Problem 1 i ph o 34
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(1) The document describes a theoretical physics problem involving a pendulum bob suspended from a rigid cylindrical rod. (2) It asks a series of questions about the motion of the pendulum bob and its velocity and acceleration at different points in its swing. (3) It also considers a variation where a heavier weight is attached below the pendulum, causing it to fall as the pendulum swings, and asks for the critical ratio of the weight's fall distance to the total string length needed for the pendulum to swing all the way around the rod.
![Theoretical Question 1
A Swing with a Falling Weight
A rigid cylindrical rod of radius R is held horizontal above the ground. With a
string of negligible mass and length L ( RL π2> ), a pendulum bob of mass m is
suspended from point A at the top of the rod as shown in Figure 1a. The bob is raised
until it is level with A and then released from rest when the string is taut. Neglect any
stretching of the string. Assume the pendulum bob may be treated as a mass point
and swings only in a plane perpendicular to the axis of the rod. Accordingly, the
pendulum bob is also referred to as the particle. The acceleration of gravity is g
.
Let O be the origin of the coordinate system. When the particle is at point P, the
string is tangential to the cylindrical surface at Q. The length of the line segment QP
is called s. The unit tangent vector and the unit radial vector at Q are given by tˆ and
rˆ , respectively. The angular displacement θ of the radius OQ, as measured
counterclockwise from the vertical x-axis along OA, is taken to be positive.
When 0=θ , the length s is equal to L and the gravitational potential energy U of
the particle is zero. As the particle moves, the instantaneous time rates of change of θ
and s are given by θ and s , respectively.
Unless otherwise stated, all the speeds and velocities are relative to the fixed point
O.
Part A
In Part A, the string is taut as the particle moves. In terms of the quantities
introduced above (i.e., s, θ, s ,θ , R, L, g, tˆ and rˆ ), find:
(a) The relation between θ and s . [0.5 point]
(b) The velocity Qv
of the moving point Q relative to O. [0.5 point]
(c) The particle’s velocity v′
relative to the moving point Q when it is at P.
[0.7 point]
(d) The particle’s velocity v
relative to O when it is at P. [0.7 point]
(e) The tˆ -component of the particle’s acceleration relative to O when it is at P.
m
R
θ
Figure 1a
s
x
O
AL
Q
P
rˆ
tˆg
](https://image.slidesharecdn.com/problem1ipho34-130920235903-phpapp02/85/Problem-1-i-ph-o-34-1-320.jpg)
![[0.7 point]
(f) The particle’s gravitational potential energy U when it is at P. [0.5 point]
(g) The speed vm of the particle at the lowest point of its trajectory. [0.7 point]
Part B
In Part B, the ratio L to R has the following value:
886.6352.3534.3
16
cot
3
2
8
9
=+=+=
ππ
R
L
(h) What is the speed sv of the particle when the string segment from Q to P is both
straight and shortest in length? (in terms of g and R) [2.4 points]
(i) What is the speed Hv of the particle at its highest point H when it has swung to
the other side of the rod? (in terms of g and R [1.9 points]
Part C
In Part C, instead of being suspended from A, the pendulum bob of mass m is
connected by a string over the top of the rod to a heavier weight of mass M, as shown
in Figure 1b. The weight can also be treated as a particle.
Initially, the bob is held stationary at the same level as A so that, with the weight
hanging below O, the string is taut with a horizontal section of length L. The bob is
then released from rest and the weight starts falling. Assume that the bob remains in
a vertical plane and can swing past the falling weight without any interruption.
The kinetic friction between the string and the rod surface is negligible. But the
static friction is assumed to be large enough so that the weight will remain stationary
once it has come to a stop (i.e. zero velocity).
