum A (assume free to rotate) ( )0: 4.8 N 48 N 0x A B kEF T T µΣ = − − − =
( )4.8 N + 48 NB A kET T µ= + (1)
( )0: 0,A A A T T AM r T T T TΣ = − = = (2)
FBD Drive drum B
( )0: 0B B T BM M r T TΣ = + − =
2.4 N m
96 N
0.025 N
B T TT T T
⋅
= + = +
Impending slip on drum, 0.35s
B T TT T e T eµ β π
= =
so 0.35
96 N , 47.932 NT T TT T e Tπ
+ = =
143.932 NBT =
From (2) above, ,A TT T= so (a) min lower 47.9 NT =
From (1) above, ( )143.932 N 47.932 N + 4.8 N + 48 NkEµ=
So (b) 1.900kEµ =

FBD Flywheel:
( )( )0: 0.225 m 12.60 N m 0C B AM T TΣ = − − ⋅ =
56 N, 56 NB A B AT T T T− = = +
Also, since the belt doesn’t change length, the additional stretch in
spring B equals the decrease in stretch of spring A. Thus the increase
in BT equals the decrease in .AT
Thus ( ) ( )70 N 70 N 140 NB AT T T T+ = + ∆ + − ∆ =
( )56 N 140 N, 42 NA A AT T T+ + = =
42 N 56 N 98 NBT = + =
(a) 42.0 NAT =
98.0 NBT =
For slip ,k
B AT T eµ β
= or
1
ln B
k
A
T
T
µ
β
=
1 98
ln 0.2697
42
kµ
π
= =
(b) 0.230kµ =

FBD Flywheel:
Slip of belt: 0.20k
B A AT T e T eµ β π
= =
Also, since the belt doesn’t change length, the increase in stretch of
spring B equals the decrease in stretch of spring A. Therefore the
increase in BT equals the decrease in ,AT and the sum is unchanged,
so 80 N 80 N 160 NA BT T+ = + =
( )0.20
1 160 N,AT e π
∴ + = so 55.663 NAT =
104.337 NBT = (a) 55.7 NAT =
104.3 NBT =
( )( )0: 0.225 m 0C B A CM T T MΣ = − − =
( )( )0.225 m 104.337 N 55.663 NCM = −
(b) 10.95 N mCM = ⋅

FBD Lever:
( )( ) ( )0: 60 mm 240 N 40 mm cos30 0E BDM FΣ = − ° =
415.69 NBDF =
FBD Drum:
Belt slip: 2 1
kT T eµ β
=
( ) ( )0.25 5.5851
415.69 N e=
1679.44 N=
( )2 10: 0CM r T T MΣ = − − =
( )( )0.08 m 1679.44 N 415.69 N 0M− − =
101.1 N mM = ⋅

FBD Drum:
(a) With 125 lb ftEM = ⋅
( )( ) ( )0: 7 in. 125 lb ft 0E A CM T TΣ = − − ⋅ =
214.29 lbA CT T− =
Belt slip:
( )7
6
0.30
3.0028k
A C C CT T e T e T
π
µ β
= = =
so 2.0028 214.9 lb,CT = 106.995 lbCT =
321.28 lbAT =
FBD Lever:
( ) ( ) ( )0: 15 in. 2 in. 7.5 in. 0B C AM P T TΣ = + − = (1)
( )( ) ( )( )7.5 in. 321.28 lb 2 in. 106.995 lb
17 in.
P
−
=
129.2 lbP =
(b) With 125 lb ftEM = ⋅ , the drum analysis will be reversed, and will yield 106.995 lb,AT =
321.28 lbCT =
Eqn. (1) will remain the same, so
( )( ) ( )( )7.5 in. 106.995 lb 2 in. 321.28 lb
17 in.
P
−
=
9.41 lbP =

