10.3 Dynamics of Rotational Motion: Rotational Inertia
A piece of wood can be carved by spinning it on a motorized lathe and holding a sharp chisel to the edge of the wood as it spins. How does the angular velocity of a piece of wood with a radius of 0.2 m spinning on a lathe change when a chisel is held to the wood's edge with a force of 50 N?
- It increases by 0.1 N•m multiplied by the moment of inertia of the wood.
- It decreases by 0.1 N•m divided by the moment of inertia of the wood-and-lathe system.
- It decreases by 0.1 N•m multiplied by the moment of inertia of the wood.
- It decreases by 0.1 m/s2.
A Ferris wheel is loaded with people in the chairs at the following positions: 4 o'clock, 1 o'clock, 9 o'clock, and 6 o'clock. As the wheel begins to turn, what forces are acting on the system? How will each force affect the angular velocity and angular momentum?
A lever is placed on a fulcrum. A rock is placed on the left end of the lever and a downward (clockwise) force is applied to the right end of the lever. What measurements would be most effective to help you determine the angular momentum of the system? (Assume the lever itself has negligible mass.)
- the angular velocity and mass of the rock
- the angular velocity and mass of the rock, and the radius of the lever
- the velocity of the force, the radius of the lever, and the mass of the rock
- the mass of the rock, the length of the lever on both sides of the fulcrum, and the force applied on the right side of the lever
You can use the following setup to determine angular acceleration and angular momentum: A lever is placed on a fulcrum. A rock is placed on the left end of the lever and a known downward (clockwise) force is applied to the right end of the lever. What calculations would you perform? How would you account for gravity in your calculations?
Consider two sizes of disk, both of mass M. One size of disk has radius R; the other has radius 2R. System A consists of two of the larger disks rigidly connected to each other with a common axis of rotation. System B consists of one of the larger disks and a number of the smaller disks rigidly connected with a common axis of rotation. If the moment of inertia for system A equals the moment of inertia for system B, how many of the smaller disks are in system B?
You are given a thin rod of length 1.0 m and mass 2.0 kg, a small lead weight of 0.50 kg, and a not-so-small lead weight of 1.0 kg. The rod has three holes, one in each end and one through the middle, which may either hold a pivot point or one of the small lead weights. How do you arrange these objects so that the resulting system has the maximum possible moment of inertia? What is that moment of inertia? do you arrange these objects so that the resulting system has the maximum possible moment of inertia? What is that moment of inertia?
10.4 Rotational Kinetic Energy: Work and Energy Revisited
Gear A, which turns clockwise, meshes with gear B, which turns counterclockwise. When more force is applied through gear A, torque is created. How does the angular velocity of gear B change as a result?
- It increases in magnitude.
- It decreases in magnitude.
- It changes direction.
- It stays the same.
Which will cause a greater increase in the angular velocity of a disk: doubling the torque applied or halving the radius at which the torque is applied? Explain.
Which measure would not be useful to help you determine the change in angular velocity when the torque on a fishing reel is increased?
- the radius of the reel
- the amount of line that unspools
- the angular momentum of the fishing line
- the time it takes the line to unspool
What data could you collect to study the change in angular velocity when two people push a merry-go-round instead of one, providing twice as much torque? How would you use the data you collect?
10.5 Angular Momentum and Its Conservation
Which rotational system would be best to use as a model to measure how angular momentum changes when forces on the system are changed?
- a fishing reel
- a planet and its moon
- a figure skater spinning
- a person's lower leg
You are collecting data to study changes in the angular momentum of a bicycle wheel when a force is applied to it. Which of the following measurements would be least helpful to you?
- the time for which the force is applied
- the radius at which the force is applied
- the angular velocity of the wheel when the force is applied
- the direction of the force
Which torque applied to a disk with radius 7.0 cm for 3.5 s will produce an angular momentum of 25 N•m•s?
- 7.1 N•m
- 357.1 N•m
- 3.6 N•m
- 612.5 N•m
Which of the following would be the best way to produce measurable amounts of torque on a system to test the relationship between the angular momentum of the system, the average torque applied to the system, and the time for which the torque is applied?
- having different numbers of people push on a merry-go-round
- placing known masses on one end of a seesaw
- touching the outer edge of a bicycle wheel to a treadmill that is moving at different speeds
- hanging known masses from a string that is wound around a spool suspended horizontally on an axle
The diagram above shows a top view of a child of mass M on a circular platform of mass 2M that is rotating counterclockwise. Assume the platform rotates without friction. Which of the following describes an action by the child that will increase the angular speed of the platform-child system and why?