(j) Assume that the weight indeed comes to a stop after falling a distance D and that
(L-D) >> R. If the particle can then swing around the rod to θ = 2π while both
segments of the string free from the rod remain straight, the ratio α = D/L must
not be smaller than a critical value αc. Neglecting terms of the order R/L or
higher, obtain an estimate on αc in terms of M/m. [3.4 points]
A
M
m
L
R
Figure 1b
x
θ
O](https://image.slidesharecdn.com/problem1ipho34-130920235903-phpapp02/85/Problem-1-i-ph-o-34-2-320.jpg)
![[Answer Sheet] Theoretical Question 1
A Swing with a Falling Weight
(a) The relation between θ and s is
(b) The velocity of the moving point Q relative to O is
(c) When at P, the particle’s velocity relative to the moving point Q is
(d) When at P, the particle’s velocity relative to O is
(e) When at P, the tˆ -component of the particle’s acceleration relative to O is
(f) When at P, the particle’s gravitational potential energy is
(g) The particle’s speed when at the lowest point of its trajectory is
Qv
=
v′
=
v
=
U =
mv =](https://image.slidesharecdn.com/problem1ipho34-130920235903-phpapp02/85/Problem-1-i-ph-o-34-3-320.jpg)
Problem 1 i ph o 34
- 1. Theoretical Question 1 A Swing with a Falling Weight A rigid cylindrical rod of radius R is held horizontal above the ground. With a string of negligible mass and length L ( RL π2> ), a pendulum bob of mass m is suspended from point A at the top of the rod as shown in Figure 1a. The bob is raised until it is level with A and then released from rest when the string is taut. Neglect any stretching of the string. Assume the pendulum bob may be treated as a mass point and swings only in a plane perpendicular to the axis of the rod. Accordingly, the pendulum bob is also referred to as the particle. The acceleration of gravity is g . Let O be the origin of the coordinate system. When the particle is at point P, the string is tangential to the cylindrical surface at Q. The length of the line segment QP is called s. The unit tangent vector and the unit radial vector at Q are given by tˆ and rˆ , respectively. The angular displacement θ of the radius OQ, as measured counterclockwise from the vertical x-axis along OA, is taken to be positive. When 0=θ , the length s is equal to L and the gravitational potential energy U of the particle is zero. As the particle moves, the instantaneous time rates of change of θ and s are given by θ and s , respectively. Unless otherwise stated, all the speeds and velocities are relative to the fixed point O. Part A In Part A, the string is taut as the particle moves. In terms of the quantities introduced above (i.e., s, θ, s ,θ , R, L, g, tˆ and rˆ ), find: (a) The relation between θ and s . [0.5 point] (b) The velocity Qv of the moving point Q relative to O. [0.5 point] (c) The particle’s velocity v′ relative to the moving point Q when it is at P. [0.7 point] (d) The particle’s velocity v relative to O when it is at P. [0.7 point] (e) The tˆ -component of the particle’s acceleration relative to O when it is at P. m R θ Figure 1a s x O AL Q P rˆ tˆg
- 2. [0.7 point] (f) The particle’s gravitational potential energy U when it is at P. [0.5 point] (g) The speed vm of the particle at the lowest point of its trajectory. [0.7 point] Part B In Part B, the ratio L to R has the following value: 886.6352.3534.3 16 cot 3 2 8 9 =+=+= ππ R L (h) What is the speed sv of the particle when the string segment from Q to P is both straight and shortest in length? (in terms of g and R) [2.4 points] (i) What is the speed Hv of the particle at its highest point H when it has swung to the other side of the rod? (in terms of g and R [1.9 points] Part C In Part C, instead of being suspended from A, the pendulum bob of mass m is connected by a string over the top of the rod to a heavier weight of mass M, as shown in Figure 1b. The weight can also be treated as a particle. Initially, the bob is held stationary at the same level as A so that, with the weight hanging below O, the string is taut with a horizontal section of length L. The bob is then released from rest and the weight starts falling. Assume that the bob remains in a vertical plane and can swing past the falling weight without any interruption. The kinetic friction between the string and the rod surface is negligible. But the static friction is assumed to be large enough so that the weight will remain stationary once it has come to a stop (i.e. zero velocity). (j) Assume that the weight indeed comes to a stop after falling a distance D and that (L-D) >> R. If the particle can then swing around the rod to θ = 2π while both segments of the string free from the rod remain straight, the ratio α = D/L must not be smaller than a critical value αc. Neglecting terms of the order R/L or higher, obtain an estimate on αc in terms of M/m. [3.4 points] A M m L R Figure 1b x θ O
- 3. [Answer Sheet] Theoretical Question 1 A Swing with a Falling Weight (a) The relation between θ and s is (b) The velocity of the moving point Q relative to O is (c) When at P, the particle’s velocity relative to the moving point Q is (d) When at P, the particle’s velocity relative to O is (e) When at P, the tˆ -component of the particle’s acceleration relative to O is (f) When at P, the particle’s gravitational potential energy is (g) The particle’s speed when at the lowest point of its trajectory is Qv = v′ = v = U = mv =
- 4. (h) When line segment QP is straight with the shortest length, the particle‘s speed is (Give expression and value in terms of g and R ) (i) At the highest point, the particle’s speed is (Give expression and value in terms of g and R) (j) In terms of the mass ratio M/m, the critical value αc of the ratio D/L is sv = Hv = αc =