FBD Lever:
If brake is self-locking, no force P is required
( ) ( )0: 2 in. 7.5 in. 0B C AM T TΣ = − =
3.75C AT T=
For impending slip on drum: s
C AT T eµ β
=
3.75,seµ β
∴ = or
1
ln3.75sµ
β
=
With
7
,
6
π
β = 0.361sµ =

FBD Lever:
( ) ( )0: 40 mm 100 mm 0, 2.5B C A C AM T T T TΣ = − = =
FBD Drum:
(a) For impending slip ccw: max 4.5 kNCT T= =
so 1.8 kN
2.5
C
A
T
T = =
( )( )0: 0.16 m 1.8 kN 4.5 kN 0D DM MΣ = + − =
0.432 kN mDM = ⋅
432 N mDM = ⋅
(b) For impending slip ccw, s
C AT T eµ β
=
or
1 3
ln ln 2.5 0.21875
4
C
s
A
T
T
µ
β π
= = =
0.219sµ =

(a) For minimum Cm with blocks at rest, impending slip of A is down/left.
Note: 1 1
tan tan 0.30 16.7 30 ,s sφ µ− −
= = = ° < ° so min 0Cm >
FBD A:
( )( )2
6 kg 9.81 m/s 58.86 NAW = =
0: cos30 0, cos30y A A A AF N W N WΣ = − ° = = °
Impending slip: 0.30 cos30A s A AF N Wµ= = °
( )0: sin30 0, sin30 0.30cos30x A A A A AF T F W T WΣ = + − ° = = ° − °
14.1377 N=
FBD Drum:
If blocks don’t move, belt slips on drum, so
( )0.2 0.87266
14.1377 N 1.19069k
A C C CT T e T e Tµ β
= = = =
so 11.8735 NCT =
FBD C:
0: cos20 0, cos20y C C C CF N W N W′Σ = − ° = = °
Impending slip: 0.30 cos20C s C CF N Wµ= = °
0: 0.30 cos20 sin 20 11.8735 N 0x C CF W W′Σ = ° + ° − =
2
19.0302 N, 1.93988 kg
9.81 m/s
C
C C
W
W m= = =
1.940 kgCm =
continued

(b) For motion of A to impend up/right
FBD A:
As in part (a) cos30 , 0.30 cos30A A A AN W F W= ° = °
( )0: sin30 0.30cos30 0x A AF T WΣ = − ° + ° =
44.722 NAT =
Also, as in part (a) 1.19069 ,k
A C CT T e Tµ β
= = so
44.722 N
1.19069
CT =
37.560 NCT =
FBD C:
As in part (a) 0.30 cos20C CF W= °
( )0: sin 20 0.30cos20 37.560 N 0x CF W′Σ = ° − ° − =
2
624.83 N, 63.69 kg
9.81 m/s
C
C C
W
W m= = =
63.7 kgCm =
(c) For uniform motion of A up and B down, and minimum ,Cm there will be impending slip of the rope on
the drum.
FBD A is same as in (b) but 0.20 cos30A k A AF N Wµ= = °
and ( )0: sin30 0.20cos30 0, 39.625 Nx A A AF T W TΣ = − ° + ° = =
Drum analysis, with impending slip, s
A CT T eµ β
=
( )0.30 0.87266
39.625 N 1.29926C CT e T= =
or 30.498 NCT =
FBD C is same as in (b), but 0.20 cos20C k C CF N Wµ= = °
and ( )0: sin 20 0.20cos20 0x C CF W T′Σ = ° − ° − =
2
30.498 N 197.934 N
197.933 N,
0.154082 9.81 m/s
C CW m= = =
20.2 kgCm =

