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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedExam QuestionsUnit 6: Energy & Momentum of Rotating SystemsRotational Kinetic Energy, Torque & Work

Rotational Kinetic Energy, Torque & Work (College Board AP® Physics 1: Algebra-Based): Exam Questions

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1
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The rotational kinetic energy of a wheel rotating at 20 space rev divided by straight s is 1600 straight pi squared space straight J.

Which of the following is most nearly the rotational inertia of the wheel?

  • 1 space kg times straight m squared

  • 2 space kg times straight m squared

  • 4 space kg times straight m squared

  • 4 straight pi squared space kg times straight m squared

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2
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A solid disc and a solid sphere of massM and radius R spin about their central axes with the same angular velocity. The rotational inertia of a solid disc is 1 half M R squared , and rotational inertia of a solid sphere is 2 over 5 M R squared.

Which of the following statements is true about the rotational kinetic energies of the disc and sphere?

  • The disc has a greater rotational kinetic energy than the sphere.

  • The sphere has a greater rotational kinetic energy than the disc.

  • The disc and the sphere have the same rotational kinetic energy.

  • There is not enough information to compare the rotational kinetic energies of the disc and sphere.

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3
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A cylindrical pulley with a rotational inertia of 2.0 space kg times straight m squared rotates with an angular acceleration of 5.0 space rad divided by straight s squared when a constant torque is applied to its rim.

Which of the following is most nearly the work done by the torque on the pulley after it rotates through an angle of 6 straight pi space rad?

  • 30 space straight J

  • 60 space straight J

  • 100 space straight J

  • 200 space straight J

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4
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A solid disc rotates at a constant angular speed about an axis through its center. When an identical disc is placed on top of the first disc, both discs rotate together at half the first disc's initial angular speed.

How do the rotational inertia and rotational kinetic energy of the system change after the second disc is placed on top of the first?

Rotational Inertia

Rotational Kinetic Energy

A

Increases

Decreases

B

Decreases

Increases

C

Increases

Increases

D

Decreases

Decreases

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    5
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    Diagram of a block on an inclined plane connected to a pulley.

    A block is initially held at rest on a frictionless inclined plane and is wound around the rim of a pulley by a light cord, as shown in the figure. When the block is released, it begins to slide down the inclined plane.

    Which of the following graphs could represent the net torque tau on the pulley as a function of angular position theta?

    • Graph showing torque (τ) in Nm on the vertical axis and angular position (θ) in radians on the horizontal axis, with a horizontal line.
    • Graph showing torque (τ) in Nm on the vertical axis and angular position (θ) in radians on the horizontal axis, with a line showing a positive linear relationship.
    • Graph showing torque (τ) in Nm on the vertical axis and angular position (θ) in radians on the horizontal axis, with a line showing a negative linear relationship.
    • Graph showing torque (τ) in Nm on the vertical axis and angular position (θ) in radians on the horizontal axis, with a curve which increases initially and gradually plateaus.

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    1
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    A fairground wheel of radius 15 space straight m and mass 4 cross times 10 to the power of 5 space kg is mounted on the fixed, frictionless axle of a motor. The wheel can be modeled as a hoop of rotational inertia M R squared. The motor increases the angular speed of the wheel from rest to a maximum value of 0.35 space rad divided by straight s in 25.0 space seconds.

    Which of the following is most nearly the average power output of the motor?

    • 2.2 cross times 10 to the power of 5 space straight W

    • 4.4 cross times 10 to the power of 5 space straight W

    • 6.3 cross times 10 to the power of 5 space straight W

    • 1.3 cross times 10 to the power of 6 space straight W

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    2
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    When a force of 60 space straight N is applied normally to the end of the wrench to tighten a bolt, it turns through an angle of 45 degree.

    Which of the following is most nearly the work done on the bolt if the distance from the end of the wrench to the center of the bolt is 22 space cm?

