Rotational motion adds a new spin to physics, introducing angular quantities that measure circular movement. These concepts parallel linear motion but focus on rotation around an axis, like a spinning top or rotating fan blades.

Centripetal acceleration is key to understanding circular motion. It's always directed toward the center, changing the velocity's direction without altering its magnitude. This concept explains everything from roller coaster loops to planetary orbits.

Angular Quantities and Rotational Motion

Angular vs linear quantities

• Angular displacement measures rotation in radians (θ) represents change in angular position (spinning top)
• Angular velocity (ω) rad/s rate of change of angular displacement $\omega_{avg} = \frac{\Delta \theta}{\Delta t}$ and $\omega = \frac{d\theta}{dt}$ (rotating fan blades)
• Angular acceleration (α) rad/s² rate of change of angular velocity $\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$ and $\alpha = \frac{d\omega}{dt}$ (accelerating merry-go-round)
• Linear displacement measures straight-line distance traveled (m)
• Linear velocity (v) m/s rate of change of linear displacement
• Linear acceleration (a) m/s² rate of change of linear velocity

Relationships in circular motion

• Arc length $s = r\theta$ relates linear and angular displacement (Ferris wheel)
• Tangential velocity $v = r\omega$ connects linear and angular velocity (car on a circular track)
• Tangential acceleration $a_t = r\alpha$ links linear and angular acceleration (spinning figure skater)
• Angle conversion $1 \text{ revolution} = 360° = 2\pi \text{ radians}$
• Frequency and angular velocity $\omega = 2\pi f$ (f in Hz) (bicycle wheel)

Rotational kinematics equations

• $\omega = \omega_0 + \alpha t$ final angular velocity
• $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ angular displacement
• $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$ angular velocity-displacement relation
• $\theta = \theta_0 + \frac{1}{2}(\omega + \omega_0)t$ average angular velocity
• Analogous to linear kinematics replace x with θ, v with ω, and a with α
• Problem-solving steps:
  1. Identify known and unknown quantities
  2. Select appropriate equation(s)
  3. Solve for desired variable

Centripetal Acceleration

Centripetal acceleration concept

• Directed toward center of circular motion changes direction of velocity vector (roller coaster loop)
• Magnitude constant in uniform circular motion
• $a_c = \frac{v^2}{r} = r\omega^2$ relates to angular velocity
• Centripetal force $F_c = ma_c = m\frac{v^2}{r} = mr\omega^2$ causes centripetal acceleration
• Applications include planetary orbits, banked curves, artificial satellites