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5.1 Rotational Kinematics

Syllabus
2024
Topic
5.1
Level
—

Learning objectives

5.1A—Describe the rotation of a system with respect to time using angular displacement, angular velocity, and angular…Describe the rotation of a system with respect to time using angular displacement, angular velocity, and angular acceleration. BOUNDARY STATEMENT Descriptions of the directions of rotation for a point or object are limited to clockwise and counterclockwise with respect to a given axis of rotation. AP Physics 1: Algebra-Based Course and Exam Description Torque and Rotational Dynamics UNIT 5 TOPIC 5.2 Connecting Linear and Rotational Motion• Angular displacement is the measurement of the angle, in radians, through which a point on a rigid system rotates about a specified axis. Relevant equation:- i. A rigid system is one that holds its shape but in which different points on the system move in different directions during rotation. A rigid system cannot be modeled as an object.- ii. One direction of angular displacement about an axis of rotation—clockwise or counterclockwise—is typically indicated as mathematically positive, with the other direction becoming mathematically negative.- iii. If the rotation of a system about an axis may be well described using the motion of the system’s center of mass, the system may be treated as a single object. For example, the rotation of Earth about its axis may be considered negligible when considering the revolution of Earth about the center of mass of the Earth–Sun system.• Average angular velocity is the average rate at which angular position changes with respect to time. Relevant equation:• Average angular acceleration is the average rate at which the angular velocity changes with respect to time. Relevant equation:• Angular displacement, angular velocity, and angular acceleration around one axis are analogous to linear displacement, velocity, and acceleration in one dimension and demonstrate the same mathematical relationships.- i. For constant angular acceleration, the mathematical relationships between angular displacement, angular velocity, and angular acceleration can be described with the following equations:- ii. Graphs of angular displacement, angular velocity, and angular acceleration as functions of time can be used to find the relationships between those quantities.

Angular motion follows signed kinematics

Define signed angular quantities

Angular displacement measures rotation about a specified axis in radians. Choose either clockwise or counterclockwise as positive; the opposite direction is negative.

Δθ=θ−θ0ωavg=ΔθΔtαavg=ΔωΔt\begin{aligned}\Delta\theta&=\theta-\theta_0\\[3pt]\omega_{\text{avg}}&=\frac{\Delta\theta}{\Delta t}\\[3pt]\alpha_{\text{avg}}&=\frac{\Delta\omega}{\Delta t}\end{aligned}

Units are rad, rad/s, and rad/s².

Reuse the linear-kinematics structure

One-dimensional linear motion Rotation about one axis
Position xx Angular position θ\theta
Velocity vv Angular velocity ω\omega
Acceleration aa Angular acceleration α\alpha

ω=ω0+αtθ=θ0+ω0t+12αt2ω2=ω02+2α(θ−θ0)\begin{aligned}\omega&=\omega_0+\alpha t\\[3pt]\theta&=\theta_0+\omega_0t+\frac12\alpha t^2\\[3pt]\omega^2&=\omega_0^2+2\alpha(\theta-\theta_0)\end{aligned}

Read slopes and areas

Graph Slope gives Signed area gives
θ\theta versus tt ω\omega —
ω\omega versus tt α\alpha Δθ\Delta\theta
α\alpha versus tt Rate of change of α\alpha Δω\Delta\omega

Calculate constant angular acceleration

Constant-α\alpha example: choose counterclockwise as positive. A wheel has ω0=+2.0 rad/s\omega_0=+2.0\,\text{rad/s} and α=−0.50 rad/s2\alpha=-0.50\,\text{rad/s}^2 for 4.0 s4.0\,\text{s}.

ω=ω0+αt=2.0+(−0.50)(4.0)=0 rad/s\omega=\omega_0+\alpha t=2.0+(-0.50)(4.0)=0\,\text{rad/s},

Δθ=ω0t+12αt2=(2.0)(4.0)+12(−0.50)(4.0)2=4.0 rad\Delta\theta=\omega_0t+\tfrac12\alpha t^2=(2.0)(4.0)+\tfrac12(-0.50)(4.0)^2=4.0\,\text{rad}.

The wheel slows to rest but still turns 4.04.0 rad counterclockwise during the interval.

Keep the rigid-system model bounded

A rigid system keeps its shape, but different points move in different directions during rotation, so it is not generally a point object. Treat it as a single object only when its rotation is negligible for the motion being analyzed. Direction descriptions here are limited to clockwise and counterclockwise about the stated axis.