Rotational Motion, Moment of Inertia, and Torque: Study Notes

Rotational Motion and Rigid Bodies

Angular Velocity and Angular Acceleration

In rotational motion, all masses in a rigid body rotate together with the same angular velocity ($\omega$) and angular acceleration ($\alpha$). This uniformity is a key property of rigid bodies.

• Angular velocity ($\omega$): The rate at which an object rotates or spins.

• Angular acceleration ($\alpha$): The rate of change of angular velocity.

• Helper equations:

  • $S = R \theta$ (arc length)

  • $v_t = R \omega$ (tangential velocity)

  • $a_t = R \alpha$ (tangential acceleration)

• Total acceleration: $a = \sqrt{a_t^2 + a_r^2}$, where $a_t$ is tangential and $a_r$ is radial (centripetal) acceleration.

Rigid Bodies and Rotation

Forces and Kinematics in Rotational Motion

To analyze rotational motion, we apply a net force to produce acceleration, then use kinematic equations. For rotation, angular acceleration depends on how mass is distributed relative to the axis of rotation.

• Key point: The farther a mass is from the axis, the more force is needed to rotate it.

• Application: Masses farther from the axis must move faster to maintain the same angular velocity, resulting in higher kinetic energy.

Location of Mass and Rotational Effects

Distribution of Mass and Rotational Kinetic Energy

The position of mass relative to the axis affects rotational motion. Masses farther from the axis have greater tangential velocity and kinetic energy.

• Kinetic energy: $K = \frac{1}{2} m v^2$

• For rotation: $K = \frac{1}{2} m (R \omega)^2 = \frac{1}{2} m R^2 \omega^2$

• All rotational quantities (v, K, $a_t$) depend on R (distance from axis).

Moment of Inertia

Definition and Physical Meaning

The moment of inertia ($I$) is the rotational analog of mass. It quantifies how difficult it is to change an object's rotational motion.

• Moment of inertia ($I$): $I = \sum m_i r_i^2$ (sum over all masses $m_i$ at distance $r_i$ from axis)

• Rotational kinetic energy: $K = \frac{1}{2} I \omega^2$

• Physical meaning: Larger $I$ means harder to accelerate rotationally, just as larger mass means harder to accelerate linearly.

Moment of Inertia for Common Shapes

Different shapes have different moments of inertia, depending on how mass is distributed.

• Point mass at distance R: $I = mR^2$

• Thin rod (center): $I = \frac{1}{12} ML^2$

• Solid sphere: $I = \frac{2}{5} MR^2$

• Hollow sphere: $I = \frac{2}{3} MR^2$

Parallel Axis Theorem

The parallel axis theorem allows calculation of the moment of inertia about any axis parallel to one through the center of mass.

• $I_{\text{new}} = I_{\text{center of mass}} + Md^2$

• Where $d$ is the distance between axes.

• Example: For a rod of length $L$ about its end: $I = \frac{1}{12}ML^2 + M(\frac{L}{2})^2 = \frac{1}{3}ML^2$

Angular Kinematics

Equations of Rotational Motion

Rotational kinematics parallels linear kinematics, with angular variables replacing linear ones.

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

• $\omega = \omega_0 + \alpha t$

• $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

• Helper equations: $s = R\theta$, $v = R\omega$, $a = R\alpha$

Torque

Definition and Physical Meaning

Torque ($\tau$) is the rotational analog of force. It measures the tendency of a force to rotate an object about an axis.

• Definition: $\tau = rF$ (for force perpendicular to lever arm)

• General definition: $\tau = rF \sin \theta$

• Units: Newton-meter (N·m)

• Direction: Counterclockwise torque is positive; clockwise is negative.

Torque and Angular Acceleration

Torque causes angular acceleration, analogous to force causing linear acceleration.

• Newton's Second Law for Rotation: $\sum \tau = I \alpha$

• Where $I$ is moment of inertia and $\alpha$ is angular acceleration.

Lever Arm and Line of Action

The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force.

• To analyze torque:

  1. Draw the free-body diagram.

  2. Fix the axis of rotation.

  3. Draw the line of action of the force.

  4. Measure the lever arm.

  5. Determine the direction (positive/negative).

Static Equilibrium

Conditions for Equilibrium

An object is in static equilibrium if the sum of all forces and the sum of all torques acting on it are zero.

• $\sum F = 0$ (no net force)

• $\sum \tau = 0$ (no net torque)

• Application: Used to analyze structures, levers, and balance problems.

Summary Table: Rotational vs Linear Quantities

Linear Quantity	Rotational Quantity
Displacement ($x$)	Angular Displacement ($\theta$)
Velocity ($v$)	Angular Velocity ($\omega$)
Acceleration ($a$)	Angular Acceleration ($\alpha$)
Mass ($m$)	Moment of Inertia ($I$)
Force ($F$)	Torque ($\tau$)
$F = ma$	$\tau = I\alpha$

Additional info: These notes expand on the original slides and handwritten notes, providing full definitions, equations, and context for rotational motion, moment of inertia, torque, and static equilibrium, suitable for college-level physics study.