Section 1.2: Dynamics of Rigid Bodies

This section deals with the motion of rigid bodies, including rotation, torque, and moment of inertia. We analyze how forces cause linear and rotational acceleration.

Key Equations & Concepts:
  • Torque: \( \tau = r F \sin\theta \)
  • Moment of Inertia (for point mass): \( I = \sum m_i r_i^2 \)
  • Newton’s 2nd law (rotation): \( \tau = I \alpha \)
  • Rotational kinetic energy: \( K_r = \frac{1}{2} I \omega^2 \)
  • Parallel axis theorem: \( I = I_{CM} + M d^2 \)

Example 1

A solid disk of mass 2 kg and radius 0.5 m rotates under a torque of 5 N·m. Find its angular acceleration.

Moment of inertia: \( I = \frac{1}{2}MR^2 = \frac{1}{2} \cdot 2 \cdot 0.5^2 = 0.25 \text{ kg·m²} \)

Angular acceleration: \( \alpha = \tau / I = 5 / 0.25 = 20 \text{ rad/s²} \)

Example 2

A uniform rod 1.5 m long and 3 kg rotates about one end. Find its moment of inertia.

\( I = \frac{1}{3} M L^2 = \frac{1}{3} \cdot 3 \cdot 1.5^2 = 2.25 \text{ kg·m²} \)

Practice Problems

  1. Find the torque on a 10 N force applied at 0.5 m from pivot at 60°.
  2. A solid cylinder of mass 5 kg and radius 0.4 m rotates with 10 N·m torque. Compute angular acceleration.
  3. Calculate moment of inertia of a thin rod 2 m long, pivoted at center.
  4. Wheel of radius 0.3 m and mass 4 kg rotates under 8 N·m torque. Find α.
  5. A uniform disk of 6 kg, radius 0.2 m, rotates. Find rotational KE at ω = 10 rad/s.
  6. Use parallel axis theorem: rod 1 m long, 2 kg, rotation about end. Compute I.
  7. A seesaw: two 30 kg children 2 m apart. Find torque on pivot if one child pushes down 50 N.
  8. A solid sphere of mass 3 kg, radius 0.1 m rotates. Find rotational inertia about center.
  9. Torque applied to wheel: 12 N·m, I = 0.6 kg·m². Find angular acceleration.
  10. Rod of length 1 m, mass 2 kg rotates about end. Find KE if ω = 5 rad/s.
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