Solution baupc 2003

Solutions
9th Annual
Boston Area Undergraduate
Physics Competition
April 26, 2003
1. At time t, the free end of the rope is moving at speed gt, and it has fallen a distance
gt2/2. This distance gets “doubled up” below the support. So at time t, a length gt2/4
is hanging at rest, and a length L − gt2/4 is moving at speed gt. The momentum
of the entire rope is therefore p = −ρ(L − gt2/4)(gt), where ρ ≡ M/L is the mass
density, and the minus sign signiﬁes downward motion.
The forces on the entire rope are Mg = ρLg downward, and N (the force from the
support) upward. F = dp/dt applied to the entire rope therefore gives
N − ρLg =
d
dt
−ρLgt +
ρg2t3
4
=⇒ N =
3ρg2t2
4
. (1)
(Note that this equals 3ρv2/4.) This result holds until t = 4L/g, which is the time
it takes the free end to fall a distance 2L. After this time, the force from the support
is simply the weight of the entire rope, Mg.
Remark: At time t, the part of the rope that is hanging at rest has weight ρ(gt2
/4)g =
ρg2
t2
/4. From eq. (1), we see that the support must apply a force that is three times the
weight of this motionless part of the rope. The extra force is necessary because the support
must do more than hold up the motionless part. It must cause the change in momentum of
the atoms in the rope that are abruptly brought to rest from their freefall motion.
2. Because the pole is very thin, we can approximate the motion of the mass at all times
by a circle. Let θ be the desired angle of the string when it becomes completely
unwound. Then the total change in height of the mass is L cos θ. So conservation of
energy gives
mgL cos θ =
mv2
2
. (2)
The vertical component of the tension in the string is essentially mg, because the
height of the mass changes so slowly (because the pole is so thin). Therefore, the
horizontal component of the tension is mg tan θ. The mass travels in a horizontal
circle of radius L sin θ, so the horizontal F = ma equation gives
mg tan θ =
mv2
L sin θ
. (3)
Dividing eq. (3) by eq. (2) gives
tan θ =
√
2. (4)
The numerical value turns out to be θ ≈ 54.7◦.
1

3. Let the desired resistance between opposite corners be xR, where x is a numerical
factor to be determined. Consider all the squares except the largest two. Label this
subset of resistors as S3. S3 is identical to the original circuit, except that it is shrunk
by a factor of 2. Therefore, the resistance between the right and left corners of S3 is
xR/2 (because all the cross sections are the same, and resistance is proportional to
length).
From left-right symmetry, all the points on the vertical bisector of the circuit below
are at the same potential. Therefore, we can separate the top and bottom corners of
S3 from the horizontal lines they touch, as shown.
L
We can then think of S3 as an eﬀective resistor of resistance xR/2, as shown.
A
B
C D
F
E
We can now simplify this circuit by noting the top-bottom symmetry, which tells us
that we can identify A with C, and also D with F. (Note that we cannot identify B
with A, C; or E with D, F.) We arrive at:
R/
R
B
A,C
E
D,F
R/
2
R/ 22R/ 222
xR/2
R/2
2

Reducing things a bit more gives:
R
BA,C E D,F
R/
2
R/ 24R/ 244
xR/2
R/4
_______
2 +1( )
Reducing one last time, and setting the result equal to xR, gives:
xR =
R
2
+


1
R
2(
√
2+1)
+
1
R
2
√
2
+ xR
2


−1
(5)
We must now solve for x. Gradual simplification gives:
2x − 1 =
1
√
2 + 1 +
√
2√
2x+1
=⇒ (2x − 1) (2 +
√
2)x + 2
√
2 + 1 =
√
2x + 1
=⇒ (2 +
√
2)x2
+
√
2x − (
√
2 + 1) = 0
=⇒ x2
+ (
√
2 − 1)x −
1
√
2
= 0
=⇒ x =
1
2
(
√
3 −
√
2 + 1) ≈ 0.659. (6)
4. Integrating F = dp/dt gives
∆p = F dt. (7)
Before the magnetic field is turned on, the force on the particle takes the form, F =
−bv. Therefore,
∆p = (−bv) dt. (8)
But v dt = ∆x, where ∆x is the total displacement (which equals 10 cm straight
into the region here). So we have
∆p0 = −b∆x0, (9)
where the subscript denotes the zero-B case.
Let us now turn on the magnetic field. We have F = −bv + qv × B, so
∆p = (−bv + qv × B) dt
= −b∆x + q∆x × B. (10)
3

