Which expression gives the rotational kinetic energy of a rigid body with moment of inertia I rotating at angular speed omega?
(1/2) I omega^2
I omega
(1/2) I omega
I omega^2
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WhyRotational kinetic energy is K = (1/2) I omega^2, the rotational analog of (1/2) m v^2.
2Multiple choice · Easy
The angular momentum of a rigid body about a fixed axis is given by which of the following?
L = I omega
L = I omega^2
L = (1/2) I omega
L = I / omega
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WhyFor a rigid body rotating about a fixed axis, L = I omega, analogous to linear momentum p = m v.
3Multiple choice · Easy
A wheel has moment of inertia 2 kg m^2 and spins at 3 rad/s. What is its angular momentum?
6 kg m^2/s
3 kg m^2/s
1.5 kg m^2/s
9 kg m^2/s
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WhyL = I omega = (2)(3) = 6 kg m^2/s.
4Multiple choice · Easy
A disk with moment of inertia 4 kg m^2 rotates at 2 rad/s. What is its rotational kinetic energy?
8 J
4 J
16 J
2 J
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WhyK = (1/2) I omega^2 = (1/2)(4)(2^2) = (1/2)(4)(4) = 8 J.
5Fill in the blank · Easy
For a system with no net external torque, the total angular momentum is (conserved or not conserved).
Answer:
conserved
WhyWith zero net external torque, angular momentum is conserved.
6Fill in the blank · Easy
An object rolls without slipping when the contact point with the surface has velocity relative to the surface.
Answer:
zero / no
WhyRolling without slipping means the instantaneous velocity of the contact point is zero relative to the surface.
7Multiple choice · Medium
A merry-go-round (I = 100 kg m^2) spins at 1.0 rad/s. A child adds 50 kg m^2 of moment of inertia by stepping on. If no external torque acts, what is the new angular speed?
0.67 rad/s
1.5 rad/s
1.0 rad/s
0.50 rad/s
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WhyConservation of L: I1 omega1 = I2 omega2, so (100)(1.0) = (150) omega2, giving omega2 = 0.67 rad/s.
8Multiple choice · Medium
A constant torque of 5 N m acts on a wheel that turns through 4 rad. How much work does the torque do?
20 J
1.25 J
9 J
40 J
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WhyRotational work W = tau * theta = (5)(4) = 20 J.
9Multiple choice · Medium
A motor delivers a torque of 10 N m to a shaft rotating at 6 rad/s. What is the power output?
60 W
16 W
1.67 W
600 W
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WhyRotational power P = tau * omega = (10)(6) = 60 W.
10Multiple choice · Medium
A solid sphere and a hollow sphere of equal mass and radius are released from rest at the same height and roll without slipping down an incline. Which reaches the bottom first?
The solid sphere
The hollow sphere
They arrive at the same time
It depends on the incline angle
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WhyThe solid sphere has a smaller moment of inertia (2/5 m r^2 vs 2/3 m r^2), so more of its energy goes to translation, giving it greater linear speed and a faster descent.
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Key terms in Energy and momentum of rotating systems
Rotational kinetic energy: The energy an object possesses due to its rotation, equal to one-half its moment of inertia times the square of its angular speed.
Moment of inertia: A measure of an object's resistance to changes in its rotation, depending on its mass and how that mass is distributed about the axis (units kg m^2).
Angular speed: The rate at which an object rotates through angle, measured in radians per second.
Angular velocity: A vector describing the rate of change of angular position, with direction given by the right-hand rule along the rotation axis.
Angular acceleration: The rate of change of angular velocity with respect to time, measured in radians per second squared.
Torque: The rotational influence of a force, equal to the force times its lever arm, that tends to cause angular acceleration (units N m).
Lever arm: The perpendicular distance from the axis of rotation to the line of action of an applied force.
Rotational work: The energy transferred when a torque acts through an angular displacement, equal to torque times the angle turned (units J).