Torque, Angular Momentum & Rotational Dynamics Explained | Chapter 10 of University Physics

Chapter 10 of University Physics develops the rotational analogs of Newton’s laws by introducing torque, rotational dynamics, and angular momentum. We explore Στ=Iα, combined translational and rotational motion, work and power in rotation, conservation of angular momentum, and gyroscopic precession.

Torque: The Rotational Force

Torque (τ) measures a force’s ability to cause rotation about an axis. Defined as τ = r × F = r F sin(φ), torque uses the lever arm (perpendicular distance) and the right-hand rule to determine its direction. The SI unit is the newton-meter (N·m).

Rotational Form of Newton’s Second Law

For a rigid body about a fixed axis, the net torque equals rotational inertia times angular acceleration:

Στ = I α

This equation parallels F = m a in linear dynamics. Solving problems involves drawing free-body diagrams, choosing rotation axes, writing ΣF = m a and Στ = I α, and solving component equations.

Combined Translation & Rotation

The total kinetic energy of a rigid body combines translation of its center of mass and rotation about that point:

K_total = ½ M v_cm² + ½ I_cm ω²

In rolling without slipping, the no-slip condition v_cm = R ω links linear and angular velocities.

Work & Power in Rotational Motion

Analogous to linear work, torque does work through angular displacement:

• W = τ Δθ
• P = τ ω

The rotational work-energy theorem states that the work done by torque equals the change in rotational kinetic energy: W = Δ(½ I ω²).

Angular Momentum & Its Conservation

Angular momentum (L) for a rigid body is L = I ω; for a particle, L = r × p. When net external torque is zero, angular momentum is conserved (dL/dt = 0), making it invaluable for analyzing isolated systems and collisions in rotational contexts.

Gyroscopes & Precession

Gyroscopic precession occurs when an applied torque changes the direction of L without altering its magnitude. The precession rate is given by:

Ω = τ / L = (m g r) / (I ω)

This phenomenon underpins applications in navigation and stabilization.

Conclusion

By mastering torque, Στ=Iα, and angular momentum conservation, you gain powerful methods to analyze rotating systems — from spinning wheels to gyroscopes. Energy and work in rotation complement these tools for comprehensive problem solving.

Watch the full video summary here for detailed derivations, examples, and problem walkthroughs.

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