Geometry and force rotation:
Let 1 50 mm
cos 60 s , ,
100 mm
EBC DBE FAE GAEα −
= = = ° =
Then contact angles are
4
360 120 240 rad
3
B
π
β = ° − ° = ° = for cord on
upper cylinder, and 30 rad
6
A
π
β = ° = for each cord
contact on lower cylinder.
Let the force in section FFC T=
Let the force in section GDG T=
With A fixed and the cord moving,
( )6
0.25
1.13985k A
GT We We W
π
µ β
= = =
For maximum W, slip impends on drum B, so
s B
B FT T eµ β
= or s B
F GT T e µ β−
=
( )4
3
0.30
1.13985 0.32441FT We W
π−
= =
For slip at F
( )6
0.25
1 0.32441 0.36978k A
FW T e We W
π
µ β
= = =
so 12.7043W W= and 2.7043 in.m =
( )2.7043 75 kg=
203 kgm =

Geometry and force notation:
Note: 1
sin 30 rad,
2 6
r
r
π
θ −
= = ° = so contact angles are:
2
,
2 6 3
C D E
π π π
β β β π= = + = =
(a) For all pulleys locked, slip impends at all contacts
If AW impends downward, ( )1 2 1 216 lb , ,s E s D s C
AT e T T e W T eµ β µ β µ β
= = =
so ( ) ( )
( ) ( )7
3
0.20
16 lb 16 lb 69.315 lbs C D E
AW e e
π
µ β β β+ +
= = =
If AW impends upward all ratios are inverted, so ( ) ( )7
3
0.20
16 lbAW e
π−
=
3.6933 lb=
For equilibrium, 3.69 lb 69.3 lbAW≤ ≤
(b) If pulley D is free to rotate, 1 2T T= while the other ratios remain as in (a)
For AW impending down ( ) ( )
( ) ( )5
3
0.20
16 lb 16 lbs C E
AW e e
π
µ β β+
= =
45.594 lbAW =
For AW impending upward, ( ) ( )3
0.2
16 lb 5.6147 lbAW e
π5−
= =
For equilibrium 5.61 lb 45.6 lbAW≤ ≤

1
sin 30 ,
2 6
r
r
π
θ −
2
,
2 6 3
C D E
π π π
β β β π= = + = =
(a) D and E fixed, so slip on these surfaces. For maximum ,AN slip impends on pulley C
2 ,s C
AW T eµ β
= and ( )1 2 1, 16 lbk D k ET T e T eµ β µ β
= =
so ( ) ( )
( ) ( ) ( )5 2
3 3
0.15 0.20
16 lb 16 lb 11.09 lbk E D s C
AW e e e e
π π
µ β β µ β −− +
= = =
(b) C and D fixed, so slip there. For maximum ,AW slip impends on E
so ( )1 1 2 216 lb , ,s E k D k C
AT e T T e T W eµ β µ β µ β
= = =
so ( ) ( )
( ) ( )4
3
0.150.20
16 lb 16 lb 16 lbk C Ds E
AW e e e e
π
µ β βµ β π −− +
= = =
16.00 lbAW =

1 5 in.
sin 30 rad,
10 in. 6
π
θ −
5 2
, ,
6 6 2 6 3 2
C D E
π π π π π π
β π β β= − = = + = =
(a) All pulleys locked with impending slip at all.
If AW impends upward, 1 ,s C
AT W eµ β
=
( )2 1 2, 16 lb ,s D s ET T e T eµ β µ β
= = so
( ) ( )
( ) ( )5 34
6 6 6
0.20
16 lb 16 lbs C D E
AW e e
πµ β β β − + +− + +
= =
4.5538 lbAW =
If AW impends downward all ratios are inverted
so ( ) ( )0.20 2
16 lb 56.217 lbAW e
π+
= =
For equilibrium, 4.55 lb 56.2 lbAW≤ ≤
(b) Pulley D is free to rotate so 1 2T T= , other ratios are the same
If AW impends upward, ( ) ( )
( ) ( )4
3
0.20
16 lb 16 lbs C E
AW e e
π
µ β β −− +
= =
6.9229 lbAW =
If AW impends downward, ratios are inverted, ( ) ( )4
3
0.20
16 lbAW e
π+
=
36.979 lbAW =
For equilibrium 6.92 lb 37.0 lbAW≤ ≤

