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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 5.25 Practice, Study, and Apply - Rotational Motion

Subsection Practice

Calculation 5.25.1. Crankshaft.

The crankshaft (\(r = 3 \mathrm{~cm}\)) in a race car accelerates uniformly from rest to \(3000 \mathrm{~rpm}\) in \(2.0 \mathrm{~s}\text{.}\) How many revolutions does it make?
  1. 10 revolutions
  2. 25 revolutions
  3. 42 revolutions
  4. 50 revolutions
  5. 63 revolutions
  6. 101 revolutions
Answer.
D.

Calculation 5.25.2. Angular motion.

A 60 cm diameter wheel accelerates from rest at a rate of \(7 \mathrm{~rad/s^2}\text{.}\)
  1. What is the tangential acceleration of a point on the edge of the wheel?
  2. How long does it take for the wheel to turn through 14 rotations?
  3. After the wheel has undergone 14 rotations, what is the radial component of the acceleration on the edge the wheel?
Answer 1.
\(2.1 \mathrm{~m/s^2}\)
Answer 2.
\(5.01 \mathrm{~s}\)
Answer 3.
\(739 \mathrm{~m/s^2}\)

Calculation 5.25.3. Army the Armadillo.

The Brazilian three-banded armadillo has a hard outer exterior, and when it senses danger, it rolls into a ball for protection. Suppose that Army the armadillo rolls into a ball (with a \(5 \mathrm{~cm}\) radius) while on the slope of a hill. As a result, they begin to roll down the hill; and after \(15 \mathrm{~s}\text{,}\) they have undergone \(45.8\) revolutions. What is Army’s angular speed at this time?
  1. \(\displaystyle 12.7 \mathrm{~rad/s}\)
  2. \(\displaystyle 15.5 \mathrm{~rad/s}\)
  3. \(\displaystyle 22.0 \mathrm{~rad/s}\)
  4. \(\displaystyle 29.1 \mathrm{~rad/s}\)
  5. \(\displaystyle 38.4 \mathrm{~rad/s}\)
How fast is Army traveling at this point?
  1. \(\displaystyle 1.92 \mathrm{~m/s}\)
  2. \(\displaystyle 2.23 \mathrm{~m/s}\)
  3. \(\displaystyle 3.49 \mathrm{~m/s}\)
  4. \(\displaystyle 4.24 \mathrm{~m/s}\)
  5. \(\displaystyle 5.56 \mathrm{~m/s}\)
Answer 1.
D.
Answer 2.
A.

Calculation 5.25.4. Torque.

A force is applied on a lever arm \(1 \mathrm{~m}\) away from the pivot point, and it produces torque. How much force would have to be applied to produce the same amount of torque if it were applied \(4 \mathrm{~m}\) from the pivot point? Assume that both forces are applied perpendicularly to the lever arm.
  1. Four times the initial force
  2. Sixteen times the initial force
  3. One fourth of the initial force
  4. One sixteenth of the initial force
  5. Same as the initial force
Answer.
C.

Calculation 5.25.5. See-Saw.

Two children of different weights are riding a seesaw. How do they position themselves with respect to the pivot point (the fulcrum) so that they are balanced?
  1. The heavier child sits closer to the fulcrum.
  2. The heavier child sits farther from the fulcrum.
  3. Both children sit at equal distance from the fulcrum.
  4. Since both have different weights, they will never be in balance.
Answer.
A.

Calculation 5.25.6. Moment of Inertia.

Which of the following statements concerning the moment of inertia I are false?
  1. I may be expressed in units of \(\mathrm{~kg m^2}\)
  2. I depends on the angular acceleration of the object as it rotates
  3. I depends on the location of the rotation axis relative to the particles that make up the object.
  4. I depends on the orientation of the rotation axis relative to the particles that make up the object.
Answer.
B.

Calculation 5.25.7. Angular Acceleration.

A uniform, solid disk with a mass of \(24.3 \mathrm{~kg}\) and a radius of \(0.314 \mathrm{~m}\) is oriented vertically and is free to rotate about a frictionless axle. Forces of \(90 \mathrm{~N}\) and \(125 \mathrm{~N}\) are applied to the disk in the same horizontal direction, but one force is applied to the top and the other is applied to the bottom. What is the magnitude of the angular acceleration of the disk?
Answer.
\(9.17 \mathrm{~rad/s^2}\)

Calculation 5.25.8. Olympic High Diver.

An Olympic high diver in midair pulls her legs inward toward her chest. Doing so changes which of these quantities?
  1. Angular momentum
  2. Rotational inertia about her center of mass
  3. Angular velocity
  4. Translational (linear) momentum
  5. Translational (linear) kinetic energy
  6. Rotational kinetic energy
Answer.
B., C., E.

Calculation 5.25.9. Helicopter I.

