Lecture on Angular Acceleration of a Merry-Go-Round

Key Concepts

• Angular Acceleration: The rate of change of angular velocity.
• Angular Speed: Equivalent to linear speed in rotational dynamics.
• Moment of Inertia: A measure of an object's resistance to any change in its state of rotation.
• Torque: A measure of the force that can cause an object to rotate about an axis.

Given Values

• Moment of Inertia: 100 kg·m²
• Force Applied by Father: 200 N
• Distance from Axis of Rotation: 0.5 m

Calculating Torque

• Torque is calculated using: [ \tau = r \times F \times \sin(\theta) ]
  • Here, ( r = 0.5 \text{ m} ), ( F = 200 \text{ N} ), and ( \theta = 90^\circ ).
  • ( \sin(90^\circ) = 1 ), so Torque = 100 N·m.

Calculating Angular Acceleration

• Using the formula from Newton's second law for rotation: [ \alpha = \frac{\tau}{I} ]
  • Given ( \tau = 100 \text{ N·m} ) and ( I = 100 \text{ kg·m}^2 ),
  • ( \alpha = 1 \text{ rad/s}^2 ).

Calculating Angular Speed

• Using the equation similar to linear motion: [ \omega = \omega_0 + \alpha t ]
  • Initial angular speed ( \omega_0 = 0 ) (merry-go-round is initially at rest).
  • ( \alpha = 1 \text{ rad/s}^2 ), ( t = 20 \text{ s} ).
  • ( \omega = 0 + 1 \times 20 = 20 \text{ rad/s} ).

Calculating Number of Revolutions

• Use the equation for rotational motion: [ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 ]
  • ( \theta = 0 + 0.5 \times 1 \times 20^2 = 200 \text{ rad} ).
• Convert radians to revolutions: [ \theta = \frac{200}{2\pi} \approx 31.8 \text{ revolutions} ].

Assumptions

• No frictional forces are considered in this scenario.