![12
Vertically
R=W
_______ (ii)
From (i) & (ii), we get
S = µW
_______ (iii)
Taking moment of all forces about A
S(BC) = W(AAԢ)
⇒
S(ABcosα) = W(AGsinα)
⇒
S(2Lcosα) = W(Lsinα)
⇒
2Scosα = Wsinα
⇒
2µWcosα = Wsinα
⇒
tanα = 2µ
By (iii)
Now a man ascends the ladder and let
α
weight of man be WԢ. Let M be the
position of the man when ladder is in
limiting
equilibrium.
Now
G
normal
reactions are R Ԣ and S Ԣ and force of
M
RԢ
friction is µR Ԣ.
α
α
AԢԢ
For limiting equilibrium,
A
Horizontally
SԢ = µR Ԣ
B
SԢ
AԢ
μRԢ
C
WԢ W
_______ (iv)
Vertically
R Ԣ = W + WԢ
_______ (v)
From (iv) & (v), we get
SԢ = µW + µW Ԣ
_______ (vi)
Taking moment of all forces about A
SԢ(BC) = W(AAԢ) + W Ԣ(AA")
⇒
SԢ(ABcosα) = W(AGsinα) + WԢ(AMsinα)
⇒
SԢ(2Lcosα) = W(Lsinα) + WԢ(xsinα)
⇒
⇒
⇒
⇒
2 SԢLcosα = (WL + W Ԣx)sinα
2SԢL = (WL + W Ԣx)tanα
2SԢL = (WL + W Ԣx) 2µ
2L(µW + µW Ԣ) = (WL + W Ԣx) 2µ
Available At: mathcity.org
[ tanα = 2µ]
By (vi)
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Ch02 friction
- 1. M th 2 FRICTION CHAPTER INTRODUCTION When two smooth surfaces are in contact with each other at the point P. The reaction of one body on the other body is along the common normal. But if the bodies are not smooth then the reaction of the one body on the other is not along the normal but inclined at an angle θ to the common normal at the point of the contact P. ሬሬԦ R A θ ሬԦ RԢ ሬԦ F B Let ሬሬԦ′ has components ሬሬԦ along the common normal at P and ሬԦ along the tangent plane at P. R R F ሬሬԦ Then R ൌ R′cosθ and ሬԦ ൌ R′sinθ F Where ሬሬԦ is called normal reaction or normal pressure and ሬԦ is called the force of friction or R F simply friction. The force of friction prevents the sliding of one body on the surface of other ሬሬԦ ሬሬԦ body. R′ is the resultant of normal reaction R and force of friction ሬԦ and also called resultant F friction. TYPES OF FRICTION Now we discuss some important types of friction: Statical Friction When one body in contact with another is in equilibrium and is not on the point of sliding along the other, the force of friction in this case is called statical friction and the equilibrium in this case is non-limiting. Limiting Friction When one body is just at the point of sliding on the other, the friction in this case is maximum and is called limiting friction. The equilibrium in this case is called limiting equilibrium. Available At: mathcity.org Contact At: qadri86@yahoo.com
- 2. 2 Dynamical Friction When one body is sliding on the other body the friction in this case is slightly less than the limiting friction and is called dynamical friction or kinetic friction. coefficient of friction The magnitude of the limiting friction bears a constant ratio µ to the magnitude R of the normal reaction between the surfaces. The constant µ is called coefficient of friction. i.e. F = R ⇒ F= R angle of friction The angle λ, which the direction of the resultant reaction ሬԦԢ makes with normal reaction ሬԦ R R when the body is just on the point of the motion, is called the angle of friction. From fig ሬሬԦ R = R′cosλ and ሬԦ = R′sinλ F F = R R′sinλ = R′cosλ As ⇒ tanλ = ⇒ ሬሬԦ R A ሬԦ RԢ λ ሬԦ F B Question 1 A body weighing 40lb is resting on a rough plane and can just be moved by a force acting horizontally. Find the coefficient of friction. Solution For limiting equilibrium, Horizontally F = µR _______ (i) Vertically R=W Using value of R in (i), we get F = µW ⇒ = R µR F F 10 ൌ ൌ 0.25 W 40 W Available At: mathcity.org Contact At: qadri86@yahoo.com
