Question

A passenger compartment of a rotating amusement park ride contains a bench on which a book

of mass mb is placed, as indicated by the dot in the left figure above. The compartment moves

with a constant angular speed about the center of the ride along a circular path or radius R. The

bench remains horizontal throughout the compartment’s motion. The right figure above shows a

magnified view of the compartment.

The graph below shows the horizontal (x) component of the book’s position as a function of

time, where the +x direction is to the right.


(a) Explain briefly how the graph could be used to determine the period of revolution of the

book.

(b) Can the tangential speed (not the angular speed) be determined from the information given?

Briefly explain why or why not.

(c) What forces act on the book at the lowest point of its circular path? Indicate in words both

the direction and the source of each force.

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Question 1 continued

At the lowest point of the circular path, the book is moving only in the horizontal direction.

Three students are discussing the force of friction on the book. Here are their arguments:

Student 1: At the instant the box is at its lowest position, the book experiences no force of

friction at all, because the net force is only in the vertical direction.

Student 2: At the instant the box is at its lowest position, the book experiences a force of

kinetic friction, because the book and the bench are moving to the right.

Student 3: At the instant the box is at its lowest position, the book experiences a force of

static friction, because the book doesn’t slip relative to the bench.

(d) What is one aspect of student 2’s reasoning that is incorrect? Explain why.

(e) Who is correct, student 1 or student 3? Explain why.

(f) At the lowest point of the circular path, the book is moving only in the horizontal direction.

In what direction, if any, is the net vertical force on the book? Briefly explain your reasoning

in terms of the book’s motion without deriving any equations.


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Question 1 continued

(g) A student is asked to derive an equation for the vertical force Fb that the bench exerts on the

book at the lowest point of the circular path in terms of the book’s mass mb, tangential speed

vb, radius R of the path, and physical constants. Here is the student’s INCORRECT

derivation:


Briefly explain one way in which this student’s derivation is incorrect, and why it is

incorrect.

(h) At the lowest point of the circular path, is the force that the bench exerts on the book greater

than, less than, or equal to the weight of the book? Briefly explain how your answers in (e)

and/or (f) support your choice.

(i) Consider the position of the compartment at the top of the path, and at the bottom of the path.

At which of these positions, if either, is the total mechanical energy of the book greater?

Justify your answer. In your justification, either explain the source of the work done on the

book which changes its mechanical energy, or explain why the book’s mechanical energy did

not change

Answer 1

Here are the answers to the physics questions:

(a) The period of revolution of the book can be determined from the graph by measuring the time interval between successive peaks or successive troughs. This time interval represents the period.

(b) No, the tangential speed cannot be determined from the information given. The graph shows only the horizontal position vs time. It does not provide any information about the tangential or angular speed.

(c) At the lowest point of the circular path, the forces acting on the book are:

- Weight of the book, downward force from gravity

- Normal force, upward force from the bench supporting the book's weight

- Friction force, horizontal force from the bench on the book opposing its motion

(d) Student 2 is incorrect because kinetic friction acts to oppose motion between two surfaces sliding past each other. At the instant the book is at the lowest point, it is not sliding relative to the bench, so kinetic friction does not apply.

(e) Student 3 is correct. Static friction acts between two surfaces in contact that are not sliding past each other. At its lowest point, the book is in contact with the bench and not sliding, so static friction opposes its horizontal motion.

(f) At its lowest point, the net vertical force on the book is zero. The upward normal force from the bench exactly balances the downward gravitational force. So there is no net vertical force in either direction.

(g) One way the student's derivation is incorrect is that the equation contains the tangential speed vb. However, the tangential speed cannot be determined from the information provided in the question, as explained in (b). So it is invalid to include vb in the derived equation.

(h) The force the bench exerts on the book is equal to the book's weight. As explained in (e) and (f), at the lowest point the net vertical force is zero due to the normal force balancing weight. So the normal force must equal the book's weight.

(i) The book's mechanical energy is greater at the top of the path. As it moves from top to bottom, work is done by friction to reduce its kinetic energy. So it has more kinetic (and thus total mechanical) energy at the top.