But the ∆p here is the same as the ∆p0 in eq. (9), because the the particle has the
same initial velocity and the same ﬁnal velocity (namely zero). Therefore, we have
(after dividing through by −b)
∆x0 = ∆xB − (q/b)∆xB × B. (11)
The two terms on the righthand side represent orthogonal vectors. Since the sum
of these two orthogonal vectors equals the vector on the lefthand side, and since
|∆xB|/|∆x0| = 6/10, we see that we have the following 6-8-10 right triangle.
∆x0
∆xB
-(q/b)∆xB B
6
8
10
If we now double the magnetic ﬁeld, we have
∆x0 = ∆x2B − (q/b)∆x2B × 2B. (12)
The ratio of the magnitudes of the two vectors on the righthand side is twice the ratio
of the two vectors in eq. (11). That is, it is 8/3 instead of 8/6. So we now have the
following right triangle.
∆x0
∆x2B
-(q/b)∆x2B 2B
10
3a
8a
The Pythagorean theorem gives the value of a as 10/
√
73. Therefore, the net dis-
placement of the particle is |∆x2B| = 3a = 30/
√
73 ≈ 3.51 cm.
5. Let us ignore the tilt of the plane for a moment and determine how the ωf and vf
after a bounce are related to the ωi and vi before the bounce (where v denotes the
velocity component parallel to the plane). Let the positive directions of velocity and
force be to the right along the plane, and let the positive direction of angular velocity
4

be counterclockwise. If we integrate (over the small time of a bounce) the friction
force and the resulting torque, we obtain
F =
dp
dt
=⇒ F dt = ∆p,
τ =
dL
dt
=⇒ τ dt = ∆L. (13)
But τ = RF. And since R is constant, we have
∆L = RF dt = R F dt = R∆p. (14)
Therefore,
I(ωf − ωi) = Rm(vf − vi). (15)
But conservation of energy gives
1
2
mv2
f +
1
2
Iω2
f =
1
2
mv2
i +
1
2
Iω2
i
=⇒ I(ω2
f − ω2
i ) = m(v2
i − v2
f ). (16)
Dividing this equation by eq. (15) gives1
R(ωf + ωi) = −(vf + vi). (17)
We can now combine this equation with eq. (15), which can be rewritten (using
I = (2/5)mR2) as
2
5
R(ωf − ωi) = vf − vi. (18)
Given vi and ωi, the previous two equations are two linear equations in the two
unknowns, vf and ωf . Solving for vf and ωf , and then writing the result in matrix
notation, gives
vf
Rωf
=
1
7
3 −4
−10 −3
vi
Rωi
≡ A
vi
Rωi
. (19)
Note that
A2
=
1
49
49 0
0 49
= I. (20)
Let us now consider the eﬀects of the tilted plane. Since the ball’s speed perpendicular
to the plane is unchanged by each bounce, the ball spends the same amount of time in
the air between any two successive bounces. This time equals T = 2V/g cos θ, because
the component of gravity perpendicular to the plane is g cos θ. During this time, the
speed along the plane increases by (g sin θ)T = 2V tan θ ≡ V0.
1
We have divided out the trivial ωf = ωi and vf = vi solution, which corresponds to slipping motion
on a frictionless plane. The nontrivial solution we will ﬁnd shortly is the non-slipping one. Basically, to
conserve energy, there must be no work done by friction. But since work is force times distance, this means
that either the plane is frictionless, or that there is no relative motion between ball’s contact point and the
plane. Since we are given that the plane has friction, the latter (non-slipping) case must be the one we are
concerned with.
5