- 3. 3 Question 2 Find the least force which will set into motion a particle at rest on a rough horizontal plane. Solution Let F be the required least force which makes an angle θ with horizontal plane and W be the weight of the particle. R F For limiting equilibrium, Horizontally Fcosθ = µR Fsinθ ___________ (i) θ µR Vertically Fcosθ W = R + Fsinθ ⇒ R = W െ Fsinθ Using value of R in (i), we get Fcosθ = µ(W െ Fsinθ) ⇒ Fcosθ + µ Fsinθ = µW ⇒ F= W ⇒ F(cosθ + µsinθ) = µW W cosθ + sinθ tanλW μ ൌ tanλ cosθ + tanλsinθ sinλ Wsinλ Wsinλ cosλ W cosλ = = = sinλ cosθcosλ + sinλsinθ cosθcosλ + sinλsinθ cosθ + cosλ sinθ cosλ Wsinλ = cosሺθ െ λሻ = F will be least when cos(θ െ λ) is maximum and maximum value of cos(θ െ λ) is 1. So F will be least iff cos(θ െ λ) = 1 Thus Fleast = Wsinλ Equilibrium of a Particle on a Rough Inclined Plane Show that a particle is in equilibrium on a rough inclined plane under its own weight if the inclination of the plane is equal to the angle of friction. Solution Let α be the inclination of the plane and W be the weight acting downward, R is normal reaction and µR is its force of friction. Available At: mathcity.org Contact At: qadri86@yahoo.com
- 4. 4 For limiting equilibrium, Horizontally R Wsinα = µR µR ___________ (i) Vertically R = Wcosα α Using value of R in (i), we get Wsinα = µWcosα ⇒ tanα = µ ⇒ tanα = tanλ ⇒ α=λ Wcosα α Wsinα W tanλ = µ Where λ is the angle of friction. Question 2 Find the least force to drag up a particle on a rough inclined plane. Solution Let F be the required least force which make angle θ with inclined plane. Let α be the inclination of the plane and W be the weight acting downward , R is normal reaction and µR is its force of friction. For limiting equilibrium, F R θ Fsinθ Fcosθ Horizontally Fcosθ = Wsinα + µR µR _______ (i) α Vertically R + Fsinθ = Wcosα ⇒ α W R = Wcosα െ Fsinθ Wcosα Wsinα Using value of R in (i), we get Fcosθ = Wsinα + µ(Wcosα െ Fsinθ) ⇒ Fcosθ + µFsinθ = Wsinα + µWcosα ⇒ F(cosθ + µsinθ) = W(sinα + µcosα) ⇒ F(cosθ + tanλsinθ) = W(sinα + tanλcosα) ⇒ F cosθ + ⇒ F ∵ tanλ = µ sinλ sinλ sinθ൨ = W sinα + cosα൨ cosλ cosλ cosθcosλ + sinθsinλ sinαcosλ + sinλcosα ൨ = W ൨ cosλ cosλ Available At: mathcity.org Contact At: qadri86@yahoo.com
- 5. ⇒ ⇒ cosሺθ െ λሻ sinሺα + λ) ൨ = W ൨ cosλ cosλ Wsinሺα + λ) F= cosሺθ െ λሻ 5 F F will be least when cos(θ െ λ) is maximum and maximum value of cos(θ െ λ) is 1. So F will be least iff cos(θ െ λ) = 1 Thus Fleast = Wsin(α + λ) Question 3 The least force which will move a weight up on an inclined plane is of magnitude P. Show that the least force acting parallel to the plane which will move the weight upward is: Pඥ1+ 2 Where µ is the coefficient of friction. Solution We know that P = Wsin(α + λ) Let F be the required force acting parallel to ___________________(i) inclined plane. Let α be the inclination of the R F plane and W be the weight acting downward , R is normal reaction and µR is its force of friction. For limiting equilibrium, µR Horizontally F = Wsinα + µR α Wcosα α _____ (ii) Vertically W Wsinα R = Wcosα Using value of R in (ii), we get F = Wsinα + µWcosα = W(sinα + µcosα) = W(sinα + tanλcosα) sinλ cosα൨ cosλ sinαcosλ + sinλcosα = W ൨ cosλ = W sinα+ Available At: mathcity.org tanλ = µ Contact At: qadri86@yahoo.com
- 6. ൌ Wsinሺα + λ) P ൌ cosλ cosλ 6 By(i) ൌ Psecλ = Pඥ1 + tan2 λ = Pඥ1 + 2 tanλ ൌ μ Question 4 Find the least force to drag a particle on a rough inclined plane in downward direction. Solution Let F be the required least force which make angle θ with inclined plane. Let α be the inclination of the plane and W be the weight acting downward, R is normal reaction and µR is its force of friction. R µR F Fsinθ For limiting equilibrium, ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ α Wcosα Fcosθ α Horizontally Fcosθ + Wsinα = µR Vertically R + Fsinθ = Wcosα ⇒ R = Wcosα െ Fsinθ Using value of R in (i), we get θ _____ (i) W Wsinα Fcosθ + Wsinα = µ(Wcosα െ Fsinθ) Fcosθ + Wsinα = µWcosα െ µFsinθ F(cosθ + µsinθ) = W(µcosα െ sinα) F(cosθ + tanλsinθ) = W(tanλcosα െ sinα) F cosθ + sinλ sinλ sinθ൨ = W cosα െ sinα൨ cosλ cosλ cosθcosλ + sinθsinλ sinλcosα െ sinαcosλ F ൨ = W ൨ cosλ cosλ cosሺλ െ θሻ sinሺλ െ α) F ൨ = W ൨ cosλ cosλ Wsinሺλ െ α) F= cosሺλ െ θሻ tanλ ൌ μ F will be least when cosሺλ െ θሻ is maximum and maximum value of cosሺλ െ θሻ is 1. So F will be least iff cosሺλ െ θሻ = 1 Thus Fleast = Wsin(λ െ α) Available At: mathcity.org Contact At: qadri86@yahoo.com