1 5 in.
sin 30 rad,
10 in. 6
π
θ −
5 2
, ,
6 6 2 6 3 2
C D E
π π π π π π
β π β β= − = = + = =
(a) D and E fixed, so slip at these surfaces,
For maximum ,AW slip impends on C.
1 2 1 2, , 16 lbs C k D k E
AW T e T T e T eµ β µ β µ β
= = =
so ( ) ( )16 lb k D E s C
AW e e
µ β β µ β− +
=
( ) ( ) ( )7 5
6 6
0.15 0.20
16 lb 15.5866 lbe e
π π−
= =
max 15.59 lbAW =
(b) C and D fixed, so slip at these surfaces—impending slip on E
( )1 2 1 2, , 16 lbk C k D s E
AT W e T T e T eµ β µ β µ β
= = =
so ( ) ( )
( ) ( ) ( )3
2 2
0.15 0.20
16 lb 16 lbk C D s E
AW e e e e
π π
µ β β µ β −− +
= =
10.8037 lb,AW = max 10.80 lbAW =

FBD drum B:
( )( )0: 0.02 m 0.30 N m 0B AM T TΣ = − − ⋅ =
0.30 N m
15 N
0.02 m
AT T
⋅
− = =
Impending slip: 0.40s
AT Te Teµ β π
= =
Solving; ( )0.40
1 15 NT e π
− =
5.9676 NT =
If C is free to rotate P T=
min 5.97 NP =

FBD drum B:
( )( )0: 0.02 m 0.3 N m 0B AM T TΣ = − − ⋅ =
15 NAT T− =
Impending slip: 0.40s B
AT Te Teµ β π
= =
Solving, ( )0.40
1 15 NT e π
− =
5.9676 NT =
If C is frozen, tape must slip there, so
( ) ( )2
0.30
5.9676 N 9.5599 Nk CP Te e
π
µ β
= = =
min 9.56 NP =

FBD pin B:
Drum:
Lever:
(a) By symmetry: 1 2T T=
1 1 2
2
0: 2 0 or 2 2
2
yF B T B T T
 
Σ = − = = =  
 
(1)
For impending rotation :
3 1 2 4 3 max, so 5.6 kNT T T T T T> = > = =
Then ( ) ( )4 6
0.25
1 3 5.6 kNs LT T e e
π π
µ β − +−
= =
or 1 24.03706 kNT T= =
and ( ) ( )3
4
0.25
4 2 4.03706 kNs RT T e e
π
µ β −−
= =
or 4 2.23998 kNT =
( )0 4 3 2 10: 0FM M r T T T TΣ = + − + − =
or ( )( )0 0.16 m 5.6 kN 2.23998 kN 0.5376 kN mM = − = ⋅
0 538 N m= ⋅M
(b) Using Equation (1)
( )12 2 4.03706 kNB T= =
5.70927 kN=
( )( ) ( )0: 0.05 m 5.70927 kN 0.25 m 0DM PΣ = − =
1.142 kN=P

FBD pin B:
FBD Drum:
FBD Lever:
(a) By symmetry: 1 2T T=
1 1
2
0: 2 0 or 2
2
yF B T B T
 
Σ = − = =  
 
(1)
For impending rotation :
4 2 1 3 4 max, so 5.6 kNT T T T T T> = > = =
Then ( ) ( )3
4
0.25
2 4 5.6 kNs RT T e e
π
µ β −−
= =
or 2 13.10719 kNT T= =
and ( ) ( )4 6
0.25
3 1 3.10719 kNs LT T e e
π π
µ β − +−
= =
or 3 2.23999 kNT =
( )0 2 1 3 40: 0FM M r T T T TΣ = + − + − =
( )( )0 160 mm 5.6 kN 2.23999 kN 537.6 N mM = − = ⋅
0 538 N m= ⋅M
(b) Using Equation (1)
( )12 2 3.10719 kNB T= =
4.3942 kNB =
( )( ) ( )0: 0.05 m 4.3942 kN 0.25 m 0DM PΣ = − =
879 N=P

