Rolling - AP Physics C: Mechanics
Card 1 of 30
State the equation for the translational kinetic energy of a rolling object.
State the equation for the translational kinetic energy of a rolling object.
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$KE_{trans} = \frac{1}{2} m v^2$. Standard kinetic energy formula using linear velocity.
$KE_{trans} = \frac{1}{2} m v^2$. Standard kinetic energy formula using linear velocity.
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Calculate the final speed of a rolling object down an incline.
Calculate the final speed of a rolling object down an incline.
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$v = \sqrt{\frac{2gh}{1 + \frac{I}{mr^2}}}$$. Energy conservation with rotational inertia term.
$v = \sqrt{\frac{2gh}{1 + \frac{I}{mr^2}}}$$. Energy conservation with rotational inertia term.
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Determine the ratio of rotational to translational kinetic energy for a solid sphere.
Determine the ratio of rotational to translational kinetic energy for a solid sphere.
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$\frac{KE_{rot}}{KE_{trans}} = \frac{2}{5}$. For solid sphere, $I = \frac{2}{5}mr^2$ gives this ratio.
$\frac{KE_{rot}}{KE_{trans}} = \frac{2}{5}$. For solid sphere, $I = \frac{2}{5}mr^2$ gives this ratio.
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Calculate the acceleration of a disk rolling down an incline.
Calculate the acceleration of a disk rolling down an incline.
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$a = \frac{2}{3} g \text{sin}\theta$. For disk, $I = \frac{1}{2}mr^2$ gives this acceleration.
$a = \frac{2}{3} g \text{sin}\theta$. For disk, $I = \frac{1}{2}mr^2$ gives this acceleration.
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What is the condition for rolling motion in terms of angular displacement?
What is the condition for rolling motion in terms of angular displacement?
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$s = r \theta$. Linear displacement equals radius times angular displacement.
$s = r \theta$. Linear displacement equals radius times angular displacement.
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Calculate the final speed of a rolling object down an incline.
Calculate the final speed of a rolling object down an incline.
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$v = \sqrt(\frac{2gh}{1 + \frac{I}{mr^2}})$. Energy conservation with rotational inertia term.
$v = \sqrt(\frac{2gh}{1 + \frac{I}{mr^2}})$. Energy conservation with rotational inertia term.
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Determine the linear acceleration of a rolling object down an incline.
Determine the linear acceleration of a rolling object down an incline.
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$a = \frac{g \text{sin}\theta}{1 + \frac{I}{mr^2}}$. Derived from applying Newton's laws to rolling motion.
$a = \frac{g \text{sin}\theta}{1 + \frac{I}{mr^2}}$. Derived from applying Newton's laws to rolling motion.
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What is the formula for the moment of inertia of a disk?
What is the formula for the moment of inertia of a disk?
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$I = \frac{1}{2} m r^2$. For uniform disk, mass distributed over area.
$I = \frac{1}{2} m r^2$. For uniform disk, mass distributed over area.
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State the formula for the moment of inertia of a solid cylinder (disk) about its central axis.
State the formula for the moment of inertia of a solid cylinder (disk) about its central axis.
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$I_{\mathrm{cm}}=\frac{1}{2}MR^2$. Mass is distributed uniformly throughout the solid cylinder.
$I_{\mathrm{cm}}=\frac{1}{2}MR^2$. Mass is distributed uniformly throughout the solid cylinder.
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State the formula for the moment of inertia of a solid sphere about a diameter.
State the formula for the moment of inertia of a solid sphere about a diameter.
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$I_{\mathrm{cm}}=\frac{2}{5}MR^2$. Mass distribution in 3D gives factor $\frac{2}{5}$ for solid sphere.
$I_{\mathrm{cm}}=\frac{2}{5}MR^2$. Mass distribution in 3D gives factor $\frac{2}{5}$ for solid sphere.
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State the formula for the moment of inertia of a thin spherical shell about a diameter.
State the formula for the moment of inertia of a thin spherical shell about a diameter.
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$I_{\mathrm{cm}}=\frac{2}{3}MR^2$. All mass is at distance $R$ from center, integrated over sphere surface.
$I_{\mathrm{cm}}=\frac{2}{3}MR^2$. All mass is at distance $R$ from center, integrated over sphere surface.
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What is $I$ for a rolling body about the instantaneous contact point (use radius $R$)?
What is $I$ for a rolling body about the instantaneous contact point (use radius $R$)?
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$I_{\mathrm{contact}}=I_{\mathrm{cm}}+MR^2$. Apply parallel-axis theorem with $d=R$.
$I_{\mathrm{contact}}=I_{\mathrm{cm}}+MR^2$. Apply parallel-axis theorem with $d=R$.
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What is the center-of-mass acceleration down an incline for rolling without slipping, using $I_{\mathrm{cm}}$?
What is the center-of-mass acceleration down an incline for rolling without slipping, using $I_{\mathrm{cm}}$?
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$a=\frac{g\sin\theta}{1+\frac{I_{\mathrm{cm}}}{MR^2}}$. Derived from $F=Ma$ and $\tau=I\alpha$ with rolling constraint.
$a=\frac{g\sin\theta}{1+\frac{I_{\mathrm{cm}}}{MR^2}}$. Derived from $F=Ma$ and $\tau=I\alpha$ with rolling constraint.