- The child moves toward the center of the platform, increasing the total angular momentum of the system.
- The child moves toward the center of the platform, decreasing the rotational inertia of the system.
- The child moves away from the center of the platform, increasing the total angular momentum of the system.
- The child moves away from the center of the platform, decreasing the rotational inertia of the system.
A moon is in an elliptical orbit about a planet as shown above. At point A the moon has speed uA and is at distance RA from the planet. At point B the moon has speed uB. Has the moon's angular momentum changed? Explain your answer.
A hamster sits 0.10 m from the center of a lazy Susan of negligible mass. The wheel spins with a frequency of 1.0 rev/s. How will the frequency of the lazy Susan change if the hamster walks to 0.30 m from the center of rotation? Assume zero friction and no external torque.
- It will speed up to 2.0 rev/s.
- It will speed up to 9.0 rev/s.
- It will slow to 0.11 rev/s.
- It will slow to 0.22 rev/s.
Earth has a mass of 6.0 × 1024 kg, a radius of 6.4 × 106 m, and a rotational frequency of 1.2 × 10–5 rev/s. How would the planet's rotational frequency change if a layer of Earth with mass 1.0 × 1023 kg broke off of the Earth, decreasing Earth's radius by 0.2 × 106 m? Assume no friction.
Consider system A, consisting of two disks of radius R, with both rotating clockwise. Now consider system B, consisting of one disk of radius R rotating counterclockwise and another disk of radius 2R rotating clockwise. All of the disks have the same mass, and all have the same magnitude of angular velocity.
Which system has the greatest angular momentum?
- They're equal.
- Not enough information
Assume that a baseball bat being swung at 3π rad/s by a batting machine is equivalent to a 1.1 m thin rod with a mass of 1.0 kg. How fast would a 0.15 kg baseball that squarely hits the very tip of the bat have to be going for the net angular momentum of the bat-ball system to be zero?
10.6 Collisions of Extended Bodies in Two Dimensions
A box with a mass of 2.0 kg rests on one end of a seesaw. The seesaw is 6.0 m long, and we can assume it has negligible mass. Approximately what angular momentum will the box have if someone with a mass of 65 kg sits on the other end of the seesaw quickly, with a velocity of 1.2 m/s?
- 702 kg•m2/s
- 39 kg•m2/s
- 18 kg•m2/s
- 1.2 kg•m2/s
A spinner in a board game can be thought of as a thin rod that spins about an axis at its center. The spinner in a certain game is 12 cm long and has a mass of 10 g. How will its angular velocity change when it is flicked at one end with a force equivalent to 15 g travelling at 5.0 m/s if all the energy of the collision is transferred to the spinner? (You can use the table in Figure 10.12 to estimate the rotational inertia of the spinner.)
A cyclist pedals to exert a torque on the rear wheel of the bicycle. When the cyclist changes to a higher gear, the torque increases. Which of the following would be the most effective strategy to help you determine the change in angular momentum of the bicycle wheel?
- multiplying the ratio between the two torques by the mass of the bicycle and rider
- adding the two torques together, and multiplying by the time for which both torques are applied
- multiplying the difference in the two torques by the time for which the new torque is applied
- multiplying both torques by the mass of the bicycle and rider
An electric screwdriver has two speeds, each of which exerts a different torque on a screw. Describe what calculations you could use to help you compare the angular momentum of a screw at each speed. What measurements would you need to make in order to calculate this?
Why is it important to consider the shape of an object when determining the object's angular momentum?
- The shape determines the location of the center of mass. The location of the center of mass in turn determines the angular velocity of the object.
- The shape helps you determine the location of the object's outer edge, where rotational velocity will be greatest.
- The shape helps you determine the location of the center of rotation.
- The shape determines the location of the center of mass. The location of the center of mass contributes to the object's rotational inertia, which contributes to its angular momentum.
How could you collect and analyze data to test the difference between the torques provided by two speeds on a tabletop fan?
Describe a rotational system you could use to demonstrate the effect on the system's angular momentum of applying different amounts of external torque.
How could you use simple equipment such as balls and string to study the changes in angular momentum of a system when it interacts with another system?
10.7 Gyroscopic Effects: Vector Aspects of Angular Momentum
A globe (model of the Earth) is a hollow sphere with a radius of 16 cm. By wrapping a cord around the equator of a globe and pulling on it, a person exerts a torque on the globe of 120 N • m for 1.2 s. What angular momentum does the globe have after 1.2 s?
How could you use a fishing reel to test the relationship between the torque applied to a system, the time for which the torque was applied, and the resulting angular momentum of the system? How would you measure angular momentum?