Rotational Kinetic Energy - AP Physics C: Mechanics
Card 1 of 30
Identify the rotational equivalent of force in linear motion.
Identify the rotational equivalent of force in linear motion.
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Torque. Both cause acceleration in their respective motions.
Torque. Both cause acceleration in their respective motions.
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What happens to rotational kinetic energy if the moment of inertia doubles?
What happens to rotational kinetic energy if the moment of inertia doubles?
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It doubles if angular velocity is constant. KE is proportional to moment of inertia.
It doubles if angular velocity is constant. KE is proportional to moment of inertia.
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Determine the kinetic energy for $I = 0.1 \text{ kg m}^2$ and $\theta' = 10 \text{ rad/s}$.
Determine the kinetic energy for $I = 0.1 \text{ kg m}^2$ and $\theta' = 10 \text{ rad/s}$.
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$K = 5 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.1)(100) = 5$ J.
$K = 5 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.1)(100) = 5$ J.
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What is the effect of increasing radius on the moment of inertia?
What is the effect of increasing radius on the moment of inertia?
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Moment of inertia increases. Larger radius increases moment of inertia.
Moment of inertia increases. Larger radius increases moment of inertia.
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What is the change in kinetic energy when angular velocity is halved?
What is the change in kinetic energy when angular velocity is halved?
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It decreases by a factor of four. KE proportional to $\theta'^2$, so halving gives 1/4.
It decreases by a factor of four. KE proportional to $\theta'^2$, so halving gives 1/4.
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What is the moment of inertia of a rod of length $L$ and mass $m$ about an end?
What is the moment of inertia of a rod of length $L$ and mass $m$ about an end?
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$I = \frac{1}{3} m L^2$. Standard formula for thin rod about end.
$I = \frac{1}{3} m L^2$. Standard formula for thin rod about end.
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What symbol represents angular velocity in rotational motion?
What symbol represents angular velocity in rotational motion?
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$\theta'$. Standard notation for angular velocity in mechanics.
$\theta'$. Standard notation for angular velocity in mechanics.
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What is the formula for the total kinetic energy of a rolling object?
What is the formula for the total kinetic energy of a rolling object?
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$K_{\text{total}} = K_{\text{trans}} + K_{\text{rot}}$. Rolling objects have both translational and rotational energy.
$K_{\text{total}} = K_{\text{trans}} + K_{\text{rot}}$. Rolling objects have both translational and rotational energy.
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What is the moment of inertia for a thin cylindrical shell about its axis?
What is the moment of inertia for a thin cylindrical shell about its axis?
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$I = m r^2$. All mass concentrated at radius $r$ from axis.
$I = m r^2$. All mass concentrated at radius $r$ from axis.
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Determine the kinetic energy for $I = 0.1 \text{ kg m}^2$ and $\theta' = 10 \text{ rad/s}$.
Determine the kinetic energy for $I = 0.1 \text{ kg m}^2$ and $\theta' = 10 \text{ rad/s}$.
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$K = 5 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.1)(100) = 5$ J.
$K = 5 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.1)(100) = 5$ J.
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What is the moment of inertia of a rod of length $L$ and mass $m$ about an end?
What is the moment of inertia of a rod of length $L$ and mass $m$ about an end?
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$I = \frac{1}{3} m L^2$. Standard formula for thin rod about end.
$I = \frac{1}{3} m L^2$. Standard formula for thin rod about end.
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Identify the direction of angular momentum for a rotating body.
Identify the direction of angular momentum for a rotating body.
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Perpendicular to the plane of rotation. Angular momentum vector follows right-hand rule.
Perpendicular to the plane of rotation. Angular momentum vector follows right-hand rule.
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Determine the moment of inertia for a point mass $m$ at radius $r$.
Determine the moment of inertia for a point mass $m$ at radius $r$.
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$I = m r^2$. Definition of moment of inertia for point mass.
$I = m r^2$. Definition of moment of inertia for point mass.
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Calculate the rotational kinetic energy for $I = 0.5 \text{ kg m}^2$ and $\theta' = 4 \text{ rad/s}$.
Calculate the rotational kinetic energy for $I = 0.5 \text{ kg m}^2$ and $\theta' = 4 \text{ rad/s}$.
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$K = 4 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.5)(16) = 4$ J.
$K = 4 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.5)(16) = 4$ J.
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What is the moment of inertia of a rod of length $L$ and mass $m$ about its center?
What is the moment of inertia of a rod of length $L$ and mass $m$ about its center?
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$I = \frac{1}{12} m L^2$. Standard formula for thin rod about center.
$I = \frac{1}{12} m L^2$. Standard formula for thin rod about center.