FBD wrench:
Note:
( )0.2 m
, 0.03 m
sin65
EC EA EC= = −
°
65θ = °
so 295 5.1487 radβ = ° =
0.20 m 0.20 m
0: 0.03 m cos65 0.03 m 0
sin65 sin65
EM F T
   
Σ = − − ° − =   
° °   
3.01408T F=
0: sin65 cos65 0xF N F TΣ = ° + ° − =
Impending slip: ,
s
F
N
µ
= so
sin65
cos65
s
F T
µ
 °
= + ° = 
 
or
sin65
cos65 3.01408
sµ
°
+ ° =
0.3497sµ =
Must still check slip of belt on pipe
FBD small portion of belt at A:
1 20: 0nF N NΣ = − =
Impending slip, both sides: 1 1 2 2,s sF N F Nµ µ= =
so 1 2F F F= =
0: 2 0, 2t A AF F T T FΣ = − = =
Impending slip of belt on pipe: s
AT T eµ β
=
or
1 1 3.01408
ln ln 0.0797
2 5.1487 2
s
T
F
µ
β
= = =
Above controls, so for self-locking, need 0.350sµ =

FBD wrench
Note:
( )0.20 m
, 0.03 m
sin75
EC EA EC= = −
°
75θ = °
so 285 4.9742 radβ = ° =
0.20 m 0.20 m
0: 0.03 m cos75 0.03 m 0
sin75 sin 75
EM F T
   
Σ = − − ° − =   
° °   
7.5056T F=
0: sin75 cos75 0xF N F TΣ = ° + ° − =
Impending slip: ,
s
F
N
µ
= so
sin75
cos75 7.5056
s
F T F
µ
 °
= + ° = = 
 
sin 75
cos75 7.5056
sµ
°
+ ° =
0.1333sµ =
Must still check impending slip of belt on pipe
FBD small portion of belt at A
1 20: 0nF N NΣ = − =
Impending slip 1 1 2 2,s sF N F Nµ µ= =
so 1 2F F F= =
0: 2 0, 2t A AF F T T FΣ = − = =
Impending slip of belt on pipe s
AT T eµ β
=
or
1 1 7.5056
ln ln 0.2659
2 4.9742 2
s
T
F
µ
β
= = =
This controls, so for self locking, min 0.267sµ =

( )0: sin 0
2
nF N T T T
θ∆
 Σ = ∆ − + + ∆ = 
or ( )2 sin
2
N T T
θ∆
∆ = + ∆
( )0: cos 0
2
tF T T T F
θ∆
 Σ = + ∆ − − ∆ = 
or cos
2
F T
θ∆
∆ = ∆
Impending slipping: sF Nµ∆ = ∆
So
sin
cos 2 sin
2 2 2
s sT T T
θ θ θ
µ µ
∆ ∆ ∆
∆ = + ∆
In limit as 0: , ors s
dT
dT Td d
T
θ µ θ µ θ∆ → = =
So 2
1 0
;
T
sT
dT
d
T
β
µ θ=∫ ∫
and 2
1
ln s
T
T
µ β=
or 2 1
sT T eµ β
=
Note: Nothing above depends on the shape of the surface, except it is
assumed to be a smooth curve.

