Unit 5: Torque and Rotational Dynamics

Angular Motion

• Angular Velocity: $$\bar{\omega} = \frac{\Delta \theta}{\Delta t} \quad \left( \frac{\text{rad}}{s} \text{ or } \frac{\text{rev}}{\text{min}} \right)$$
• Angular Acceleration: $$\bar{\alpha} = \frac{\Delta \omega}{\Delta t} \quad \left( \frac{s^2}{j} \right)$$
• One full revolution: $$1 \text{ rev} = 360^\circ = 2\pi \text{ radians}$$

Uniformly Angularly Accelerated Motion (UaM)

These equations are valid when $\alpha$ is constant.

Rotational Kinematics

• Tangential velocity: The linear velocity of an object moving along a circular path. $$v_t = r\omega$$
  • Direction is tangent to the circle and normal to the radius.
  • Same units as linear velocity: $\frac{m}{s}$.

Centripetal Force and Centripetal Acceleration

• Centripetal force: The net force acting towards the center of circular motion. $$\sum F_r = m a_c$$
  • Not a new force.
  • Never drawn in a Free Body Diagram.
  • “Inward” is positive; “outward” is negative.
• Centripetal acceleration: $$a_c = \frac{v^2}{r} = r\omega^2$$

Period and Frequency

• Period (T): Time for one full revolution. Units: seconds per cycle.
• Frequency (f): Number of cycles per second (Hz). $$f = \frac{\text{cyc}}{\text{sec}} = Hz$$
• Relationship between period and frequency: $$T = \frac{1}{f}$$

Torque and Rotational Dynamics

• Torque ($\tau$): The ability to cause angular acceleration. $$\vec{\tau} = \vec{r}_\perp \times \vec{F} = rF\sin\theta$$
  • Moment arm: The perpendicular distance from the axis of rotation to the force. $$\vec{r}_\perp = \vec{r} \sin\theta$$
  • A larger moment arm results in greater torque.
  • Maximum torque is achieved at a $90^\circ$ angle: $$(\sin\theta)_{\max} = \sin(90^\circ) = 1$$
  • Units: Newton-meters (N·m).
  • Torque is a vector:
    • Use **clockwise** and **counterclockwise** for direction.

Newton’s Second Law for Rotation

$$\sum \vec{\tau} = I \alpha$$

• Where $I$ is the **moment of inertia** and $\alpha$ is angular acceleration.
• This is the rotational equivalent of $\sum F = ma$.

The Conical Pendulum

• For a pendulum moving in circular motion: $$F_T \cos\theta = mg \Rightarrow F_T = \frac{mg}{\cos\theta}$$
• Using centripetal force: $$F_T \sin\theta = m \left(\frac{v_t^2}{r} \right)$$
• Solving for period: $$T^2 = \frac{4\pi^2 L \cos\theta}{g}$$