A typical small rescue helicopter has four blades. Each blade is \(4.0 \mathrm{~m}\) long and has a mass of \(50.0 \mathrm{~kg}\text{.}\) The blades can be approximated as thin rods that rotate about one end of an axis perpendicular to their length. The helicopter has a total loaded mass of \(1000 \mathrm{~kg}\text{.}\) Calculate the rotational kinetic energy in the blades when they rotate at \(300 \mathrm{~rpm}\text{.}\)
Answer.
\(5.26 \times 10^5 \mathrm{~J}\)

Calculation 5.25.10. Helicopter II.

A typical small rescue helicopter has four blades. Each blade is \(4.00 \mathrm{~m}\) long and has a mass of \(50.0 \mathrm{~kg}\text{.}\) The blades can be approximated as thin rods that rotate about one end of an axis perpendicular to their length. The helicopter has a total loaded mass of \(1000 \mathrm{~kg}\text{.}\) What is the ratio of translational kinetic energy of the helicopter over the rotational kinetic energy of its blades when it flies at \(20.0 \mathrm{~m/s}\text{?}\)
Answer.
0.38

Calculation 5.25.11. Helicopter III.

A typical small rescue helicopter has four blades. Each blade is \(4.00 \mathrm{~m}\) long and has a mass of \(50.0 \mathrm{~kg}\text{.}\) The blades can be approximated as thin rods that rotate about one end of an axis perpendicular to their length. The helicopter has a total loaded mass of \(1000 \mathrm{~kg}\text{.}\) To what height could the helicopter be raised if all of the rotational kinetic energy could be used to lift it?
Answer.
\(53.7 \mathrm{~m}\)

Subsection Study

A*R*C*S 5.25.12. Swinging a Bucket.

You swing a bucket of water with total mass m in a vertical circle with a radius given by the length of your arm, \(L\text{.}\) Determine the slowest speed at which you can swing the bucket without water falling on you. For that speed, determine the tension in your arm \(T\) at the bottom of the swing.

Explanation 5.25.13. Acceleration of a Pendulum.

Suppose a pendulum is pulled to one side and released at \(t_1\text{.}\) At \(t_2\text{,}\) the pendulum has swung halfway back to a vertical position. At \(t_3\text{,}\) the pendulum has swung all the way back to a vertical position. Rank the three instants in time by the magnitude of the centripetal acceleration, from greatest to least.

A*R*C*S 5.25.14. Conical Pendulum.

A conical pendulum is formed by attaching a ball of mass \(m\) to a string of length \(L\text{,}\) then allowing the ball to move in a horizontal circle of radius \(r\text{.}\)
Part A. Find an expression for the tension \(T\) in the string.
Part B. Find an expression for the ball’s angular speed \(\omega\text{.}\)

A*R*C*S 5.25.15. Car Around a Curve.

A car starts from rest on a curve with a radius of \(120 \mathrm{~m}\) and tangential acceleration of \(1.5 \mathrm{~m/s^2}\text{.}\) Through what angle will the car have traveled when the magnitude of its total acceleration is \(3 \mathrm{~m/s^2}\text{?}\)
Tip.
For your representation (part 1c), sketch careful velocity and change in velocity vectors for a few different points.

A*R*C*S 5.25.16. Friction on a Baseball.

Find a symbolic expression for the frictional force exerted by a baseball pitcher (tangent to the surface of the baseball) when throwing a fastball that spins at \(2500 \mathrm{~rpm}\text{.}\)

A*R*C*S 5.25.17. Center of Mass.

You have a triangular piece of metal with total mass \(M\text{,}\) base length \(L\text{,}\) and height \(H\text{.}\) You know the mass density is uniform. Find the center of mass of the piece of metal.

Explanation 5.25.18. Tipping Truck.

You observe a truck driving along a road at a constant speed \(v\text{.}\) As the truck begins to turn along a flat, circular part of the road, it also begins to tip over. Use an extended free body diagram to explain why the truck begins to tip.

A*R*C*S 5.25.19. Moment of Inertia of a Tennis Ball.

A tennis ball can be modeled as a spherical shell with total mass \(M\) concentrated at radius \(R\text{.}\) Calculate the moment of inertia about an axis passing through the center of the tennis ball.

A*R*C*S 5.25.20. String Around a Tennis Ball.

A string is wrapped many times around a tennis ball that has mass \(m = 60 \mathrm{~g}\) and radius \(r = 7 \mathrm{~cm}\text{.}\) For a hollow ball, \(I = \frac{2}{3}mr^2\) about the center or \(I = \frac{5}{3}mr^2\) about the edge — calculating these for yourself is a good challenge! The string is attached to the ceiling and the ball is allowed to fall (see the figure below). Determine the tension in the string.

Explanation 5.25.21. String Around a Tennis Ball II.

Suppose you were to drop another tennis ball with the same mass and radius as in the previous problem, but a smaller moment of inertia (this tennis ball might be solid instead of hollow). Is the magnitude of the translational acceleration of the new tennis ball greater than, less than, or equal to the magnitude of the translational acceleration of the original tennis ball?