    • 5 space straight J

    • 9 space straight J

    • 10 space straight J

    • 20 space straight J

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    3
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    Line graph showing torque (τ, in Newton-meters) decreasing linearly from 14 N·m at 0 degrees to 0 N·m at 35 degrees, with grid lines for precision.

    The graph shows the net torque applied to a rotating body as a function of angular position.

    Which of the following is most nearly the change in kinetic energy of the body after it has rotated through an angle of 35°?

    • 4.3 space straight J

    • 8.6 space straight J

    • 245 space straight J

    • 490 space straight J

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    4
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    A system with two wheels connected by a cord passing over their rims. The smaller wheel on the left is labeled "R, I" and rotates at ω. The larger wheel on the right is labeled "2R, 16I".

    Two wheels mounted on separate frictionless axles are joined by a loop of cord, as shown in the figure. The smaller wheel of radius R has a rotational inertia of I, and the larger wheel of radius 2 R has a rotational inertia of 16 I. The axis of the smaller wheel is attached to a motor that rotates it at angular speed omega, which in turn causes the larger wheel to rotate.

    Assuming the loop does not slip on the wheels, which of the following expressions represents the total kinetic energy of the system?

    • 1 half I omega squared

    • 2 I omega squared

    • 5 over 2 I omega squared

    • 3 I omega squared

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    5
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    A cake of mass 8 m and radius R rotates on a decorating turntable of negligible mass at an angular velocity of omega. The cake can be modeled as a solid disc of rotational inertia 1 half M R squared and rotational kinetic energy K. When four balls of frosting, each of mass m, are placed onto the outer edge of the cake, the new angular velocity of the system is omega over 2.

    Which of the following correctly describes the new rotational kinetic energy of the system and the work done on the cake by the balls of frosting?

    New Rotational Kinetic Energy

    Work Done

    A

    K over 2

    Positive

    B

    K over 2

    Negative

    C

    2 K

    Positive

    D

    2 K

    Negative

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      1
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      A rod of length L initially stood vertically which then rotates as it falls to the ground into a horizontal position.

      A uniform rod of length L and mass M is initially held vertically with one end resting on the horizontal ground, as shown in the figure. When the rod is released from rest, the upper end falls to the ground, and the lower end does not slip. The rotational inertia of the rod about one of its ends is 1 third M L squared.

      Which of the following expressions represents the linear velocity of the upper end of the rod immediately before it strikes the floor?

      • square root of fraction numerator 2 g over denominator L end fraction end root

      • square root of fraction numerator 3 g over denominator L end fraction end root

      • square root of 2 g L end root

      • square root of 3 g L end root

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      2
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      Diagram of a pulley system with a mass M on a wheel of radius R, connected by a string to a mass m at height h above the ground.

      A mass m is attached to a light cable which is wrapped around the rim of a cylindrical pulley of radius R, as shown in the figure. The cylinder rotates about an axle at its center with rotational inertia 1 half M R squared. When the cylinder rotates, a constant frictional torque acts on the system. Mass m is initially held at rest at a height h above the ground. After the mass is released, it strikes the ground with speed v.

      Which of the following expressions represents the work done by the system against friction as the mass m falls through height h?

      • m g h space minus space 1 half m v squared

      • m g h space minus space 1 half open parentheses m space plus space M close parentheses v squared

      • m g h space minus space open parentheses 1 half m space plus space 1 fourth M close parentheses v squared

      • m g h space minus space open parentheses 1 half m space plus space 1 fourth M R squared close parentheses v squared

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      3
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      A horizontal rod of length L and mass M, pivoted on the left, with a string attached to the right, extending vertically to a fixed point.

      A uniform rod of mass M and length L is free to rotate about a frictionless pivot at its left end and held horizontally by a string of negligible mass at its right end, as shown in the figure. The string is cut and releases the rod, causing it to rotate about the axis and swing down from the horizontal position.