Let Q denote the (v, Rω) vector at a given time (where v denotes the velocity compo-
nent parallel to the plane). The ball is initially projected with Q = 0. Therefore, right
before the first bounce, we have Qbefore
1 = (V0, 0) ≡ V0. (We have used the fact that
ω doesn’t change while the ball is in the air.) Right after the first bounce, we have
Qafter
1 = AV0. We then have Qbefore
2 = AV0 + V0, and so Qafter
2 = A(AV0 + V0).
Continuing in this manner, we see that
Qbefore
n = (An−1
+ · · · + A + I)V0, and
Qafter
n = (An
+ · · · + A2
+ A)V0. (21)
However, A2 = I, so all the even powers of A equal I. The value of Q after the nth
bounce is therefore given by
n even =⇒ Qafter
n =
n
2
(A + I)V0.
n odd =⇒ Qafter
n =
1
2
(n + 1)A + (n − 1)I V0. (22)
Using the value of A defined in eq. (19), we find
n even =⇒
vn
Rωn
=
n
7
5 −2
−5 2
V0
0
.
n odd =⇒
vn
Rωn
=
1
7
5n − 2 −2n − 2
−5n − 5 2n − 5
V0
0
. (23)
Therefore, the speed along the plane after the nth bounce equals (using V0 ≡ 2V tan θ)
vn =
10nV tan θ
7
(n even),
vn =
(10n − 4)V tan θ
7
(n odd). (24)
Remark: Note that after an even number of bounces, eq. (23) gives v = −Rω. This is the
“rolling” condition. That is, the angular speed exactly matches up with the translation speed,
so v and ω are unaffected by the bounce. (The vector (1, −1) is an eigenvector of A.) At the
instant that an even-n bounce occurs, the v and ω are the same as they would be for a ball
that simply rolls down the plane. At the instant after an odd-n bounce, the v is smaller than
it would be for a rolling ball, but the ω is larger. (And right before an odd-n bounce, the v is
larger but the ω is smaller.)
6. It turns out that the ball can move arbitrarily fast around the cone. As we will see,
the plane of the contact circle (represented by the chord in the figure below) will need
to be tilted downward from the contact point, so that the angular momentum has a
rightward horizontal component when it is at the position shown. In what follows, it
will be convenient to work with the angle φ ≡ 90◦ − θ.
6

φ
θ
φ
β
L
r
F
R
Let’s ﬁrst look at F = ma along the plane. Let Ω be the angular frequency of the
ball’s motion around the cone. Then the ball’s horizontal acceleration is m Ω2 to the
left. So F = ma along the plane gives (where Ff is the friction force)
mg sin φ + Ff = m Ω2
cos φ. (25)
Now let’s look at τ = dL/dt. To get a handle on how fast the ball is spinning,
consider what the setup looks like in the rotating frame in which the center of the ball
is stationary (so the ball just spins in place as the cone spins around). Since there is
no slipping, the contact points on the ball and the cone must have the same speed.
That is,
ωr = Ω =⇒ ω =
Ω
r
, (26)
where ω is the angular speed of the ball in the rotating frame, and r is the radius of
the contact circle on the ball.2 The angular momentum of the ball in the lab frame is
L = Iω (at least for the purposes here3), and it points in the direction shown above.
The L vector precesses around a cone in L-space with the same frequency, Ω, as the
ball moves around the cone. Only the horizontal component of L changes, and it
traces out a circle of radius Lhor = L sin β, at frequency Ω. Therefore,
dL
dt
= LhorΩ = (Iω sin β)Ω =
IΩ2 sin β
r
, (27)
and the direction of dL/dt is into the page.
The torque on the ball (relative to its center) is due to the friction force, Ff . Hence,
|τ| = Ff R, and its direction is into the page. Therefore, τ = dL/dt gives (with
2
If the center of the ball travels in a circle of radius , then the here should actually be replaced with
+ R sin φ, which is the radius of the contact circle on the cone. But since we’re assuming that R , we
can ignore the R sin φ part.
3
This L = Iω result isn’t quite correct, because the angular velocity of the ball in the lab frame equals
the angular velocity in the rotating frame (which tilts downwards with the magnitude ω we just found)
plus the angular velocity of the rotating frame with respect to the lab frame (which points straight up with
magnitude Ω). This second part of the angular velocity simply yields an additional vertical component of
the angular momentum. But the vertical component of L doesn’t change with time as the ball moves around
the cone. It is therefore irrelevant, since we will be concerned only with dL/dt in what follows.
7