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What is $a$ down an incline for a solid sphere rolling without slipping?
What is $a$ down an incline for a solid sphere rolling without slipping?
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$a=\frac{5}{7}g\sin\theta$. Substitute $I_{\mathrm{cm}}=\frac{2}{5}MR^2$ into general formula.
$a=\frac{5}{7}g\sin\theta$. Substitute $I_{\mathrm{cm}}=\frac{2}{5}MR^2$ into general formula.
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What is the formula for the moment of inertia of a solid sphere?
What is the formula for the moment of inertia of a solid sphere?
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$I = \frac{2}{5} m r^2$. For a solid sphere, mass distributed throughout volume.
$I = \frac{2}{5} m r^2$. For a solid sphere, mass distributed throughout volume.
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What is the equation for angular displacement in rolling?
What is the equation for angular displacement in rolling?
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$\theta = \frac{s}{r}$. Arc length equals radius times angle in radians.
$\theta = \frac{s}{r}$. Arc length equals radius times angle in radians.
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Determine the linear acceleration of a rolling object down an incline.
Determine the linear acceleration of a rolling object down an incline.
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$a = \frac{g \sin \theta}{1 + \frac{I}{mr^2}}$. Derived from applying Newton's laws to rolling motion.
$a = \frac{g \sin \theta}{1 + \frac{I}{mr^2}}$. Derived from applying Newton's laws to rolling motion.
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Identify the formula for the frictional force in rolling without slipping.
Identify the formula for the frictional force in rolling without slipping.
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$f = \frac{I a}{r^2}$. Friction provides torque for rolling without slipping.
$f = \frac{I a}{r^2}$. Friction provides torque for rolling without slipping.
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What is the formula for the moment of inertia of a disk?
What is the formula for the moment of inertia of a disk?
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$I = \frac{1}{2} m r^2$. For uniform disk, mass distributed over area.
$I = \frac{1}{2} m r^2$. For uniform disk, mass distributed over area.
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What is the formula for the moment of inertia of a solid sphere?
What is the formula for the moment of inertia of a solid sphere?
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$I = \frac{2}{5} m r^2$. For a solid sphere, mass distributed throughout volume.
$I = \frac{2}{5} m r^2$. For a solid sphere, mass distributed throughout volume.
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Calculate the mechanical energy of a rolling object.
Calculate the mechanical energy of a rolling object.
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$E_{mech} = KE_{trans} + KE_{rot} + PE$. Conservation of energy includes all energy forms.
$E_{mech} = KE_{trans} + KE_{rot} + PE$. Conservation of energy includes all energy forms.
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Find the total kinetic energy of a rolling object.
Find the total kinetic energy of a rolling object.
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$KE_{total} = KE_{trans} + KE_{rot}$. Sum of translational and rotational kinetic energies.
$KE_{total} = KE_{trans} + KE_{rot}$. Sum of translational and rotational kinetic energies.
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State the equation for the translational kinetic energy of a rolling object.
State the equation for the translational kinetic energy of a rolling object.
Tap to reveal answer
$KE_{trans} = \frac{1}{2} m v^2$. Standard kinetic energy formula using linear velocity.
$KE_{trans} = \frac{1}{2} m v^2$. Standard kinetic energy formula using linear velocity.
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Find the total kinetic energy of a rolling object.
Find the total kinetic energy of a rolling object.
Tap to reveal answer
$KE_{total} = KE_{trans} + KE_{rot}$. Sum of translational and rotational kinetic energies.
$KE_{total} = KE_{trans} + KE_{rot}$. Sum of translational and rotational kinetic energies.
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What is the equation for angular displacement in rolling?
What is the equation for angular displacement in rolling?
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$\theta = \frac{s}{r}$. Arc length equals radius times angle in radians.
$\theta = \frac{s}{r}$. Arc length equals radius times angle in radians.
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Identify the formula for the frictional force in rolling without slipping.
Identify the formula for the frictional force in rolling without slipping.
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$f = \frac{I a}{r^2}$. Friction provides torque for rolling without slipping.
$f = \frac{I a}{r^2}$. Friction provides torque for rolling without slipping.
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Calculate the acceleration of a disk rolling down an incline.
Calculate the acceleration of a disk rolling down an incline.
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$a = \frac{2}{3} g \sin \theta$. For disk, $I = \frac{1}{2}mr^2$ gives this acceleration.
$a = \frac{2}{3} g \sin \theta$. For disk, $I = \frac{1}{2}mr^2$ gives this acceleration.
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What is the condition for rolling motion in terms of angular displacement?
What is the condition for rolling motion in terms of angular displacement?
Tap to reveal answer
$s = r \theta$. Linear displacement equals radius times angular displacement.
$s = r \theta$. Linear displacement equals radius times angular displacement.
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Calculate the mechanical energy of a rolling object.
Calculate the mechanical energy of a rolling object.
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$E_{mech} = KE_{trans} + KE_{rot} + PE$. Conservation of energy includes all energy forms.
$E_{mech} = KE_{trans} + KE_{rot} + PE$. Conservation of energy includes all energy forms.
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What is the formula for the moment of inertia of a hoop?
What is the formula for the moment of inertia of a hoop?
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$I = m r^2$. All mass concentrated at rim, distance $r$ from center.
$I = m r^2$. All mass concentrated at rim, distance $r$ from center.
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