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

Calculate torque given $F = 10N$ , $r = 2m$ , and $\theta = 90^\circ$ .

Torque = 20 Nm. Using $\tau = rF\sin\theta = (2)(10)(1) = 20$ Nm.
What happens to torque if the angle $\theta$ changes from $90^\circ$ to $0^\circ$ ?

Torque becomes zero. When $\theta = 0°$ , $\sin\theta = 0$ , so $\tau = 0$ .
What is the effect on torque if force is applied perpendicular to lever arm?

Torque is maximized. Perpendicular force gives maximum torque since $\sin(90°) = 1$ .
What symbol represents angular acceleration?

$\alpha$ . Greek letter alpha represents the rate of change of angular velocity.
How can torque be increased other than increasing force?

Increase lever arm length. Torque equals force times lever arm, so increase either factor.
What is the formula for net torque in terms of multiple forces?

$\tau_{net} = \sum (r_i F_i \sin\theta_i)$ . Vector sum of individual torques from each applied force.
What condition must $\theta$ satisfy for maximum torque?

$\theta = 90^\circ$ . When force is perpendicular to lever arm, $\sin\theta = 1$ .
What is the effect of increasing the lever arm on torque?

Torque increases. Torque is proportional to lever arm length in $\tau = rF\sin\theta$ .
What does the symbol $I$ represent in rotational dynamics?

Moment of inertia. Rotational inertia - resistance to change in rotational motion.
Determine angular acceleration if $\tau = 30 Nm$ and $I = 5 kg·m²$ .

$\alpha = 6 \frac{rad}{s^2}$ . Using $\alpha = \frac{\tau}{I} = \frac{30}{5} = 6$ rad/s².
What is the effect of reducing force to zero on angular acceleration?

Angular acceleration becomes zero. No torque means no angular acceleration from $\tau = I\alpha$ .
What is the unit of torque in the SI system?

Newton meter (Nm). Force times distance units, equivalent to joules for torque.
What is Newton's Second Law in rotational form?

$\tau = I\alpha$ . Rotational analog of $F = ma$ , relating torque, inertia, and angular acceleration.
What is the relationship between net torque and rotational motion?

Net torque causes angular acceleration. Unbalanced torque produces change in rotational motion state.
What is the moment of inertia for a solid cylinder rotating about its central axis?

$I = \frac{1}{2}mr^2$ . Standard formula for uniform solid cylinder about central axis.
What does the angle $\theta$ in the torque formula signify?

Angle between force vector and lever arm. Determines the component of force that creates rotation.
What is the effect of increasing the moment of inertia on angular acceleration?

Angular acceleration decreases. From $\alpha = \frac{\tau}{I}$ , larger $I$ means smaller $\alpha$ .
What is the rotational analog of mass in linear motion?

Moment of inertia ( $I$ ). Measures resistance to rotational acceleration, like mass for linear motion.
If the moment of inertia is halved, what happens to angular acceleration?

Angular acceleration doubles. From $\alpha = \frac{\tau}{I}$ , halving $I$ doubles $\alpha$ .
What happens to angular acceleration if torque is doubled?

Angular acceleration doubles. From $\tau = I\alpha$ , doubling $\tau$ doubles $\alpha$ when $I$ is constant.
What is the moment of inertia for a solid sphere about its diameter?

$I = \frac{2}{5}mr^2$ . Standard moment of inertia formula for uniform solid sphere.
Calculate the moment of inertia for two masses, $m_1 = 2kg$ and $m_2 = 3kg$ , at $r_1 = 1m$ , $r_2 = 2m$ .

$I = 2(1)^2 + 3(2)^2 = 14 kg·m²$ . Sum individual contributions: $I = m_1r_1^2 + m_2r_2^2$ .
What does $r$ represent in the torque formula $\tau = rF\sin\theta$ ?

Lever arm length. Perpendicular distance from rotation axis to line of action of force.
What is the effect of mass distribution on moment of inertia?

Further mass increases moment of inertia. Mass farther from axis contributes more to rotational inertia.
What is the formula for the moment of inertia for a point mass?

$I = mr^2$ . Mass times distance squared for a particle at distance $r$ from axis.
What is the rotational analog of Newton's first law?

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

Torque ( $\tau$ ) is the rotational equivalent of force. It causes angular acceleration just like force causes linear acceleration.
How is angular acceleration calculated from torque and moment of inertia?

$\alpha = \frac{\tau}{I}$ . Derived from rearranging Newton's second law for rotation.
What is the moment of inertia for a thin hoop rotating about its central axis?

$I = mr^2$ . All mass concentrated at radius $r$ from rotation axis.
Identify the relationship between torque and angular acceleration.

Directly proportional. Greater torque produces greater angular acceleration when $I$ is constant.
What is the formula for torque involving force and lever arm?

$\tau = rF\sin\theta$ . Torque equals perpendicular force component times lever arm distance.
Identify the unit of moment of inertia in the SI system.

Kilogram meter squared (kg·m²). Mass times distance squared units from rotational inertia definition.
What is the moment of inertia for a solid disk about an axis through its center?

$I = \frac{1}{2}mr^2$ . Same as solid cylinder - uniform mass distribution about central axis.
What does a non-zero net torque indicate about an object's motion?

The object is undergoing angular acceleration. Net torque causes change in angular velocity over time.
How is rotational inertia affected by distance from rotation axis?

Increases with distance squared. Moment of inertia proportional to $r^2$ in $I = mr^2$ .
What is the effect of balanced torques on an object?

The object is in rotational equilibrium. Equal and opposite torques cancel, producing no angular acceleration.
What is the role of friction in rotational motion?

Can provide torque to change rotational motion. Friction forces can create torques about rotation axes.
For rotational equilibrium, what must the net torque be?

Zero. No net torque means no angular acceleration occurs.

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