Small belt section:
Side view: End view:
( )0: 2 sin sin 0
2 2 2
y
N
F T T T
α θ∆ ∆
 Σ = − + + ∆ = 
( )0: cos 0
2
xF T T T F
θ∆
 Σ = + ∆ − − ∆ = 
Impending slipping:
2
cos sin
2 2sin
2
s s
T T
F N T
θ θ
µ µ
α
∆ + ∆ ∆
∆ = ∆ ⇒ ∆ =
In limit as 0:θ∆ → or
sin sin
2 2
s sTd dT
dT d
T
µ θ µ
θ
α α
= =
So 2
1 0
sin
2
T s
T
dT
d
T
βµ
θ
α
=∫ ∫
or 2
1
ln
sin
2
sT
T
µ β
α
=
or 2
/sin
2 1
s
T T e
αµ β
=

Impending belt slip, cw rotation
2
sin
2 1
s
T T e
α
µ β
=
( )0.40
sin18
2 1 158.356T T e T
π
°= =
( )( ) ( ) ( )1 20: 12 in. 175 lb 13 in. 7 in. 0DM T TΣ = − − =
( )( ) 12100 lb 13 in. 7 in. 58.356 T = + 
1 2 14.9823 lb, 58.356 290.75 lbT T T= = =
FBD drum at B:
( )( )0: 3 in. 4.9823 lb 290.75 lb 0B BM MΣ = + − =
857 lb in.BM = ⋅
(Compare to 421 lb in.⋅ using flat belt, Problem 8.107)

Geometry:
1 2 in.
sin 7.1808 0.12533 rad
16 in.
θ −
= = ° =
2 2.8909 radAβ π θ= − =
Since ,B Aβ β> impending slip on A will control the
maximum couple transmitted
FBD A:
( )( )1 20: 60 lb in. 2 in. 0AM T TΣ = ⋅ + − =
2 1 30 lbT T− =
Impending slip: 2
sin
2 1
s
T T e
α
µ β
=
so
( )( )0.35 2.8909
sin18
1 1 30 lbT e °
 
 − =
 
 
1 1.17995 lbT =
2 31.180 lbT =
FBD B:
( )0: 31.180 lb 1.17995 lb cos7.1808 0xF PΣ = − + ° =
32.1 lbP =

FBD block:
( ) ( )0: 1000 N cos30 200 N sin30 0nF NΣ = − ° − ° =
966.03 NN =
Assume equilibrium:
( ) ( )0: 200 N cos30 1000 N sin30 0tF FΣ = + ° − ° =
eq.326.8 NF F= =
But ( )max 0.3 966 N 290 NsF Nµ= = =
eq. max impossibleF F> ⇒ Block moves
and kF Nµ=
( )( )0.2 966.03 N=
Block slides down 193.2 N=F

FBD block (impending
motion to the right)
( )1 1
tan tan 0.25 14.036s sφ µ− −
= = = °
( )sin sins s
P W
φ θ φ
=
−
( )sin sins s
W
W mg
P
θ φ φ− = =
(a)
( )( )2
1
30 kg 9.81m/s
30 kg: sin sin14.036
120 N
sm θ φ −
 
 = − = °
  
36.499= °
36.499 14.036θ∴ = ° + ° or 50.5θ = °
(b)
( )( )2
1
40 kg 9.81m/s
40 kg: sin sin14.036
120 N
sm θ φ −
 
 = − = °
  
52.474= °
52.474 14.036θ∴ = ° + ° or 66.5θ = °

FBDs
Top block:
Bottom block:
FBD blocks:
(a) Note: With the cable, motion must impend at both contact surfaces.
1 10: 40 lb 0 40 lbyF N NΣ = − = =
Impending slip: ( )1 1 0.4 40 lb 16 lbsF Nµ= = =
10: 0 16 lb 0 16 lbxF T F T TΣ = − = − = =
2 20: 40 lb 60 lb 0 100 lbyF N NΣ = − − = =
Impending slip: ( )2 2 0.4 100 lb 40 lbsF Nµ= = =
0: 16 lb 16 lb 40 lb 0xF PΣ = − + + + =
72.0 lb=P
(b) Without the cable, both blocks will stay together and motion will
impend only at the floor.
0: 40 lb 60 lb 0 100 lbyF N NΣ = − − = =
Impending slip: ( )0.4 100 lb 40 lbsF Nµ= = =
0: 40 lb 0xF PΣ = − =
40.0 lb=P