- 7. 7 Question 5 Find the force which is necessary just to support a heavy body on an inclined plane with inclination α; (α > λ ) Solution Let F be the required force which make angle θ with inclined plane. Let α be the inclination of the plane F Fsinθ and W be the weight acting downward, R is normal R reaction and µR is its force of friction. θ µR For limiting equilibrium, Horizontally Wsinα = Fcosθ + µR α α _______ (i) Wcosα Wsinα Vertically W R + Fsinθ = Wcosα ⇒ Fcosθ R = Wcosα െ Fsinθ Using value of R in (i), we get Wsinα = Fcosθ + µ(Wcosα െ Fsinθ) ⇒ Wsinα = Fcosθ + µWcosα െ µFsinθ ⇒ Fcosθ െ µFsinθ = Wsinα െ µWcosα ⇒ F(cosθ െ µsinθ) = W(sinα െ µcosα) ⇒ F(cosθ െ tanλsinθ) = W(sinα െ tanλcosα) ⇒ F cosθ െ ⇒ F ⇒ ⇒ tanλ = µ sinλ sinλ sinθ൨ = W sinα െ cosα൨ cosλ cosλ cosθcosλ െ sinθsinλ sinαcosλ െ sinλcosα ൨ = W ൨ cosλ cosλ cosሺθ + λሻ sinሺα െ λ) F ൨ = W ൨ cosλ cosλ Wsinሺα െ λ) F= cosሺθ + λሻ F will be least when cos(θ + λ) is maximum and maximum value of cosሺθ + λሻ is 1. So F will be least iff cosሺθ + λሻ = 1 Available At: mathcity.org Thus Fleast = Wsinሺα െ λ) Contact At: qadri86@yahoo.com
- 8. 8 Question 6 A light ladder is supported on a rough floor and leans against a smooth wall. How far a up a man can climb without slipping taking place ? Solution S B α M R α AԢ A C µR W Let AB be a light ladder which makes angle α with vertical. Let M be the position of the man of weight W when the ladder is in limiting equilibrium. R and S are normal reactions of ground and wall respectively, µR is force of friction of ground. Let AM = x and AB = L For limiting equilibrium, Horizontally S = µR _______ (i) Vertically R=W _______ (ii) From (i) & (ii), we get S = µW _______ (iii) Taking moment of all forces about A S(BC) = W(AAԢ) ⇒ S(ABcosα) = W(AMsinα) ⇒ SLcosα = Wxsinα ⇒ µWLcosα = Wxsinα ⇒ x = µLcotα Available At: mathcity.org By (iii) Contact At: qadri86@yahoo.com
- 9. 9 Question 7 A light rod is supported on a rough floor and leans against a rough wall. The coefficient of friction of both wall and floor is µ. How far a up a man can climb without slipping taking place ? Solution µS S B α M R α AԢ A µR C W Let AB be a light ladder which makes angle α with vertical. Let M be the position of the man of weight W when the ladder is in limiting equilibrium. R and S are normal reactions of ground and wall respectively, µR and µR are forces of friction of ground and wall respectively. Let AM = x and AB = L For limiting equilibrium, Horizontally S = µR _______ (i) Vertically R + µS = W R = W െ µS _______ (ii) From (i) & (ii), we get S = µW െ µ 2S ⇒ S= S + µ 2S = µW S(1+ µ 2) = µW ⇒ ⇒ W 1+ 2 _______ (iii) Taking moment of all forces about A S(BC) + µS(AC) = W(AAԢ) Available At: mathcity.org Contact At: qadri86@yahoo.com
- 10. 10 ⇒ S(ABcosα) + µS(ABsinα) = W(AMsinα) ⇒ SLcosα + µS(Lsinα) = Wxsinα ⇒ Wxsinα = SLcosα + µS(Lsinα) = SL(cosα + µsinα) = = ⇒ x= L 1+ W L(cosα + sinα) 1+ 2 Wsinα L(cotα + ) 1+ 2 2( + cotα) By (iii) Question 8 A uniform ladder rests in limiting equilibrium with one end on rough floor whose coefficient of friction is µ and with other end on a smooth vertical wall. Show that its inclination to the vertical is tan–1(2µ). Solution S B α G R α AԢ A µR C W Let AB be a ladder of weight W is its weight is acting downward from midpoint G. R and S are normal reactions of floor and wall respectively, µR is force of friction of floor. Let AB = 2L then AG = BG = L For limiting equilibrium, Horizontally S = µR Available At: mathcity.org _______ (i) Contact At: qadri86@yahoo.com
- 11. 11 Vertically R=W _______ (ii) From (i) & (ii), we get S = µW _______ (iii) Taking moment of all forces about A S(BC) = W(AAԢ) ⇒ S(ABcosα) = W(AGsinα) ⇒ S(2Lcosα) = W(Lsinα) ⇒ 2Scosα = Wsinα ⇒ 2µWcosα = Wsinα ⇒ 2µ = tanα ⇒ By (iii) α = tan–1(2µ) Question 9 A uniform ladder rests in limiting equilibrium with one end on rough horizontal plane and other end on a smooth vertical wall. A man ascends the ladder. Show that he cannot go more than half way up. Solution S B α G R α AԢ A µR C W Let AB be a ladder of weight W. Its weight is acting downward from midpoint G. R and S are normal reactions of floor and wall respectively, µR is force of friction of floor. Let AB = 2L then AG = BG = L For limiting equilibrium, Horizontally S = µR Available At: mathcity.org _______ (i) Contact At: qadri86@yahoo.com