I = ηmR2, where η = 2/5 in this problem)
Ff R =
IΩ2 sin β
r
=⇒ Ff =
ηmRΩ2 sin β
r
. (28)
Using this Ff in eq. (25) gives
mg sin φ +
ηmRΩ2 sin β
r
= m Ω2
cos φ. (29)
Solving for Ω gives
Ω2
=
g sin φ
cos φ − ηR sin β
r
. (30)
We see that it is possible for the ball to move around the cone infinitely fast if
cos φ =
η sin β
r/R
. (31)
If we now define γ ≡ φ − β in the above figure, we have r/R = sin γ, and β = φ − γ.
So eq. (31) becomes
cos φ =
η sin(φ − γ)
sin γ
. (32)
Using the sum formula for the sine in the numerator, we can rewrite this as
tan γ =
η
1 + η
tan φ =
2
7
tan φ
=
2
7
cot θ, (33)
where we have used η = 2/5. We finally obtain
r
R
= sin γ =
1
1 + cot2 γ
=
1
1 + 49
4 tan2 θ
. (34)
Remarks:
(1) In the limit θ ≈ 0 (that is, a very thin cone), we obtain r/R ≈ 1, which makes sense.
The contact circle is essentially a horizontal great circle.
In the limit θ ≈ 90◦
(that is, a nearly flat plane), we obtain r/R ≈ 0. The circle of
contact points is very small, but the ball can still roll around the cone arbitrarily fast
(assuming that there is sufficient friction). This isn’t entirely intuitive.
(2) What value of φ allows the largest tilt angle of the contact circle (that is, the largest
β)? From eq. (31), we see that maximizing β is equivalent to maximizing (r/R) cos φ,
8

or equivalently (r/R)2
cos2
φ. Using the value of r/R from eq. (34), we see that we want
to maximize
(r/R)2
cos2
φ =
cos2
φ
1 + 49
4 cot2
φ
. (35)
Taking the derivative with respect to φ and going through a bit of algebra, we ﬁnd that
the maximum is achieved when
tan φ =
7
2
=⇒ φ = 61.9◦
. (36)
You can then show from eq. (31) that
sin βmax =
5
9
=⇒ βmax = 33.7◦
. (37)
(3) Let’s consider three special cases for the contact circle, namely, a horizontal circle, a
great circle, and a vertical circle.
(a) Horizontal circle: In this case, we have β = 0, so eq. (30) gives
Ω2
=
g tan φ
. (38)
In this case, L points vertically, which means that dL/dt is zero, which means that
the torque is zero, which means that the friction force is zero. Therefore, the ball
moves around the cone with the same speed as a particle sliding without friction.
(You can show that such a particle does indeed have Ω2
= g tan φ/ .) The horizontal
contact-point circle (β = 0) is the cutoﬀ case between the sphere moving faster or
slower than a frictionless particle.
(b) Great circle: In this case, we have r = R and β = −(90◦
−φ). Hence, sin β = − cos φ,
and eq. (30) gives
Ω2
=
g tan φ
(1 + η)
. (39)
This reduces to the frictionless-particle case when η = 0, as it should.
(c) Vertical circle: In this case, we have r = R cos φ and β = −90◦
, so eq. (30) gives
Ω2
=
g tan φ
1 + η
cos2 φ
. (40)
Again, this reduces to the frictionless-particle case when η = 0, as it should. But
for φ → 90◦
(thin cone), Ω goes to zero, whereas in the other two cases above, Ω
goes to ∞.
9

Solution baupc 2003

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Solution baupc 2003