FBD ladder:
7.5 fta =
19.5 ftl =
5
13
a
l
=
12
13
b
l
=
Motion impends at both A and B, so
andA s A B s BF N F Nµ µ= =
7.5 ft
0: 0 or
2 2 39 ft
A B B
a a
M lN W N W W
l
Σ = − = = =
or
2.5
13
BN W=
Then
2.5
13
B s B s
W
F Nµ µ= =
5 12
0: 0
13 13
x A B BF F F NΣ = + − =
( ) ( )2 2
12.5 30
0
13 13
s A sN W Wµ µ+ − =
( )
( )
2
30 12.5
13
s
A
s
W
N
µ
µ
−
−
12 5
0: 0
13 13
y A B BF N W F NΣ = − + + =
( )2
30 12.5
30 12.5
13
s
s
s
W
W
µ
µ
µ
− 
+ + = 
 
or 2
5.6333 1 0s sµ µ− + =
2.8167 2.6332sµ = ±
or 0.1835 and 5.45s sµ µ= =
The larger value is very unlikely unless the surface is treated with
some “non-skid” material.
In any event, the smallest value for equilibrium is 0.1835sµ =

FBD window:
( )( )2
4 kg 9.81m/s 39.24 NW = =
( )( )2
2 kg 9.81m/s 19.62 N
2
W
T = = =
0: 0x A D A DF N N N NΣ = − = =
Impending motion: A s A D s DF N F Nµ µ= =
( ) ( ) ( )0: 0.36 m 0.54 m 0.72 m 0D A AM W N FΣ = − − =
3
2
2
A s AW N Nµ= +
2
3 4
A
s
W
N
µ
=
+
0: 0y A DF F W T FΣ = − + + =
A DF F W T+ = −
2
W
=
Now ( ) 2A D s A D s AF F N N Nµ µ+ = + =
Then
2
2
2 3 4
s
s
W W
µ
µ
=
+
or 0.750sµ =

FBD Collar:
Stretch of spring
cos
a
x AB a a
θ
= − = −
( )( )
1
1.5 kN/m 0.5 m 1
cos cos
s
a
F k a
θ θ
   
= − = −   
   
( ) ( )( )
1
0.75 kN 1 750 N sec 1
cos
θ
θ
 
= − = − 
 
0: cos 0y sF F W NθΣ = − + =
or ( )( )750 N 1 cosW N θ= + −
Impending slip:
sF Nµ= (F must be ,+ but N may be positive or negative)
0: sin 0x sF F FθΣ = − =
or ( )( )sin 750 N tan sinsF F θ θ θ= = −
(a) 20 :θ = ° ( )( )750 N tan 20 sin 20 16.4626 NF = ° − ° =
Impending motion:
16.4626 N
41.156 N
0.4s
F
N
µ
= = =
(Note: for 41.156 N,N < motion will occur, equilibrium for
41.156)N >
But ( )( )750 N 1 cos20 45.231 NW N N= + − ° = +
So equilibrium for 4.07 N and 86.4 NW W≤ ≥
(b) 30 :θ = ° ( )( )750 N tan30 sin30 58.013 NF = ° − ° =
Impending motion:
58.013
145.032 N
0.4s
F
N
µ
= = =
( )( )750 N 1 cos30 145.03 NW N N= + − ° = ±
( )44.55 N impossible , 245.51 N= −
Equilibrium for 246 NW ≥

