Rotational Motion: Physics with Algebra Study Guide

Rotational Motion

Circular and Rotational Motion

Rotational motion refers to the movement of objects that spin about an axis. This type of motion is fundamental in understanding the behavior of many physical systems, from playground equipment to machinery.

• Rotational motion: The motion of an object around a fixed axis.

• Circular motion: A special case where every point on the object moves in a circle around the axis.

• Rigid body: An object whose size and shape do not change as it moves.

• Combination motion: An object rotates as it moves along a trajectory.

• Translational motion: The object moves along a trajectory but does not rotate.

Angular Position, Displacement, and Velocity

Angular position describes the orientation of a particle in circular motion relative to a reference axis. Angular displacement and angular velocity quantify how this position changes over time.

• Angular position (θ): The angle measured from the positive x-axis to the radius connecting the center to the particle.

• Arc length (s): The distance traveled along the circular path.

• Relationship: $\theta = \frac{s}{r}$ (radians)

• One revolution: $\theta_{\text{full circle}} = 2\pi$ radians $= 360^\circ$

• Conversion: $1\text{ rad} = 57.3^\circ$

Angular Displacement and Angular Velocity

Angular displacement ($\Delta\theta$) is the change in angular position. Angular velocity ($\omega$) is the rate of change of angular displacement.

• Angular displacement: $\Delta\theta = \theta_f - \theta_i$

• Angular velocity: $\omega = \frac{\Delta\theta}{\Delta t}$

• Linear speed: $v = \omega r$

• Period (T): Time for one revolution, $\omega = \frac{2\pi}{T}$

• Frequency (f): $f = \frac{1}{T}$, $\omega = 2\pi f$

Example: Wind Turbine Blade

For a wind turbine with blades of length 50 m rotating at 10 rpm:

• Period: $T = 6.0$ s

• Angular speed: $\omega = \frac{2\pi}{T} = 1.05$ rad/s

• Tip speed (r = 50 m): $v_B = \omega r = 52$ m/s

• Speed at 25 m: $v_A = \omega r = 26$ m/s

Rotation of a Rigid Body

A rigid body is an object whose size and shape remain constant during motion. Every point on a rotating rigid body has the same angular velocity, but points farther from the axis move faster.

• Angular velocity ($\omega$): Same for all points on the body.

• Linear speed ($v$): $v = \omega r$; increases with distance from the axis.

Angular Acceleration

Angular acceleration ($\alpha$) measures how quickly angular velocity changes. It is defined as:

• Angular acceleration: $\alpha = \frac{\Delta\omega}{\Delta t}$

• Uniform circular motion: $\alpha = 0$

• Non-uniform circular motion: $\alpha \neq 0$

• Relationship: Tangential acceleration $a_t = \alpha r$

Synthesis: Linear and Circular Motion Analogs

Variables and equations for linear motion have direct analogs in circular motion.

• Position: $x$ (linear), $\theta$ (angular)

• Velocity: $v_x = \frac{\Delta x}{\Delta t}$, $\omega = \frac{\Delta\theta}{\Delta t}$

• Acceleration: $a_x = \frac{\Delta v_x}{\Delta t}$, $\alpha = \frac{\Delta\omega}{\Delta t}$

• Equations:

  • Linear: $\Delta x = v \Delta t + \frac{1}{2} a (\Delta t)^2$

  • Angular: $\Delta\theta = \omega \Delta t + \frac{1}{2} \alpha (\Delta t)^2$

Torque

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

• Definition: $\tau = r F \sin \phi$

• Moment arm ($r_\perp$): $r_\perp = r \sin \phi$

• Direction: Counterclockwise (+), Clockwise (–)

• Net torque: Sum of all individual torques

Gravitational Torque and Center of Gravity

Gravitational torque is the torque caused by the weight of an object. The center of gravity is the point where the total gravitational torque is zero.

• Gravitational torque: $\tau = -r_\perp m g$

• Center of gravity (xcg): $x_{cg} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$

• Y-coordinate: $y_{cg} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2}$

Rotational Dynamics and Moment of Inertia

Rotational dynamics describes how torques cause angular acceleration. Moment of inertia ($I$) quantifies an object's resistance to rotational acceleration.

• Moment of inertia: $I = \sum m_i r_i^2$

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

• For a cylinder: $I = \frac{1}{2} M R^2$

Example: Spinning Up a Flywheel

A 540 kg cylinder with radius 0.30 m is spun up by a motor providing 2.8 Nm torque. To reach 13,000 rpm:

• Angular speed: $\omega = 1360$ rad/s

• Moment of inertia: $I = 24.3$ kg·m2

• Angular acceleration: $\alpha = 0.115$ rad/s2

• Time to reach speed: $\Delta t = \frac{\Delta\omega}{\alpha} = 11,800$ s (3.3 hours)

Summary: Rotational Motion Equations

Key equations and relationships for rotational motion:

• $\theta = \frac{s}{r}$

• $v = \omega r$

• $\omega = \frac{\Delta\theta}{\Delta t}$

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

• $a_t = \alpha r$

• $\tau = I \alpha$

• $I = \sum m_i r_i^2$

• $x_{cg} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$

• $y_{cg} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2}$

Linear Motion	Circular Motion
Position (m): x	Angle (rad): θ
Velocity (m/s): vx = Δx/Δt	Angular velocity (rad/s): ω = Δθ/Δt
Acceleration (m/s2): ax = Δvx/Δt	Angular acceleration (rad/s2): α = Δω/Δt
Δx = v Δt + ½ a (Δt)2	Δθ = ω Δt + ½ α (Δt)2