      Which of the following graphs could represent the net torque tau on the rod as a function of the angular displacement theta from the initial horizontal position?

      • Graph showing torque (τ) in Nm on the vertical axis and angular displacement (θ) in radians on the horizontal axis, with a curve starting at zero, peaking at a positive torque at π/2 rad, and ending at zero at π rad.
      • Graph showing torque (τ) in Nm on the vertical axis and angular displacement (θ) in radians on the horizontal axis, with a curve starting at a positive non-zero torque, passing through zero at π/2 rad, and ending at a negative non-zero torque at π rad.
      • Graph showing torque (τ) in Nm on the vertical axis and angular displacement (θ) in radians on the horizontal axis, with a curve starting at zero, peaking at a negative torque at π/2 rad, and ending at zero at π rad.
      • Graph showing torque (τ) in Nm on the vertical axis and angular displacement (θ) in radians on the horizontal axis, with a curve starting at a negative non-zero torque, passing through zero at π/2 rad, and ending at a positive non-zero torque at π rad.

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      4
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      Energy bar chart with three labels Kr, Kt, and Ug. Ug bar has the maximum possible value, Kr and Kt are zero.

      Two identical yo-yos are initially held at rest at the same height above the ground. The energy bar chart in the figure represents the rotational kinetic energy K subscript r and translational kinetic energy K subscript t of the yo-yos, and the gravitational potential energy U subscript g of the yo-yo-Earth systems just before their release. The yo-yos are then released at the same instant, one is dropped without its string attached to anything, and the other one falls with its string tied to someone's hand, causing it to unwind and rotate.

      Both yo-yos can be modeled as two solid discs, each of mass M, radius R, and rotational inertia 1 half M R squared, which are separated by a thin string wound about an axle of radius R over 2.

      Which of the following graphs could represent the energy bar charts for the yo-yos just before they hit the ground?

      • Energy bar chart for the dropped yo-yo: height of Kr bar is 1 unit, height of Kt bar is 4 units, Ug bar is zero. Energy bar chart for the unwinding yo-yo: height of Kr bar is 5 units (maximum), Kt and Ug bars are zero.
      • Energy bar chart for the dropped yo-yo: height of Kr bar is 4 units, height of Kt bar is 1 unit, Ug bar is zero. Energy bar chart for the unwinding yo-yo: height of Kr bar is 5 units (maximum), Kt and Ug bars are zero.
      • Energy bar chart for the dropped yo-yo: height of Kt bar is 5 units (maximum), Kr and Ug bars are zero. Energy bar chart for the unwinding yo-yo: height of Kr bar is 1 unit, height of Kt bar is 4 units, Ug bar is zero.
      • Energy bar chart for the dropped yo-yo: height of Kt bar is 5 units (maximum), Kr and Ug bars are zero. Energy bar chart for the unwinding yo-yo: height of Kr bar is 4 units, height of Kt bar is 1 unit, Ug bar is zero.

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      5
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      Diagram of a yo-yo of radius R with a string wrapped around its axle of radius r. The linear velocity v points downwards from the center of the yo-yo.

      A yo-yo can be modeled as two solid discs, each of mass M, radius R, and rotational inertia 1 half M R squared, separated by a thin string wound about an axle of radius r. Initially, a yo-yo and its wound string are held at rest. When the yo-yo is released, it falls vertically and the string unwinds from the axle causing it to spin about its center of mass with linear speed v, as shown in the figure. The total length of the unwound taut string is L.

      Which of the following expressions represents the linear speed of the center of mass of the yo-yo when the string is fully unwound?

      • square root of 2 g L end root

      • square root of fraction numerator 4 g L over denominator 3 end fraction end root

      • square root of fraction numerator 2 g L over denominator 1 space plus thin space R squared over r squared end fraction end root

      • square root of fraction numerator 2 g L over denominator 1 space plus thin space fraction numerator R squared over denominator 2 r squared end fraction end fraction end root

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