- 12. 12 Vertically R=W _______ (ii) From (i) & (ii), we get S = µW _______ (iii) Taking moment of all forces about A S(BC) = W(AAԢ) ⇒ S(ABcosα) = W(AGsinα) ⇒ S(2Lcosα) = W(Lsinα) ⇒ 2Scosα = Wsinα ⇒ 2µWcosα = Wsinα ⇒ tanα = 2µ By (iii) Now a man ascends the ladder and let α weight of man be WԢ. Let M be the position of the man when ladder is in limiting equilibrium. Now G normal reactions are R Ԣ and S Ԣ and force of M RԢ friction is µR Ԣ. α α AԢԢ For limiting equilibrium, A Horizontally SԢ = µR Ԣ B SԢ AԢ μRԢ C WԢ W _______ (iv) Vertically R Ԣ = W + WԢ _______ (v) From (iv) & (v), we get SԢ = µW + µW Ԣ _______ (vi) Taking moment of all forces about A SԢ(BC) = W(AAԢ) + W Ԣ(AA") ⇒ SԢ(ABcosα) = W(AGsinα) + WԢ(AMsinα) ⇒ SԢ(2Lcosα) = W(Lsinα) + WԢ(xsinα) ⇒ ⇒ ⇒ ⇒ 2 SԢLcosα = (WL + W Ԣx)sinα 2SԢL = (WL + W Ԣx)tanα 2SԢL = (WL + W Ԣx) 2µ 2L(µW + µW Ԣ) = (WL + W Ԣx) 2µ Available At: mathcity.org [ tanα = 2µ] By (vi) Contact At: qadri86@yahoo.com
- 13. 13 ⇒ ⇒ 2µ (WL + W ԢL) = (WL + W Ԣx) 2µ W ԢL = W Ԣx ⇒ WL + W ԢL = WL + W Ԣx ⇒ x = L = (Half of the length of Ladder) Question 10 A uniform ladder of length 70 feet rest against a vertical wall with which it makes an angle of 450, the coefficient between ladder & wall & ground respectively being 1/3 and ½. If a man whose weight is one half that of ladder, ascends the ladder, where will he be when the ladder slips.? If a body now stands on the bottom rung f the ladder what must be his least weight so that the man may go to the top of the ladder. ? Solution 1 S 3 S B 450 G R M 450 450 AԢ AԢԢ A W W 2 C 1 R 2 Let AB be a ladder of length 70 feet makes an angle of 450 with vertical. W is its weight acting downward from its midpoint G. R and S are normal reactions of ground and wall 1 1 respectively. Let M be the position of man when ladder slip. 2 R and 3 S are forces of friction of ground and wall respectively. Since AB = 70 Therefore AG = BG = 35. Let AM = x For limiting equilibrium, Horizontally 1 S= R 2 Vertically 1 W R+ S=W+ 3 2 3W 1 ⇒ R= െ S 2 3 Available At: mathcity.org _______ (i) ______ (ii) Contact At: qadri86@yahoo.com
- 14. 14 From (i) & (ii), we get 1 3W 1 3W S ൬ െ S൰ = െ 2 2 3 4 6 S 3W 7S 3W 18 ⇒ S+ = ⇒ = ⇒ S= W 6 4 6 4 28 9 ⇒ S= W 14 Taking moment of all forces about A 1 W SሺBCሻ + S(AC) = (AAԢ) + WሺAAᇱᇱ ሻ 3 2 1 W ⇒ SሺABcos45ሻ + S(ABsin45) = (AMsin45) + WሺAGsin45ሻ 3 2 1 1 1 W 1 1 ⇒ S ൬70 ൰ + S(70 ) = (x ) + W ൬35 ൰ 3 2 √2 √2 √2 √2 S= ⇒ ⇒ ⇒ ⇒ 70 Wx S= + 35W 3 2 70 x ൬70 + ൰ S = ቀ + 35ቁ W 3 2 x 70 9 ൬70 + ൰ W = ቀ + 35ቁ W 2 3 14 ________(iii) 70S + By (iii) x = 50 feet 1 S 3 S B 450 G RԢ 450 AԢ A WԢ 1 R' 2 W C W 2 Suppose a man of weight of W Ԣ stands on the bottom rung of the ladder and other man reach to the top of the ladder at point B. Now R Ԣ is the normal reaction of the ground and 2R Ԣ is the 1 force of friction. Available At: mathcity.org Contact At: qadri86@yahoo.com
- 15. 15 For limiting equilibrium, Horizontally 1 S = R′ 2 Vertically 1 W R′ + S = W + W′ + 3 2 3W 1 ⇒ R′ = + W′ െ S 2 3 _______ (iv) ______ (v) From (iv) & (v), we get 9W 3W′ + 14 7 Taking moment of all forces about A S= ______ (vi) 1 W S(AC) = (AC) + WሺAAᇱ ሻ 3 2 1 W SሺABcos45ሻ + S(ABsin45) = (ABsin45) + WሺAGsin45ሻ 3 2 1 1 1 W 1 1 S ൬70 ൰ + S(70 ) = (70 ) + W ൬35 ൰ 3 2 √2 √2 √2 √2 SሺBCሻ + ⇒ ⇒ ⇒ ⇒ ⇒ 70 S = 35W + 35W 3 70 ൬70 + ൰ S = 70W 3 70 9W 3W′ + ൰ ൬70 + ൰ = 70W ൬ 7 3 14 70S + By (vi) 9W 6W′ 1 ቇ ൬1 + ൰ = W 14 3 ⇒ ቆ ⇒ 12W′ = 3W ⇒ W′ = 1 W 4 Question 11 The upper end of a uniform ladder rests against a rough vertical wall and lower end on a rough horizontal plane, the coefficient of friction in both cases is 1/3. Prove that if the inclination of the ladder to the vertical is tan – 1(1/2), a weight equal to that of ladder cannot be attached to it at a point more than 9/10 of the distance from the foot of it without destroying the equilibrium. Solution Available At: mathcity.org Contact At: qadri86@yahoo.com