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Calculate the kinetic energy of a rotating wheel with $I = 2 \text{ kg m}^2$ and $\theta' = 3 \text{ rad/s}$.
Calculate the kinetic energy of a rotating wheel with $I = 2 \text{ kg m}^2$ and $\theta' = 3 \text{ rad/s}$.
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$K = 9 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(2)(9) = 9$ J.
$K = 9 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(2)(9) = 9$ J.
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Identify the direction of angular momentum for a rotating body.
Identify the direction of angular momentum for a rotating body.
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Perpendicular to the plane of rotation. Angular momentum vector follows right-hand rule.
Perpendicular to the plane of rotation. Angular momentum vector follows right-hand rule.
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How is energy conserved in a system with both translational and rotational motion?
How is energy conserved in a system with both translational and rotational motion?
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Total kinetic energy is conserved. Sum of translational and rotational energies conserved.
Total kinetic energy is conserved. Sum of translational and rotational energies conserved.
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What is the rotational kinetic energy of a solid disk with $I = 1 \text{ kg m}^2$ and $\theta' = 2 \text{ rad/s}$?
What is the rotational kinetic energy of a solid disk with $I = 1 \text{ kg m}^2$ and $\theta' = 2 \text{ rad/s}$?
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$K = 2 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(1)(4) = 2$ J.
$K = 2 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(1)(4) = 2$ J.
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What is the formula for the total kinetic energy of a rolling object?
What is the formula for the total kinetic energy of a rolling object?
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$K_{\text{total}} = K_{\text{trans}} + K_{\text{rot}}$. Rolling objects have both translational and rotational energy.
$K_{\text{total}} = K_{\text{trans}} + K_{\text{rot}}$. Rolling objects have both translational and rotational energy.
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Determine the moment of inertia for a point mass $m$ at radius $r$.
Determine the moment of inertia for a point mass $m$ at radius $r$.
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$I = m r^2$. Definition of moment of inertia for point mass.
$I = m r^2$. Definition of moment of inertia for point mass.
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What is the rotational kinetic energy of a disk with $I = 0.2 \text{ kg m}^2$ and $\theta' = 5 \text{ rad/s}$?
What is the rotational kinetic energy of a disk with $I = 0.2 \text{ kg m}^2$ and $\theta' = 5 \text{ rad/s}$?
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$K = 2.5 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.2)(25) = 2.5$ J.
$K = 2.5 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.2)(25) = 2.5$ J.
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Which physical quantity is conserved in an isolated rotational system?
Which physical quantity is conserved in an isolated rotational system?
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Angular momentum. No external torques means angular momentum constant.
Angular momentum. No external torques means angular momentum constant.
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What is the formula for the moment of inertia of a solid cylinder about its central axis?
What is the formula for the moment of inertia of a solid cylinder about its central axis?
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$I = \frac{1}{2} m r^2$. Mass distributed uniformly throughout solid cylinder.
$I = \frac{1}{2} m r^2$. Mass distributed uniformly throughout solid cylinder.
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What is the moment of inertia for a thin cylindrical shell about its axis?
What is the moment of inertia for a thin cylindrical shell about its axis?
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$I = m r^2$. All mass concentrated at radius $r$ from axis.
$I = m r^2$. All mass concentrated at radius $r$ from axis.
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How is the moment of inertia $I$ calculated for a solid sphere rotating about its center?
How is the moment of inertia $I$ calculated for a solid sphere rotating about its center?
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$I = \frac{2}{5} m r^2$. Standard formula for uniform solid sphere about center.
$I = \frac{2}{5} m r^2$. Standard formula for uniform solid sphere about center.
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State the relationship between linear velocity $v$ and angular velocity $\theta'$.
State the relationship between linear velocity $v$ and angular velocity $\theta'$.
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$v = r \theta'$. Linear velocity equals radius times angular velocity.
$v = r \theta'$. Linear velocity equals radius times angular velocity.
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Identify the unit of rotational kinetic energy in the SI system.
Identify the unit of rotational kinetic energy in the SI system.
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Joules (J). Energy has units of joules in SI system.
Joules (J). Energy has units of joules in SI system.
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What does the symbol $I$ denote in the context of rotational motion?
What does the symbol $I$ denote in the context of rotational motion?
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Moment of inertia. Rotational analog of mass, resistance to angular acceleration.
Moment of inertia. Rotational analog of mass, resistance to angular acceleration.
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What symbol represents angular velocity in rotational motion?
What symbol represents angular velocity in rotational motion?
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$\theta'$. Standard notation for angular velocity in mechanics.
$\theta'$. Standard notation for angular velocity in mechanics.
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