Newton's Second Law in Rotational Form - AP Physics 1

Angular acceleration doubles. From $\alpha = \frac{\tau}{I}$, halving $I$ doubles $\alpha$.

Increase lever arm length. Torque equals force times lever arm, so increase either factor.

$\tau = rF\sin\theta$. Torque equals perpendicular force component times lever arm distance.

$\alpha = 6 \frac{rad}{s^2}$. Using $\alpha = \frac{\tau}{I} = \frac{30}{5} = 6$ rad/s².

Net torque causes angular acceleration. Unbalanced torque produces change in rotational motion state.

$I = \frac{1}{2}mr^2$. Standard formula for uniform solid cylinder about central axis.

$I = mr^2$. All mass concentrated at radius $r$ from rotation axis.

Torque ($\tau$) is the rotational equivalent of force. It causes angular acceleration just like force causes linear acceleration.

$\alpha = \frac{\tau}{I}$. Derived from rearranging Newton's second law for rotation.

An object in rotational equilibrium remains at rest or in uniform rotation. Objects maintain rotational state unless acted on by net torque.

$I = \frac{1}{2}mr^2$. Same as solid cylinder - uniform mass distribution about central axis.

Zero. No net torque means no angular acceleration occurs.

Further mass increases moment of inertia. Mass farther from axis contributes more to rotational inertia.

Lever arm length. Perpendicular distance from rotation axis to line of action of force.

The object is in rotational equilibrium. Equal and opposite torques cancel, producing no angular acceleration.

Angular acceleration becomes zero. No torque means no angular acceleration from $\tau = I\alpha$.

Newton meter (Nm). Force times distance units, equivalent to joules for torque.

Torque is maximized. Perpendicular force gives maximum torque since $\sin(90°) = 1$.

$\alpha$. Greek letter alpha represents the rate of change of angular velocity.

Kilogram meter squared (kg·m²). Mass times distance squared units from rotational inertia definition.

$\theta = 90^\circ$. When force is perpendicular to lever arm, $\sin\theta = 1$.

$I = 2(1)^2 + 3(2)^2 = 14 kg·m²$. Sum individual contributions: $I = m_1r_1^2 + m_2r_2^2$.

Angular acceleration doubles. From $\tau = I\alpha$, doubling $\tau$ doubles $\alpha$ when $I$ is constant.

Moment of inertia ($I$). Measures resistance to rotational acceleration, like mass for linear motion.

Angle between force vector and lever arm. Determines the component of force that creates rotation.

Can provide torque to change rotational motion. Friction forces can create torques about rotation axes.

$I = mr^2$. Mass times distance squared for a particle at distance $r$ from axis.

Increases with distance squared. Moment of inertia proportional to $r^2$ in $I = mr^2$.

The object is undergoing angular acceleration. Net torque causes change in angular velocity over time.