- 16. 16 1 S 3 S B α G P R α AԢ A W α AԢԢ 1 R 3 W C Let AB be a ladder of length 2L makes an angle of α with vertical. W is its weight acting downward from its midpoint G. R and S are normal reactions of ground and wall 1 1 respectively. 3 R and 3 S are forces of friction of ground and wall respectively. Let P be the point where weight W is attached. Since AB = 2L Therefore AG = BG = L. Let AP = x Now we have to prove that x= 9 ሺ2Lሻ 10 For limiting equilibrium, Horizontally S= 1 R 3 _______ (i) Vertically 1 S=W+W 3 1 R = 2W െ S 3 R+ ⇒ ______ (ii) From (i) & (ii), we get 1 1 ൬2W െ S൰ 3 3 1 2 S+ S= W 9 3 3 S= W 5 S= ⇒ ⇒ ______ (iii) Taking moment of all forces about A SሺBCሻ + 1 S(AC) = WሺAAᇱ ሻ + WሺAAᇱᇱ ሻ 3 Available At: mathcity.org Contact At: qadri86@yahoo.com
- 17. 17 ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ 1 S(ABsinα) = W(APsinα) + WሺAGsinαሻ 3 1 Sሺ2Lcosαሻ + S(2Lsinα) = W(xsinα) + WሺLsinαሻ 3 2 S ൬2Lcosα + Lsinα൰ = W(xsinα + Lsinα) 3 3 2 ൬ W൰ ൬2Lcosα + Lsinα൰ = W(xsinα + Lsinα) By (iii) 5 3 3 2 ൬2Lcosα + Lsinα൰ = xsinα + Lsinα 5 3 6 2 Lcosα + Lsinα = xsinα + Lsinα 5 5 6 2 Lcosα + Lsinα െ Lsinα = xsinα 5 5 6 2 Lcosα + ൬ െ 1൰ Lsinα = xsinα 5 5 6 3 L െ Ltanα = xtanα Dividing both sides by cosα 5 5 SሺABcosαሻ + Given that ⇒ ⇒ 1 1 α = tanି1 ൬ ൰ ⇒ tanα = 2 2 6 3 1 1 L െ L൬ ൰ = x൬ ൰ 5 5 2 2 12L െ 3L = 5x ⇒ x= 9 L 5 ⇒ x= 9 (2L) 10 Question 12 A uniform rod of weight W is place with its lower end on a rough horizontal floor and its upper end against equally rough vertical wall. The rod makes an angle α with the wall and is just prevented from slipping down by a horizontal force P applied at its middle point. Prove that P = W tan(α െ 2λ) α λ 1 Where λ is the angle of friction and λ < α 2 Solution Let AB be a rod of length 2L makes an angle of α with vertical. W is its weight acting downward from its midpoint G. R and S are normal reactions of ground and wall respectively. µR and µs are forces of friction of ground and wall respectively. Let P be the horizontal force applied at middle point G of rod. Since AB = 2L Therefore AG = BG = L. Available At: mathcity.org Contact At: qadri86@yahoo.com
- 18. 18 µS S B α G R D P α AԢ A C µR W For limiting equilibrium, Horizontally S = P + µR ⇒ P = S െ µR Vertically _______ (i) W = R+ µS _______ (ii) From (i) & (ii), we get P S െ µR = W R + µS ______ (iii) Taking moment of all forces about G RሺAAԢሻ = µR(GAԢ) + SሺBDሻ + µSሺGDሻ ⇒ RሺAGsinαሻ = µR(AGcosα) + SሺGBcosαሻ + µSሺGBsinαሻ ⇒ RLsinα = µRLcosα + SLcosα + µSLsinα ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ Rሺsinα െ µcosαሻ = Sሺcosα + µsinαሻ Rሺsinα െ tanλcosαሻ = Sሺcosα + tanλsinαሻ sinλ sinλ cosα൰ = S ൬cosα + sinα൰ cosλ cosλ sinαcosλ െ cosαsinλ cosαcosλ + sinαsinλ R൬ ൰ = S൬ ൰ cosλ cosλ Rsinሺα െ λሻ = Scosሺα െ λሻ S = Rtanሺα െ λሻ R ൬sinα െ tanλ ൌ µ Using value of S in (iii), we get P Rtanሺα െ λሻ െ µR = W R + µRtanሺα െ λሻ Available At: mathcity.org Contact At: qadri86@yahoo.com
- 19. 19 = ⇒ tanሺα െ λሻ െ µ tanሺα െ λሻ െ tanλ = = tanሺα െ λ െ λሻ = tanሺα െ 2λሻ 1 + µtanሺα െ λሻ 1 + tanሺα െ λሻtanλ P = Wtan(α െ 2λ) Since P is a force and force is always positive, therefore α െ 2λ must be positive. α i.e. α െ 2λ > 0 ⇒ α > 2λ ⇒ λ< 2 Question 13 A rod 4ft long rests on a rough floor against the smooth edge of a table of height 3ft. If the rod is on the point of slipping when inclined at an angle of 600 to the horizontal, find the coefficient of friction. Solution Scos600 S B 60 0 Ssin600 30 0 C 300 G R 3ft 300 600 AԢ A µR D W Let AB be a rod of length 4ft makes an angle of 600 with horizon. W is its weight acting downward from its midpoint G. R and S are normal reactions of ground and edge of the table respectively. µR is forces of friction of ground. Since AB = 4ft Therefore AG = BG = 2ft. For limiting equilibrium, Horizontally Ssin600 = µR _______ (i) Vertically W = R+ Scos600 ⇒ R = W െ Scos600 _______ (ii) From (i) & (ii), we get Ssin600 = µW െ µScos600 Available At: mathcity.org Contact At: qadri86@yahoo.com
