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11.1.6.3: Problems

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    Exercise \(\PageIndex{1}\): A hammer hits a nail

    A \(2\text{-kg}\) hammer strikes a \(1.5\)-gram nail at \(t = 1.8\text{ s}\) as shown in the animation (position is given in centimeters and time is given in seconds). Restart.

    1. Determine the work done on the hammer by the nail.
    2. Use your calculation in (a) to determine the average force exerted on the nail by the hammer.

    Exercise \(\PageIndex{2}\): A brick falls on a nail

    A \(1.5\text{-kg}\) brick falls a given height onto a \(15\)-gram spike as shown in the animation (position is given in meters and time is given in seconds). Restart.

    1. Determine the work done on the brick by the nail.
    2. Use your calculation in (a) to determine the average force exerted on the nail by the brick.

    Exercise \(\PageIndex{3}\): A block is pushed

    A woman pushes on a \(2.5\text{-kg}\) block with an unknown force as shown in the animation (position is given in meters and time is given in seconds). At \(t = 2\) seconds she doubles the force applied to the block. Restart.

    1. Determine the total work done on the block and table in the first two seconds of the animation.
    2. Determine the total work done on the block and table in the final two seconds of the animation.

    Exercise \(\PageIndex{4}\): A bowling ball is lifted to a shelf

    A bowling ball is lifted from rest onto a shelf by an external agent (position is given in meters and time is given in seconds). The bowling ball starts at rest and ends up at rest when the animation ends. For each quantity below, rank the animations (numbered \(1\) through \(4\)) from least to greatest. Restart.

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

     

    Quantity Ranking
    Work done on the bowling ball by gravity  
    Work done on the bowling ball by the external agent  
    Total work done on the bowling ball  

    Table \(\PageIndex{1}\)

     

    Exercise \(\PageIndex{5}\): Path dependence of forces and work

    A \(5.0\text{-kg}\) block (called Block \(1\)) is lifted from rest by an external agent, then returned to its original position as shown in the animation (position is given in meters and time is given in seconds). An identical block (called Block \(2\)) is pushed along the surface with a force of \(10\text{ N}\). As with Block \(1\), it is returned to its original position at the end of the animation. Both blocks start and end at rest. Restart.

    1. Determine the work done by gravity on Block \(1\) during the animation.
    2. Determine the work done by gravity on Block \(2\) during the animation.
    3. Determine the work done by the normal force on Block \(2\) during the animation.
    4. Determine the total work done by friction (done on Block \(2\) and the table) during the animation.

    Exercise \(\PageIndex{6}\): A ball in a 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 the mass 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 \(\PageIndex{7}\): A \(12\text{-kg}\) box is pushed at an angle of \(60^{\circ}\) from the vertical

    A \(12\text{-kg}\) box is pushed at constant speed (the box is already moving 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). The hand pushes on the box at an angle of \(60^{\circ}\) from the vertical. Note that there are four forces acting on the box: gravity, the force of the hand, the normal force, and friction. Restart.

    During the animation,

    1. Is the work done on the box by the external force (hand) positive, negative, or zero?
    2. Is the work done on the box by the normal force positive, negative, or zero?
    3. Is the work done on the box by gravity positive, negative, or zero?
    4. Is the work done by friction (done on the box and the table) positive, negative, or zero?
    5. Is the total work done on the box positive, negative, or zero?

    Problem authored by Melissa Dancy.

    Exercise \(\PageIndex{8}\): 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 a constant speed as shown in the animation (position is given in meters and time is given in seconds). Note that both gravity and the hand do 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 work done on the box by gravity during the animation?
    3. What is the total work done on the box during the animation?

    Exercise \(\PageIndex{9}\): 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 both gravity and friction do 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 work done on the box by gravity during the animation?
    3. What is the total work done on the box during the animation?

    Exercise \(\PageIndex{10}\): 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 work done on the mass by gravity, 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 total amount of work done on the mass, from greatest to least.

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

    1. Calculate the work done on the mass by gravity during each of the animations.
    2. Calculate the work done on the mass by the tension in the string during each of the animations.
    3. Calculate the total amount of work done on the mass during each of the animations.

    Note

    For this Problem, the mouse-down for coordinates has been disabled.

    Exercise \(\PageIndex{11}\): A modified Atwood's machine

    A \(2.5\text{-kg}\) cart on a low-friction track is connected to a string and a \(0.5\text{-kg}\) hanging mass as shown in the animation. Neglect any effects of the 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 work done on the hanging mass due to gravity?
    3. What is the work done on the cart due to the tension in the string?
    4. What is the work done on the cart due to gravity?
    5. What is the work done on the cart due to the normal force?
    6. What is the total amount of work done on the two-object system?
    7. What is the final 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 \(\PageIndex{12}\): An oscillating mass on a spring

    A ball on an air track is attached to a compressed spring (at \(x = 0\text{ m}\) the spring is unstretched) as shown in the animation (position is given in meters and time is given in seconds). Which area properly represents the work done by the spring during the animation (assume \(v = 0\text{ m/s}\) at the beginning and end of the animation)? Restart.

    Exercise \(\PageIndex{13}\): A compressed spring is stretched

    A cart sits on a track. A compressible spring is connected to the cart and to a barrier at the end of the track. At \(t = 0\text{ s}\), the spring is compressed \(0.5\text{ m}\) from its unstretched position, and you have to push on the cart to keep it in equilibrium. Then, by applying a varying force, you allow the spring to relax and then cause it to stretch while maintaining equilibrium during the entire process. The spring constant is \(50\text{ N/m}\). The frictional force of the track on the cart is negligible. Treat the cart as a point particle (position is given in meters and time is given in seconds). Restart.

    1. What is the work done by the force of your hand on the cart during the interval between \(t = 0\) and when the spring is fully stretched?
    2. What is the work done by the spring on the cart during this same interval?
    3. What is the total work done on the cart during this interval?
    4. What must the force of your hand on the cart be to keep it in equilibrium when the spring is fully compressed?
    5. What must the force of your hand on the cart be to keep it in equilibrium when the spring is fully stretched?
    6. Why is the work done by your hand on the cart not equal to the product of this force component [calculated in part (e)] and the displacement of the cart?

    Illustration authored by Aaron Titus and placed in the public domain.

    Exercise \(\PageIndex{14}\): An oscillating mass on a spring

    A \(0.50\text{-kg}\) cart resting on an air track oscillates as shown in the animation (position is given in meters and time is given in seconds). What is the spring constant of the spring? Restart.

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

    This page titled 11.1.6.3: Problems is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Wolfgang Christian, Mario Belloni, Anne Cox, Melissa H. Dancy, and Aaron Titus, & Thomas M. Colbert.