1.7.3: Problems

Exercise \(\PageIndex{1}\): A ball slides in a frictionless bowl

A mass of \(2\text{ kg}\) is in a rather large bowl and moves as depicted in the animation (position is given in meters and time is given in seconds). There is no friction between the mass and the bowl so it slides along the surface of the bowl (it does not roll at all). Determine the velocity of the mass at the bottom of the bowl. Restart.

Exercise \(\PageIndex{2}\): A \(12\text{-kg}\) box slides up a \(26.56^{\circ}\) frictionless ramp

A \(12\text{-kg}\) box slides up a \(26.56^{\circ}\) frictionless ramp at constant speed as shown in the animation (position is given in meters and time is given in seconds). Note that the hand does work on the box. Restart.

1. What is the work done on the box by the external force (hand) during the animation?
2. What is the change in gravitational potential energy of the box during the animation?
3. What is the change in kinetic energy of the box during the animation?

Exercise \(\PageIndex{3}\): A \(12\text{-kg}\) box slides down a \(26.56^{\circ}\) frictionless ramp

A \(12\text{-kg}\) box slides down a \(26.56^{\circ}\) frictionless ramp at constant speed as shown in the animation (position is given in meters and time is given in seconds). Note that the hand does work on the box. Restart.

1. What is the work done on the box by the external force (hand) during the animation?
2. What is the change in gravitational potential energy of the box during the animation?
3. What is the change in kinetic energy of the box during the animation?

Exercise \(\PageIndex{4}\): A \(12\text{-kg}\) box slides down a rough ramp

A \(12\text{-kg}\) box slides down a rough \(26.56^{\circ}\) ramp at constant speed (it is already traveling at this constant speed at \(t = 0\text{ s}\) and continues to do so even after the animation ends) as shown in the animation (position is given in meters and time is given in seconds). Note that friction does work on the box. Restart.

1. What is the work done by friction (done on the box and the ramp) during the animation?
2. What is the change in gravitational potential energy of the box during the animation?
3. What is the change in kinetic energy of the box during the animation?

Exercise \(\PageIndex{5}\): A mass is lifted by a string

A \(10\text{-kg}\) mass is attached via a massless string over a massless pulley to a hand (position is given in meters and time is given in seconds). The masses in each animation are identical. Restart.

1. Rank the animations according to the change in gravitational potential energy of the mass, from greatest to least.
2. Rank the animations according to the work done on the mass by the tension in the string, from greatest to least.
3. Rank the animations according to the change in kinetic energy of the mass, from greatest to least.

Indicate ties by placing the animation numbers in () please. For example, a suitable response could be: \(1,\: 2,\: (3,\: 4),\: 5,\: 6\).

4. Calculate the change in gravitational potential energy of the mass during each of the animations.
5. Calculate the work done on the mass by the tension in the string during each of the animations.
6. Calculate the change in kinetic energy of the mass during each of the animations.

Exercise \(\PageIndex{6}\): A modified Atwood's machine

A \(2.5\text{-kg}\) cart on a low-friction track is connected to a string and then to a \(0.5\text{-kg}\) hanging mass as shown in the animation. Neglect any effects of the massless pulley on the motion of the system (position is given in meters and time is given in seconds). Restart.

During the animation,

1. What is the work done on the hanging mass due to the tension in the string?
2. What is the change in gravitational potential energy of the hanging mass?
3. What is the work done on the cart due to the tension in the string?
4. What is the change in gravitational potential energy of the cart?
5. What is the work done on the cart due to the normal force?
6. What is the total work done by the tension on the two-object system?
7. What is the change in potential energy of the two-object system?
8. What is the change in kinetic energy of the two-object system?

Note

Note that the coordinates for each object (the positive x direction) are already chosen for you.

Problem authored by Mario Belloni.
Script authored by Aaron Titus

Exercise \(\PageIndex{7}\): Three balls are thrown off the top of a building

Three balls are thrown off the top of a building, all with the same speed but with different launch angles (position is given in meters and time is given in seconds). Restart. The components of the initial velocities are given.

• The blue ball has an initial velocity of \((6\text{ m/s},\: 8\text{ m/s})\).
• The green ball has an initial velocity of \((10\text{ m/s},\: 0\text{ m/s})\).
• The red ball has an initial velocity of \((8\text{ m/s},\: -6\text{ m/s})\).

1. Rank the three balls according to which one hits the ground first.
2. Rank the three balls according to which one has the greatest speed the instant before impact with the ground.
3. Now calculate the speed of each of the balls the instant before impact with the ground.

Exercise \(\PageIndex{8}\): A ball is dropped on a hard floor

A ball is dropped on a hard floor as shown in the animation (position is given in meters and time is given in seconds). Assume that the acceleration due to gravity is \(9.8\text{ m/s}^{2}\). Restart.

1. What is the speed of the ball the instant before it hits the ground?
2. How much energy (in % of original energy) is lost in the collision with the floor?
3. What is the coefficient of restitution for the ball?

The coefficient of restitution, for the collision where one object does not move, is the ratio \(|v_{f} | / |v_{i}|\).

Exercise \(\PageIndex{9}\): An explosive collision

Two carts are in close proximity. A massless spring is attached to the end of the red cart and is compressed. The massless spring is released such that the two carts are "pushed" apart as shown in the animation (position is given in meters and time is given in seconds). The mass of the green cart is \(1.5\text{ kg}\). Consider a system made up of the two carts and the massless spring. Restart.

1. What is the velocity of the center of mass of the carts after the massless spring is released (assume that since the spring is massless it cannot have a kinetic energy)?
2. What is the mass of the red cart?
3. What is the change in kinetic energy of the system due to the release of the spring?
4. What was the change in potential energy of the spring?

Exercise \(\PageIndex{10}\): A spring gun

A spring gun is loaded with a \(500\)-gram projectile (position is given in centimeters and time is given in seconds). The spring is massless and therefore has no kinetic energy. Restart.

1. How much potential energy is converted to kinetic energy in the spring gun?
2. How much potential energy has been converted to kinetic energy when the ball is at the following positions: \(-5\text{ cm}\), \(-4\text{ cm}\), \(-3\text{ cm}\), \(-2\text{ cm}\), \(-1\text{ cm}\), and \(0\text{ cm}\)?
3. Plot the potential energy of the spring as a function of distance.

Exercise \(\PageIndex{11}\): Collision between a mass and a mass connected to a spring

A \(1.0\text{-kg}\) projectile bounces off of an object (\(m = 1\text{ kg}\)) attached to a massless spring as shown (position is given in meters and time is given in seconds). The table entries, \(v_{1}\) and \(v_{2}\), show the velocities of the projectile and the target, respectively. Assume that the collision is elastic. Restart.

1. There are five important time intervals during the animation. What are they? Briefly describe what is happening during these intervals.
2. Draw a graph of energy vs. time for the kinetic energy of the projectile, the kinetic energy of the target, and the potential energy of the spring.

When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

Exercise \(\PageIndex{12}\): Collision between a mass and a mass connected to a spring

A \(0.5\text{-kg}\) projectile bounces off of an object (\(m = 1\text{ kg}\)) attached to a massless spring as shown (position is given in meters and time is given in seconds). The table entries, \(v_{1}\) and \(v_{2}\), show the velocities of the projectile and the target, respectively. Assume that the collision is elastic. Restart.

1. Draw a graph of energy vs. time for the kinetic energy of the projectile, the kinetic energy of the target, and the potential energy of the spring.
2. What percent of the initial energy was transferred to the target-spring system during the collision?

When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.