- 20. 20 ⇒ Ssin600 + µScos600 = µW ⇒ S(sin600 + µcos600) = µW √3 µ Sቆ + ቇ = µW 2 2 2µW ⇒ S= √3 + µ Taking moment of all forces about A ⇒ _______ (iii) SሺACሻ = W(AAԢ) ⇒ SሺACሻ = W(AGcos60) 1 ⇒ SሺACሻ = W ൬2 ൰ 2 ⇒ SሺACሻ = W From fig. 3 3 2.3 √3 sin600 = ⇒ = ⇒ AC = ⇒ AC = 2√3 AC 2 AC √3 So S൫2√3൯ = W ⇒ S= W 2√3 _______ (iv) From (iii) & (iv), we have 2µW W ൌ √3 + µ 2√3 ⇒ ⇒ ⇒ ⇒ ⇒ 4µ√3 ൌ √3 + µ 4µ√3 െ µ ൌ √3 µ൫4√3 – 1൯ ൌ √3 µൌ √3 4√3 – 1 µ ൌ 0.292 Question 14 One end of a uniform ladder of weight W rests against a smooth wall, and other end on a rough ground, which slopes down from the wall at an angle α to the horizon. Find the inclination of the ladder to the horizontal when it is at the point of slipping, and show that the reaction of the wall is then W tan (λ െ α) where λ is the angle of friction. λ Solution Let AB be a ladder of length 2L makes an angle of θ with horizon. W is its weight acting downward from its midpoint G. R and S are normal reactions of ground and wall Available At: mathcity.org Contact At: qadri86@yahoo.com
- 21. 21 respectively. µR is forces of friction of ground. Let α be the inclination of ground to the horizon. Since AB = 2L Therefore AG = BG = L. S B G Rcosα µRsinα R µR α α θ AԢ Rsinα A µRcosα C W For limiting equilibrium, Horizontally S + Rsinα = µRcosα ⇒ S = R(µcosα െ sinα) _______ (i) Vertically W = µRsinα + Rcosα ⇒ W = R(µsinα + cosα) _______ (ii) From (i) & (ii), we get S µcosα െ sinα = W µsinα + cosα tanλcosα െ sinα tanλ ൌ µ tanλsinα + cosα sinλcosα െ cosλsinα ቀ ቁ sinሺλ െ αሻ cosλ = = sinαsinλ + cosαcosλ ቀ ቁ cosሺλ െ αሻ cosλ = ⇒ S = W tanሺλ െ αሻ Thus the normal reaction of wall is W tanሺλ െ αሻ. _______ (iii) Taking moment of all forces about A SሺBCሻ = W(AAԢ) ⇒ SሺABsinθሻ = W(AGcosθ) ⇒ Sሺ2Lsinθሻ = WሺLcosθሻ Available At: mathcity.org ⇒ 2S = Wcotθ Contact At: qadri86@yahoo.com
- 22. ⇒ ⇒ ⇒ 22 2W tanሺλ െ αሻ = Wcotθ By(iii) cotθ ൌ 2tanሺλ െ αሻ θ = cotି1 ൫2tanሺλ െ αሻ൯ Question 15 Two bodies, weight W1, W2 are placed on an inclined plane are connected by a light string which coincides with a line of greatest slope of the plane. If the coefficient of friction between the bodies and the plane be respectively µ1 and µ2. Find the inclination of the plane to the horizon when the bodies are on the point of motion, it is being assumed that the smoother body is below the other. Solution S µ 2S R T µ 1R T α α α W1cosα W1 W2cosα W2 W1sinα W2sinα R and S are normal reactions and µ1R and µ2S are forces of friction. Let T be the tension in the string. Let α be the inclination of plane to the horizontal. For W1 : For limiting equilibrium, Horizontally µ1R + T = W1sinα T = W1sinα െ µ1R _______ (i) R = W1cosα ⇒ _______ (ii) Vertically From (i) & (ii), we get T = W1sinα െ µ1W1cosα _______ (iii) For W2 : For limiting equilibrium, Horizontally T + W2sinα = µ2S ⇒ T = µ2S െ W2sinα Available At: mathcity.org _______ (iv) Contact At: qadri86@yahoo.com
- 23. 23 Vertically S = W2cosα _______ (v) From (iv) & (v), we get T = µ2W2cosα െ W2sinα _______ (vi) From (iii) & (vi), we get W1sinα െ µ1W1cosα = µ2W2cosα െ W2sinα ⇒ W1sinα + W2sinα = µ1W1cosα + µ2W2cosα ⇒ (W1 + W2)sinα = (µ1W1 + µ2W2)cosα ⇒ tanα = µ1W1 + µ2W2 W1 + W2 α = tanି 1 ൬ ⇒ µ1W1 + µ2W2 ൰ W1 + W2 Question 16 A thin uniform rod passes over a peg and under another, the coefficient of friction between each peg and the rod being µ. The distance between the pegs is a, and the straight line joining them makes an angle β with the horizon. show that the equilibrium is not possible unless the length of the rod is greater than a ሺµ + tanβሻ µ β µ Solution B µS S G C β µR A β Wcosβ AԢ Wsinβ W R Let AB be a rod of length 2L makes an angle of β with horizon. W is its weight acting downward from its midpoint G. R and S are normal reactions of pegs at A and C respectively. µR and µS are forces of friction. Since AB = 2L Therefore AG = BG = L. For limiting equilibrium, Horizontally µR + µS = Wsinβ Available At: mathcity.org _______ (i) Contact At: qadri86@yahoo.com
