In-depth Notes on Rotation

Rigid Body Rotation

• Rigid Body: A body that can rotate with all parts locked together without changing shape.

• Fixed Axis: Rotation occurs about a stationary axis.

Rotational Variables

1. Angular Position

• Defined by the arc from the x-axis (zero angular position) to the reference line.

• Measured in radians (rad).

2. Angular Displacement

• Change in angular position from ( \theta1 ) to ( \theta2 ):
  Δθ=θ2−θ1

• Positive for counterclockwise rotation, negative for clockwise.

3. Angular Velocity

• Average angular velocity between two time points ( t1 ) and ( t2 ):
  $\omega_{avg} = \frac{\Delta \theta}{\Delta t}$

• Instantaneous angular velocity:
  $\omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t}$

4. Angular Acceleration

• If angular velocity changes, there is angular acceleration.

• Average angular acceleration:
  $\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$

• Instantaneous angular acceleration:
  $\alpha = \lim_{\Delta t \to 0} \frac{\Delta \omega}{\Delta t}$

Unit: ( \text{rad/s}^2 ).

Relationship Between Linear and Angular Variables

• For a point at a position ( r ) from the rotation axis:
  $s = r \Delta \theta$

• Differentiating with respect to time gives linear velocity ( v ):
  $v = r \omega$

• The period of revolution ( T ) relates to angular velocity:
  $T = \frac{2\pi}{\omega}$

Kinetic Energy of Rotation

• Total kinetic energy of a rotating body:
  K=∑21​mivi2
  where ( v_i = r\omega ), leading to:
  $K = \frac{1}{2} I \omega^2$
  where ( I ) is the moment of inertia.

Rotational Inertia (Moment of Inertia)

• Defines resistance to rotation.

• Dependent on mass and axis of rotation:
  
  I=∑miri2I=∑miri2

• Varies based on distribution of mass regarding the rotation axis.

• Moment of inertia examples:

  • Rod around longitudinal axis:
    $I = \frac{1}{12} m L^2$

• Solid sphere:
  $I = \frac{2}{5} m R^2$

• Hoop:
  $I = m R^2$

Torque

• Defined as a force's ability to cause rotation:
  $\tau = r F \sin(\theta)$
  where ( \theta ) is the angle between the force vector and the lever arm.

• Effects of torque:

  • Greater distance from pivot increases torque.

  • Torque also depends on the angle of application.

Newton's Law of Rotation

• Relation between torque and angular acceleration:
  $T_{net} = I \alpha$

• Where ( T_{net} ) is net torque and ( \alpha ) is angular acceleration.

Work and Power in Rotation

• Work done by torque:
  $W = \tau \Delta \theta$

• Power related to torque:
  $P = \frac{W}{t} = \tau \omega$

Summary of Key Formulas