#5 only APPLIED PROJECTahROLLER DERBYSuppose that a solid ball (a marble), a hollow ball (a squash ball), a solid cylinder (a steel bar),and a hollow cylinder (a lead pipe) roll down a slope. Which of these objects reaches the bottomfirst? (Make a guess before proceeding.)To answer this question, we consider a ball or cylinder with mass m, radius r, and moment ofinertia 7 (about the axis of rotation). If the vertical drop is h, then the potential energy at the top ismgh. Suppose the object reaches the bottom with velocity v and angular velocity w, so v = wr.The kinetic energy at the bottom consists of two parts: mv² from translation (moving down theslope) and Iw² from rotation. If we assume that energy loss from rolling friction is negligible,then conservation of energy givesmgh = mv² + Iw²1. Show that2gh₂₂² =1 + 1*dydtwhere I* =2. If y(t) is the vertical distance traveled at time t, then the same reasoning as used inProblem 1 shows that v² = 2gy/(1+ I*) at any time t. Use this result to show that ysatisfies the differential equationT =2g1 + 1*Imr²-(sin a) √ywhere a is the angle of inclination of the plane.3. By solving the differential equation in Problem 2, show that the total travel time is2h(1 + 1*)g sin²aThis shows that the object with the smallest value of I* wins the race.4. Show that I* = for a solid cylinder and I* = 1 for a hollow cylinder.5. Calculate /* for a partly hollow ball with inner radius a and outer radius r. Express youranswer in terms of b = a/r. What happens as a → 0 and as a → r?6. Show that I* = for a solid ball and I* = for a hollow ball. Thus the objects finish in thefollowing order:solid ball, solid cylinder, hollow ball, hollow cylinder.

As asked only 5 is solved here. 5. Hollow ball with inner radius a and outer radius r . The formula…

Answered: 5. Calculate /* for a partly hollow…

5. Calculate /* for a partly hollow ball with inner radius a and outer radius r. Express your answer in terms of b = a/r. What happens as a → 0 and as a → r?

University Physics Volume 1

18th Edition

ISBN:9781938168277

Author:William Moebs, Samuel J. Ling, Jeff Sanny

Publisher:William Moebs, Samuel J. Ling, Jeff Sanny

Chapter11: Angular Momentum

Section: Chapter Questions

Problem 59P: A bug of mass 0.020 kg is at rest on the edge of a solid cylindrical disk (M=0.10kg,R=0.10m)...

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Question

#5 only

APPLIED PROJECT
a
h
ROLLER DERBY
Suppose that a solid ball (a marble), a hollow ball (a squash ball), a solid cylinder (a steel bar),
and a hollow cylinder (a lead pipe) roll down a slope. Which of these objects reaches the bottom
first? (Make a guess before proceeding.)
To answer this question, we consider a ball or cylinder with mass m, radius r, and moment of
inertia 7 (about the axis of rotation). If the vertical drop is h, then the potential energy at the top is
mgh. Suppose the object reaches the bottom with velocity v and angular velocity w, so v = wr.
The kinetic energy at the bottom consists of two parts: mv² from translation (moving down the
slope) and Iw² from rotation. If we assume that energy loss from rolling friction is negligible,
then conservation of energy gives
mgh = mv² + Iw²
1. Show that
2gh
₂₂² =
1 + 1*
dy
dt
where I* =
2. If y(t) is the vertical distance traveled at time t, then the same reasoning as used in
Problem 1 shows that v² = 2gy/(1+ I*) at any time t. Use this result to show that y
satisfies the differential equation
T =
2g
1 + 1*
I
mr²
-(sin a) √y
where a is the angle of inclination of the plane.
3. By solving the differential equation in Problem 2, show that the total travel time is
2h(1 + 1*)
g sin²a
This shows that the object with the smallest value of I* wins the race.
4. Show that I* = for a solid cylinder and I* = 1 for a hollow cylinder.
5. Calculate /* for a partly hollow ball with inner radius a and outer radius r. Express your
answer in terms of b = a/r. What happens as a → 0 and as a → r?
6. Show that I* = for a solid ball and I* = for a hollow ball. Thus the objects finish in the
following order:solid ball, solid cylinder, hollow ball, hollow cylinder.

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