- 24. 24 Vertically S = R + Wcosβ ⇒ R = S െ Wcosβ _______ (ii) From (i) & (ii), we get µ( S െ Wcosβ) + µS = Wsinβ ⇒ 2µS െ µWcosβ = Wsinβ ⇒ S= µS െ µWcosβ + µS = Wsinβ 2µS = µWcosβ + Wsinβ ⇒ ⇒ W ሺµcosβ + sinβሻ 2µ _______ ሺiiiሻ Taking moment of all forces about A SሺACሻ = W(AAԢ) ⇒ S= ⇒ Sሺaሻ = W(AGcosβ) ⇒ Sሺaሻ = W Lcosβ _______ ሺivሻ WLcosβ a From (iii) & (iv), we get W WLcosβ ሺµcosβ + sinβሻ ൌ 2µ a ⇒ aሺµcosβ + sinβሻ ൌ 2µLcosβ aሺµ + tanβሻ ൌ 2µL a ⇒ 2L ൌ ሺtanβ + µሻ µ a ⇒ Length of rod ൌ ሺtanβ + µሻ µ Which gives the least length of the rod. For equilibrium, a 2L ሺtanβ + µሻ µ a or Length of rod ሺtanβ + µሻ µ ⇒ By dividing both sides by cosβ Question 17 A uniform rod slides with its ends on two fixed equally rough rods, one being vertical and the other inclined at an angle α to the horizontal. Show that the angle θ to the horizontal of the movable rod, when it is on the point of sliding, is given by 1 െ 2µtanα െ µ2 µ α tanθ = θ ሺ 2ሺµ + tanαሻ α Solution Available At: mathcity.org Contact At: qadri86@yahoo.com
- 25. 25 Let AB be a rod of length 2L makes an angle α with horizon. W is its weight acting downward from its midpoint G. R and S are normal reactions. µR and µS are forces of friction. Since AB = 2L Therefore AG = BG = L. µR B R θ Ssinα G S Scosα µScosα AԢԢ AԢ α θ α A µS µSsinα α W For limiting equilibrium, Horizontally R = µScosα + Ssinα _______ (i) Vertically µR + Scosα = W + µSsinα _______ (ii) From (i) & (ii), we get µ(µScosα + Ssinα) + Scosα = W + µSsinα ⇒ µ2Scosα + µSsinα + Scosα = W + µSsinα ⇒ W = S(µ2cosα + cosα) _______ (iii) Taking moment of all forces about A W(AAԢ) = R(BAԢԢ) + µR(AAԢԢ) ⇒ W(AGcosθ) = R(ABsinθ) + µR(ABcosθ) ⇒ WLcosθ = 2LRsinθ + 2µRLcosθ ⇒ Wcosθ = 2Rsinθ + 2µRcosθ ⇒ W = 2Rtanθ + 2µR ⇒ W = R(2tanθ + 2µ) By dividing both sides by cosθ _______ (iv) From (i) and (iv), we get W = (µScosα + Ssinα)(2tanθ + 2µ) Available At: mathcity.org Contact At: qadri86@yahoo.com
- 26. 26 ⇒ W = S(µcosα + sinα)(2tanθ + 2µ) _______ (v) From (iii) and (v), we get S(µ2cosα + cosα) = S(µcosα + sinα)(2tanθ + 2µ) ⇒ µ2cosα + cosα = (µcosα + sinα)(2tanθ + 2µ) ⇒ µ2cosα + cosα = tanθ(2µcosα + 2sinα) + 2µ2cosα + 2µsinα ⇒ tanθ(2µcosα + 2sinα) = µ2cosα + cosα െ2µ2cosα െ 2µsinα ⇒ ⇒ cosα െ 2µsinα െ µ2cosα tanθ = 2sinα + 2µcosα tanθ = 1 െ 2µtanα െ µ2 2ሺµ + tanαሻ Question 18 Two inclined planes have a common vertex, and a string, passing over a small smooth pulley at the vertex, supports two equal weights. If one of the plane be rough and the other is smooth, find the relation between the inclinations of the planes when the weight on the smooth plane is on the point of moving down. Solution T T R1 R2 Wcosβ µR1 α Wcosα α β β Wsinα Wsinβ W W Let W be the weight of either particle, α and β be the inclinations of the planes and R1 and R2 are normal reactions of planes. µR1 is force of friction and T is tension in string. First Particle: For limiting equilibrium, Horizontally T = µR1 + Wsinα _______ (i) Vertically R1 = Wcosα Available At: mathcity.org _______ (ii) Contact At: qadri86@yahoo.com
- 27. 27 From (i) & (ii), we get T = µWcosα + Wsinα ⇒ T = W(µcosα + sinα) _______ (iii) Second Particle: For limiting equilibrium, Horizontally T = Wsinβ _______ (iv) From (iii) & (iv), we get Wsinβ = W(µcosα + sinα) ⇒ sinβ = sinα + µcosα Which is required. Question 19 A solid cylinder rests on a rough horizontal plane with one of its flat ends on the plane, and is acted on by a horizontal force through the centre of its upper end. If this force be just sufficient to move the solid, show that it will slide, and is not topple over, if the coefficient of friction be less than the ratio of the radius of the base of the cylinder to its height. Solution F h G AԢ B µR W R r A Let r be radius of the base of the cylinder and h be the height of the cylinder. W be the weight acting downward, R is normal reaction and µR is its force of friction. Let F be a force at the top of cylinder. For limiting equilibrium, Horizontally F = µR _______ (i) Vertically R=W Using value of R in (i), we get F = µW Available At: mathcity.org _______ (ii) Contact At: qadri86@yahoo.com