A*R*C*S 5.25.22. The Merry-Go-Round I.

You are standing at the edge of a merry-go-round at the playground. You and the merry-go-round have about the same mass, and you can treat the merry-go-round as a solid disc with moment of inertia \(I = \frac{1}{2}mR^2\text{,}\) where \(R\) is the radius. The merry-go-round is initially spinning with constant angular speed \(\omega_i\text{.}\) You decide to jump off the merry-go-round; you jump in a horizontal plane tangent to its edge.
The merry-go-round comes to a complete stop. What is your speed immediately after leaving the merry-go-round?

Explanation 5.25.23. The Merry-Go-Round II.

Consider the system of both you and the merry-go-round from the previous activity. Did the kinetic energy of this system increase, decrease, or stay the same as a result of you jumping off?

Subsection Apply

Explanation 5.25.24. The Bowl.

Shown below is a motion diagram of a ball rolling along a semicircular track. The time intervals between each pair of consecutive points are equal. The ball starts from rest at point 1, and just reaches point 9 before rolling back down the right side of the track.
A motion diagram of a ball on a semicircular track.
Figure 5.25.2.
Complete this motion diagram by sketching qualitatively accurate vectors representing both (a) the instantaneous velocity and (b) the instantaneous acceleration of the ball at each instant.
Your explanation should describe how you used the diagram to determine each vector and detail why the magnitudes of the vectors are the same or different.

A*R*C*S 5.25.25. The Conical Bowl.

You are riding a \(150 \mathrm{~kg}\) bicycle at constant speed along a circular track of radius \(25 \mathrm{~m}\text{.}\) The shape of the track is a conical bowl, so the track makes a consistent angle of 15° with the horizontal. Determine the speed of the bicycle.
Tip.
For your special case analysis (part 3c), try a special case using the angle of the track.

Explanation 5.25.26. Balancing Bat.

You spend some time experimenting with a baseball bat, and eventually you are able to balance the bat on a single finger. A friend notices this, and makes the following claim:
“It looks like the balancing point isn’t in the middle of the bat—it’s a little closer to the thicker end of the bat. That balancing point must be the location where half the bat’s mass is on the left and half the bat’s mass is on the right.”
How would you respond to your friend’s claim? Is your friend’s claim correct, partially correct, or incorrect?

Explanation 5.25.27. Standing on a Ramp.

You are standing at rest on the ramp shown in the figure below. Is the magnitude of the normal force acting on your right foot greater than, less than, or equal to the magnitude of the normal force acting on your left foot? (Note: the right foot here is toward the bottom of the ramp.)
A figure on a ramp.
Figure 5.25.3.

A*R*C*S 5.25.28. The Cheese Slice.

You have a one-dimensional slice of cheese with total mass \(M\) and total length \(L\text{.}\) You know the linear mass density is \(\lambda(x) = k \sqrt{x}\text{.}\) Find the constant of proportionality \(k\) (including its units) and the center of mass of the slice of cheese.
Tip.
For your representation (part 1c), sketch a graph of the mass density vs. x. This graph might also prove useful for your sensemaking at the end!

A*R*C*S 5.25.29. The Cube I.

A wooden cube with side length \(L\) and total mass \(M\) is rotating about an axis passing through one of its edges. Use an integral to calculate the moment of inertia.

Explanation 5.25.30. The Cube II.

The cube from the previous activity is instead rotating about an axis passing through its center (and through the center of one of its faces). The parallel axis theorem can be used to demonstrate that the moment of inertia about this axis is smaller than the moment of inertia about the axis in the previous activity. Give a qualitative explanation accounting for this result.

A*R*C*S 5.25.31. Massive Pulley.

Two massive blocks are shown in the figure below, connected to each other by a rope over a pulley with nonzero mass. Find the acceleration of both masses and of the pulley.
Figure 5.25.4.
Tip.
Use a special-case analysis and the fact that you may have solved this problem previously when the pulley is massless.

Explanation 5.25.32. Spool Competition.

You and your friend are in a competition to see which spool will reach the top of a frictionless ramp first. There are two spools: One spool has a thread wrapped many times around it so that as it is pulled up the ramp, it will rotate, and the other spool has a thread fixed at one point so that it will not rotate. Assume the thread is ideal and you and your friend can pull with the same tension.
Part A: Which spool do you want to choose in order to win?
Part B: If instead the competition was about which spool has the most energy at the top of the ramp, would you choose a different spool?

A*R*C*S 5.25.33. Brick on a Disc.

A \(100 \mathrm{~kg}\) disc with radius \(1.6 \mathrm{~m}\) is spinning horizontally at \(25 \mathrm{~rad/s}\text{.}\) You place a \(20 \mathrm{~kg}\) brick quickly and gently on the disc so that it sticks to the edge of the disc. Determine the final angular speed of the disc-brick system.
Tip.
Discuss whether the kinetic energy of the system increases, decreases, or remains the same.

References References

[1]
Practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.
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