FBD pin C:
FBD block A:
FBD block B:
sin10 0.173648ABF P P= ° =
cos10 0.98481BCF P P= ° =
0: sin30 0y A ABF N W FΣ = − − ° =
or 0.173648 sin30 0.086824AN W P W P= + ° = +
0: cos30 0x A ABF F FΣ = − ° =
or 0.173648 cos30 0.150384AF P P= ° =
For impending motion at A: A s AF Nµ=
Then
0.150384
: 0.086824
0.3
A
A
s
F
N W P P
µ
= + =
or 2.413P W=
0: cos30 0y B BCF N W FΣ = − − ° =
0.98481 cos30 0.85287BN W P W P= + ° = +
0: sin30 0x BC BF F FΣ = ° − =
0.98481 sin30 0.4924BF P P= ° =
For impending motion at B: B s BF Nµ=
Then
0.4924
: 0.85287
0.3
B
B
s
F P
N W P
µ
= + =
or 1.268P W=
Thus, maximum P for equilibrium max 1.268P W=

1 1
tan tan 0.25 14.036s sφ µ− −
= = = °
FBD block A:
2 750 lb
sin104.036 sin16.928
R
=
° °
2 2499.0 lbR =
FBD wedge B:
2499.0
sin73.072 sin75.964
P
=
° °
2464 lbP =
2.46 kips=P

Block on incline:
( )
1 0.1 in.
tan 3.0368
2 0.3 in.
θ
π
−
= = °
1 1
tan tan 0.12 6.8428s sφ µ− −
= = = °
( )500 lb tan9.8796 87.08 lbQ = ° =
Couple on each side
( )( )0.3 in. 87.08 lb 26.12 lb in.M rQ= = = ⋅
Couple to turn 2 52.2 lb in.M= = ⋅

FBD pulley:
0: 103.005 N 49.05 N 98.1 N 0yF RΣ = − − − =
250.155 NR =
( )( ) ( )0: 0.12 m 103.005 N 98.1 N 250.155 N 0O fM rΣ = − − =
0.0023529 m 2.3529 mmfr = =
1
sin
f
s
s
r
r
φ −
=
1 1 2.3529 mm
tan tan sin tan sin
30 mm
f
s s
s
r
r
µ φ − −   
= = =   
  
0.0787sµ =

FBD wheel:
FBD lever:
( )( )2 10: 7.5 in. 0E EM M T TΣ = − + − =
or ( )( )2 17.5 in.EM T T= −
( )( ) ( )( )1 20: 4 in. 16 in. 25 lb 0CM T TΣ = + − =
or 1 2 100 lbT T+ =
Impending slipping: 2 1
sT T eµ β
=
or
( )3
2
0.25
2 1 13.2482T T e T
π
= =
So ( )1 1 3.2482 100 lbT + =
1 23.539 lbT =
and ( )( )( )7.5 in. 3.2482 1 23.539 lb 396.9 lb in.EM = − = ⋅
397 lb in.EM = ⋅
Changing the direction of rotation will change the direction of EM and
will switch the magnitudes of 1T and 2T .
The magnitude of the couple applied will not change.

FBD block:
FBD Drum:
( )0: 200 lb cos30 0; 100 3 lbn CF N NΣ = − ° = =
( )0: 200 lb sin30 0t C CF T FΣ = − ° =∓
100 lbC CT F= ± (1)
where the upper signs apply when CF acts
(a) For impending motion of block , CF , and
( )0.35 100 3 lb 35 3 lbC s CF Nµ= = =
So, from Equation (1): ( )100 35 3 lbCT = −
But belt slips on drum, so k
C AT W eµ β
=
( ) ( )2
3
0.25
100 35 3 lbAW e
π−
 = −
 
23.3 lbAW =
(b) For impending motion of block , CF and 35 3 lbC s CF Nµ= =
From Equation (1): ( )100 35 3 lbCT = +
Belt still slips, so ( ) ( )2
3
0.25
100 35 3 lbk
A CW T e e
π
µ β −−  = = +
 
95.1lbAW =
continued

(c) For steady motion of block , CF , and 25 3 lbC k CF Nµ= =
Then, from Equation (1): ( )100 25 3 lb.T = +
Also, belt is not slipping on drum, so
( ) ( )2
3
0.35
100 25 3 lbs
A CW T e e
π
µ β −−  = = +
 
68.8 lbAW =

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