Rotational Motion, Torque, and Angular Momentum: Study Notes

Rotational Motion and Angular Quantities

Rotation Axis and Angle θ

Rotational motion involves objects spinning about a fixed axis. The rotation axis is the line about which the object rotates, perpendicular to the plane of rotation. The angle θ describes the amount of rotation, typically measured in radians. Counter-clockwise rotation is considered positive, while clockwise is negative.

• Angle θ: The standard angle is measured from the x-axis.

• Units: Radians (rad), where 2π rad = 360°.

• Example: A disk rotating about its center.

Angular Displacement (Δθ)

Angular displacement is the change in angle as an object rotates.

• Formula: Δθ = θf - θi

• Counter-clockwise: Δθ > 0

• Clockwise: Δθ < 0

Arc Length (s)

The arc length is the distance traveled along a circular path of radius r for an angular displacement Δθ.

• Formula: $s = r\Delta\theta$

• Example: A point on a wheel moving through an angle θ.

Angular Velocity (ω)

Angular velocity describes the rate of change of angular displacement.

• Formula: $\omega = \frac{d\theta}{dt}$ (Calculus), $\omega = \frac{\Delta\theta}{\Delta t}$ (Algebraic)

• All points on a rigid body share the same ω.

Tangential Speed (v)

Tangential speed is the linear speed of a point at radius r from the axis.

• Formula: $v = r\omega$

• Increases with distance from the axis.

Angular Acceleration (α)

Angular acceleration is the rate of change of angular velocity.

• Formula: $\alpha = \frac{d\omega}{dt}$ (Calculus), $\alpha = \frac{\Delta\omega}{\Delta t}$ (Algebraic)

Tangential Acceleration (at)

Tangential acceleration is the acceleration of a point at radius r along the arc.

• Formula: $a_t = r\alpha$

• If $a_t = 0$, the object is in uniform circular motion.

Uniform Circular Motion

In uniform circular motion, an object moves at constant speed along a circle of fixed radius.

• Radial (centripetal) acceleration: $a_c = \frac{v^2}{r} = r\omega^2$

• Direction changes, but speed remains constant.

Centripetal vs. Tangential Acceleration

Any object moving along a curve has both centripetal and tangential acceleration.

• Centripetal acceleration: $a_c = \frac{v^2}{r} = r\omega^2$

• Total acceleration: $a = \sqrt{a_t^2 + a_c^2}$

Rotational Kinematics for Constant α

Equations analogous to linear kinematics apply for rotational motion with constant angular acceleration.

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

Moment of Inertia and Angular Momentum

Moment of Inertia (I)

Moment of inertia quantifies an object's resistance to rotational acceleration about an axis.

• Point mass: $I_{pm} = mr^2$

• System of N point masses: $I_{sys} = \sum_{i=1}^N m_i r_i^2$

• Rigid object: $I_{obj} = \sum dm_i r_i^2$

• Depends on mass distribution and axis location.

Angular Momentum (L)

Angular momentum is the rotational analog of linear momentum.

• Formula: $L = I\omega$

• Conserved in isolated systems.

Conservation of Angular Momentum

• Formula: $\Delta L_{sys} = 0$ if $\sum \tau_{ext} = 0$

• Angular momentum remains constant if no external torque acts.

Torque and Rotational Dynamics

Torque (τ)

Torque measures the effectiveness of a force to rotate an object about an axis.

• Formula: $\tau = rF\sin\phi$

• r: distance from axis to force application point

• F: applied force

• φ: angle between r and F

• Only the perpendicular component of F contributes to torque.

Sign of Torque

• τ > 0: counter-clockwise rotation

• τ < 0: clockwise rotation

• Right-hand rule helps determine sign.

Rotational Analog of Newton's 2nd Law

The net torque on an object changes its angular momentum.

• Formula: $\frac{dL}{dt} = \tau_{net} = \sum \tau_i$

• If I is constant: $I\alpha = \tau_{net}$

Angular vs. Linear Quantities

Pure Translation (Fixed Direction)	Pure Rotation (Fixed Axis)
Velocity: $v$ Acceleration: $a$ Mass: $m$ Newton's 2nd Law: $F_{net} = ma$ Kinetic Energy: $\frac{1}{2}mv^2$	Angular Velocity: $\omega$ Angular Acceleration: $\alpha$ Moment of Inertia: $I$ Newton's 2nd Law: $I\alpha = \tau_{net}$ Kinetic Energy: $\frac{1}{2}I\omega^2$

Conditions for Equilibrium

• No net force: $\sum F_i = 0$

• No net torque: $\sum \tau_j = 0$

• Static equilibrium: object at rest

• Dynamic equilibrium: constant velocity and angular velocity

Moment of Inertia of Rigid Objects

Rigid objects are modeled as a sum of infinitesimal point masses. The total moment of inertia is the sum over all these masses.

• Formula: $I_{obj} = \sum dm_i r_i^2$

• For continuous objects, calculus can be used: $I_{obj} = \int r^2 dm$

Shape	Moment of Inertia
Thin rod (center)	$I = \frac{1}{12}ML^2$
Thin rod (end)	$I = \frac{1}{3}ML^2$
Solid cylinder (axis)	$I = \frac{1}{2}MR^2$
Solid sphere (center)	$I = \frac{2}{5}MR^2$
Thin rectangular plate (center)	$I = \frac{1}{12}M(a^2 + b^2)$

Selected Problems (Examples)

Example 1: Acceleration Vectors in Circular Motion

Given a bead with initial velocity on a frictionless vertical hoop, determine the possible acceleration vectors at a given point.

• Use centripetal acceleration: $a_c = \frac{v^2}{r}$

Example 2: Torque Square

Five forces act on a square about a pivot. Rank the forces according to the magnitude of their torque.

• Use $\tau = rF\sin\phi$ for each force.

Example 3: Rotational Acceleration of Rigid Bodies

Compare angular accelerations of disk, hoop, and sphere with same mass and radius under equal torque.

• Use $\alpha = \frac{\tau}{I}$

• Moment of inertia differs for each shape.

Example 4: Flywheel Deceleration

Calculate angular acceleration, revolutions before stopping, and tangential speed for a flywheel slowing down due to friction.

• Use $\alpha = \frac{\Delta\omega}{\Delta t}$

• Relate angular and tangential quantities.

Example 5: Toilet Paper Roll Acceleration

Pulling on a roll with force F, observe how angular acceleration changes as the radius decreases.

• Use $\alpha = \frac{\tau}{I}$, with $I$ depending on radius.

Example 6: Push-Up Force Analysis

Analyze the forces and torques involved in a push-up, modeling the body as a rigid bar.

• Apply equilibrium conditions: $\sum F = 0$, $\sum \tau = 0$

Summary

• Rotational motion involves angular displacement, velocity, and acceleration.

• Moment of inertia quantifies resistance to rotation.

• Torque causes angular acceleration.

• Angular momentum is conserved in isolated systems.

• Equilibrium requires zero net force and torque.

Additional info: Some formulas and examples have been expanded for clarity and completeness. Table entries for moment of inertia are standard results for common shapes.