- 28. 28 Taking moment of all forces about A W(AAԢ) = P(h) ⇒ W(r) = F(h) ⇒ W(r) = µWh By (ii) ⇒ r = µh r ⇒ µ= h Which shows that the cylinder will slide and not topple over if r µ< h Question 20 A uniform rectangular block of height h whose base is a square of side a, rests on a rough horizontal plane. The plane is gradually tilted about a line parallel to two edges of the base. Show that the block will slide or topple over according as a ≷ µh, where µ is the coefficient of friction. Solution C D α G B A α W Let α be inclination of the plane to the horizon when the block is on the point of toppling over. The vertical through G must fall just within the base of length a as shown in figure. From fig. AB a tanα = = BC h Also the inclination θ of the plane to the horizontal when the block is about to slide is given by tanθ = µ The block will slide or topple over accordingly as θ≶α ⇒ ⇒ tanθ ≶ tanα a µ≶ ⇒ h a ≷ µh Which is required. Available At: mathcity.org Contact At: qadri86@yahoo.com
- 29. 29 Question 21 A uniform semi-circular wire hangs on a rough peg, the line joining its extremities making an angle of 450 with the horizontal. If it is just on the point of slipping, find the coefficient of friction between the wire and the pegs. Solution Let P be the peg and APB be the semi-circular wire with centre at O and radius r. Then OP = r. Let G be the centre of gravity. Then 2r OG = π R RԢ µR P λr B ∠AOG = 90 ∠AGO = 45 G ∠OGP = 135 O 450 A W From fig. ∠OGP = 1350 and ∠OPG = λ In triangle OGP, ⇒ ⇒ ⇒ ⇒ OG OP = sinλ sin135 2rൗ r π = sinλ 1ൗ √2 2 = √2 ⇒ πsinλ sinλ= √2 π sinλ √2 = cosλ πcosλ Available At: mathcity.org By Law of Sine sinλ= 2 π√2 Dividing both sides by cosλ Contact At: qadri86@yahoo.com
- 30. 30 ⇒ tanλ = ⇒ tanλ = ⇒ √2 secλ π µ= ⇒ ⇒ ⇒ ⇒ √2 ඥ1 + tan2 λ π + 1 tan2 λ ൌ sec2 λ √2 ඥ1 + µ2 π 2 µ2 = 2 ሺ1 + µ2 ሻ π tanλ ൌ µ µ2 π2 = 2ሺ1 + µ2 ሻ µ2 π2 െ 2µ2 ൌ 2 ⇒ 2 µൌඨ 2 ൌ 0.504 π െ2 µ2 ሺπ2 െ 2ሻ ൌ 2 Question 22 A uniform rod of length 2a and weight W rests with its middle point upon a rough horizontal cylinder whose axis is perpendicular to the rod. show that the greatest weight that can be attached to one end of the rod without slipping it off the cylinder is bλ λ W a െ bλ λ Where b is the radius of the cylinder and λ is the angle of friction. Solution B1 R µR G A B OԢ Wsinλ λ λ b W1sinλ A1 Wcosλ O λ W1cosλ W1 S = rθ GO Ԣ = bλ W Available At: mathcity.org Contact At: qadri86@yahoo.com
- 31. 31 Let AB be the rod of the length 2a, weight W and b be the radius of the cylinder. Let R is normal reaction and µR is force of friction. Let W1 is the weight attached to one end of the rod. A1 and B1 are new positions of the rod in the limiting equilibrium. Since AB = 2a Therefore AG = a Taking moment of all forces about OԢ W1cosλ(A1OԢ) = Wcosλ(GOԢ) ⇒ W1(A1OԢ) = W(GOԢ) From fig. GOԢ = bλ and A1OԢ = AG – GOԢ = a െ bλ ⇒ W1(a െ bλ) = W(bλ) ⇒ W1 = bλ W a െ bλ Question 23 A hemispherical shell rests on a rough inclined plane whose angle of friction is λ. Show that the inclination of the plane base to the horizontal cannot be greater than sin – 1(2sinλ) λ Solution D O C r G λ θ µR B A θ W Let G be the centre of the gravity of hemispherical shell, W is its weight acting downward from G. Let θ be angle that the plane base makes with the horizontal then we have to show that θ = sin – 1(2sinλ) From fig. OA = r (radius) r OG = 2 ∠OAG = λ Available At: mathcity.org Contact At: qadri86@yahoo.com
- 32. 32 ∠AGB = θ ∠AGO = π െ θ In triangle OAG, ⇒ ⇒ ⇒ OG OA = sinλ sin(π െ θ) rൗ 2 = r sin(π െ θ) ൌ sinθ sinλ sinθ sinθ ൌ 2sinλ θ = sinି1 ሺ2sinλሻ %%%%%% End of The Chapter # 2 %%%%%% Available At: mathcity.org Contact At: qadri86@yahoo.com