Premed · Premed · Physics 1
Lecture 12: Angular Momentum
Physics I — Mechanics & Thermodynamics
Learning Objectives
By the end of this lecture, students will be able to:
- Define angular momentum for a particle and for a rigid body
- Relate net torque to the rate of change of angular momentum
- Apply conservation of angular momentum to isolated systems
- Explain real-world phenomena such as figure skater spins, gyroscopic precession, and planetary orbits using angular momentum conservation
- Solve problems involving collisions and interactions of rotating objects
Lecture Content
I. Angular Momentum of a Particle
The angular momentum of a particle about a point O is defined as L = r x p = r x mv, where r is the position vector from O to the particle and p = mv is the linear momentum. The magnitude is L = r m v sin(theta) = r p_perp = r_perp p, where theta is the angle between r and v, and r_perp = r sin(theta) is the perpendicular distance from O to the line of motion. The direction of L is given by the right-hand rule, perpendicular to the plane containing r and v. The SI units are kg m^2/s. For a particle moving in a circle of radius r at speed v, the angle between r and v is 90 degrees, giving L = mvr.
II. Angular Momentum of a Rigid Body
For a rigid body rotating about a fixed axis, the angular momentum simplifies to L = I omega, where I is the moment of inertia about the axis and omega is the angular velocity. The direction is along the axis of rotation, determined by the right-hand rule. This is the rotational analog of p = mv. For a system of objects, the total angular momentum is L_total = Sum of L_i = Sum of I_i omega_i, provided all are measured about the same axis.
III. Torque and Angular Momentum
Newton's second law for rotation can be expressed in terms of angular momentum as tau_net = dL/dt, directly analogous to F_net = dp/dt. When the net external torque is zero, dL/dt = 0, and therefore L = constant. This is the law of conservation of angular momentum.
IV. Conservation of Angular Momentum
When the net external torque on a system is zero, the total angular momentum is conserved: L_i = L_f, or equivalently, I_i omega_i = I_f omega_f for rotation about a fixed axis. If the moment of inertia changes, the angular velocity must change to compensate. A decrease in I produces an increase in omega, and vice versa.
This principle explains a wide range of phenomena. A figure skater who pulls her arms in decreases her moment of inertia, causing her spin rate to increase dramatically. A diver tucks into a ball to rotate faster during a flip, then extends to slow the rotation before entering the water. A collapsing star undergoes an enormous increase in angular velocity as its radius shrinks, producing rapidly spinning neutron stars known as pulsars. When a person on a merry-go-round walks toward the center, the system's moment of inertia decreases and the platform spins faster.
<image>Panel A: A figure skater with arms extended spinning slowly (large I, small omega). An arrow shows the transition as she pulls her arms in. Panel B: The same skater with arms tucked, spinning rapidly (small I, large omega). Both panels show L = I omega with the same value. Bar charts beside each panel show I and omega as inversely related bars, with L (their product) remaining constant.</image>
V. Angular Momentum in Collisions
In rotational collisions, such as a ball of clay hitting a rotating disk, angular momentum is conserved provided no external torques act: L_before = L_after. For objects that stick together, I_1 omega_1 + I_2 omega_2 = (I_1 + I_2) omega_f. Kinetic energy is generally not conserved in such collisions, analogous to perfectly inelastic linear collisions.
Consider a person jumping onto a stationary merry-go-round. The initial angular momentum is L_i = m v r (the person's angular momentum about the center), and the final angular momentum is L_f = (I_disk + m r^2) omega_f. Setting these equal determines the final angular velocity.
VI. Angular Momentum of Orbiting Bodies
A particle in a circular orbit has angular momentum L = mvr. For elliptical orbits, L = mvr sin(theta) at any point, where theta is the angle between v and r. At perihelion and aphelion (the closest and farthest points in the orbit), the velocity is perpendicular to the position vector, so L = mvr at these points.
Conservation of angular momentum requires mv_1 r_1 = mv_2 r_2, meaning the orbiting body moves faster when it is closer to the central body and slower when farther away. This result is equivalent to Kepler's second law, which states that a line from the Sun to a planet sweeps out equal areas in equal time intervals.
<image>An elliptical orbit around a central body. The orbiting object is shown at two positions: close to the central body (perihelion, with velocity vector v_1 and distance r_1) and far from the central body (aphelion, with velocity vector v_2 and distance r_2). Shaded triangular areas swept out in equal time intervals are shown at both positions — the areas are equal (Kepler's second law). The equation mv_1 r_1 = mv_2 r_2 is written below, emphasizing that the object moves faster when closer.</image>
VII. Gyroscopic Motion and Precession (Qualitative)
A spinning gyroscope resists changes to its orientation because of its large angular momentum. When a torque is applied, for instance by gravity acting on a tilted gyroscope, the angular momentum vector L changes direction rather than magnitude. The result is precession: the axis of rotation slowly sweeps out a cone. The precession rate is Omega_p = tau / (I omega) = Mgd / (I omega), which shows that a faster spin (larger omega) produces slower precession.
Applications of gyroscopic effects include navigation gyroscopes in aircraft and ships, the stability of bicycles and motorcycles (whose spinning wheels resist tilting), and Earth's axial precession, which has a period of about 26,000 years. The direction of precession follows the direction of the applied torque, determined by the right-hand rule for dL.
<image>A gyroscope with its axis tilted at an angle to the vertical. The spin angular momentum vector L points along the axis. Gravity acts at the center of mass, creating a torque tau perpendicular to L. The resulting change dL is shown as a horizontal vector, causing the tip of L to trace a horizontal circle. The precession path of the gyroscope axis is shown as a dashed cone. Arrows indicate the direction of spin, the direction of the gravitational torque, and the direction of precession.</image>
