Torque and Rotational Motion

The resistance of an object to changes in its rotational motion.

The force required to stop a rotating object.

The measure of how fast an object is rotating.

The energy stored in a rotating object.

$I = mr$

$I = mr^2$

$I = \frac{1}{2}mr^2$

$I = 2mr^2$

$I_{tot} = m_1r_1 + m_2r_2 + m_3r_3$

$I_{tot} = (m_1 + m_2 + m_3)(r_1 + r_2 + r_3)^2$

$I_{tot} = m_1r_1^2 + m_2r_2^2 + m_3r_3^2$

$I_{tot} = (m_1 + m_2 + m_3)(r_1^2 + r_2^2 + r_3^2)$

$I = \int r dm$

$I = \int r^2 dm$

$I = \int r^3 dm$

$I = \int m dr$

An axis through its center of mass.

An axis through one of its edges.

An axis parallel to the axis through its center of mass.

An axis perpendicular to the axis through its center of mass.

$I' = I_{cm} - Md^2$

$I' = I_{cm} + Md^2$

$I' = I_{cm} + \frac{1}{2}Md^2$

$I' = I_{cm} - \frac{1}{2}Md^2$

Always use direct integration.

Always use the parallel axis theorem.

Choose between direct integration or the parallel axis theorem based on the problem's geometry.

Use the perpendicular axis theorem.

The solid sphere

The hollow sphere

They have the same rotational inertia

It depends on the material

Apply the parallel axis theorem directly.

Use the formula $I = \frac{1}{3}ML^2$.

Use the formula $I = \frac{1}{12}ML^2$.

Perform integration to account for the varying density.

The resistance of an object to changes in its linear motion.

The measure of how much torque is applied to an object.

The resistance of an object to changes in its rotational motion.

The gravitational force acting on a rotating object.