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Physics C: Mechanics Unit 7: Angular Momentum

Angular momentum is the rotational counterpart of linear momentum, and it is conserved whenever no external torque acts. This unit uses it to analyze spinning skaters, rotational collisions, orbits and gyroscopes.

Covers: Angular momentum of a particle (r × p), Angular momentum of a rigid body (Iω), Torque as dL/dt and angular impulse, Conservation of angular momentum, Rotational collisions, Orbits and central forces, Precession

Study guide

For a point particle, angular momentum about an origin is L = r × p. Its magnitude is mvr sin θ, which equals the momentum times the perpendicular distance from the origin to the particle's line of motion. So a particle moving in a straight line has angular momentum about any point not on that line, and that angular momentum is constant if no force acts. The value depends on the chosen origin, so always state the point. For a rigid body rotating about a fixed symmetry axis, adding up r × p for every bit of mass gives L = Iω, directed along the axis by the right-hand rule.

The general rotational form of Newton's second law is Στ_ext = dL/dt. For a rigid body with constant I it becomes Στ = Iα, but the general form also covers systems that change shape. Integrating gives the angular impulse: ∫τ dt = ΔL. Internal torques cancel in pairs, so only external torques change a system's total angular momentum. If the net external torque about some point is zero, angular momentum about that point is conserved, even if forces act through that point, such as a pivot force or a central gravitational pull.

Conservation of angular momentum explains why a skater spins faster when she pulls her arms in: L = Iω is fixed, so a smaller I means a larger ω. Kinetic energy is K = L²/(2I), so it rises; the extra energy comes from the work her muscles do. Rotational collisions mirror linear ones. When a disk drops onto a spinning disk, or a person jumps onto a merry-go-round, L is conserved but kinetic energy is lost. When a projectile strikes a pivoted rod, the pivot exerts a large force during impact, so linear momentum is not conserved; angular momentum about the pivot is, because the pivot force has no lever arm.

Gravity from the Sun points straight at the Sun, so it exerts no torque about the Sun. A planet's angular momentum is therefore conserved, which is the physics behind Kepler's second law: the rate of sweeping area is L/(2m), a constant. At the closest and farthest points of an elliptical orbit, velocity is perpendicular to r, so v₁r₁ = v₂r₂. The same idea governs a puck on a string pulled through a hole: the string pulls radially, giving no torque, so vr stays constant while the pull does positive work and raises the kinetic energy.

Torque changes the direction of angular momentum as well as its size. A gyroscope with a large spin angular momentum, supported at one end, feels a torque from gravity that is horizontal and perpendicular to L. That torque swings L sideways instead of tipping the gyroscope, so it precesses at Ω = τ/L = mgr/(Iω). For objects that both translate and rotate, L about any point equals the spin angular momentum about the center of mass plus the angular momentum of the center of mass itself. Choosing the contact point as the origin removes friction's torque in skidding-to-rolling problems.

Key ideas

• Angular momentum depends on the choice of origin.
• For a rigid body about a symmetry axis, L = Iω along the axis.
• Net external torque equals dL/dt.
• L is conserved about any point where the net external torque is zero.
• Changing I at constant L changes ω and kinetic energy.
• Central forces conserve angular momentum, which explains equal areas in equal times.
• A torque perpendicular to L causes precession instead of tipping.

Formulas and rules

Common mistakes

• Conserving linear momentum in a collision with a pivoted object.
• Assuming kinetic energy is conserved when angular momentum is.
• Forgetting to add the moment of inertia of an object that lands on or sticks to a rotating body.
• Thinking a particle moving in a straight line has zero angular momentum.
• Using the wrong axis for a free body; it rotates about its center of mass.

Before the test, make sure you can

• Calculate angular momentum of a particle from r × p or p times the perpendicular distance.
• Calculate L = Iω and find torque from dL/dt.
• Apply conservation of angular momentum to skaters, merry-go-rounds and stacked disks.
• Solve collisions between a projectile and a pivoted rod.
• Compute kinetic energy changes when I changes at constant L.
• Use vr = constant at the ends of an elliptical orbit.
• Find the precession rate of a gyroscope.
• Combine spin and orbital angular momentum for rolling or free objects.

Words to know

Angular momentum (particle)

L = r × p, the rotational momentum of a particle about a point.

Angular momentum (rigid body)

L = Iω for rotation about a fixed symmetry axis.

Angular impulse

The time integral of torque, equal to the change in angular momentum.

Conservation of angular momentum

With zero net external torque, a system's total angular momentum stays constant.

Right-hand rule

Curl fingers along the rotation; the thumb points along ω and L.

Central force

A force always directed toward a fixed point, which exerts no torque about it.

Kepler's second law

A planet sweeps out equal areas in equal times, a result of conserved angular momentum.

Rotational inelastic collision

A collision between rotating bodies that conserves L but loses kinetic energy.

Precession

The slow turning of a spinning object's axis caused by a torque perpendicular to L.

Gyroscope

A rapidly spinning wheel whose large angular momentum resists changes in direction.

Spin angular momentum

Angular momentum of a body about its own center of mass.

Orbital angular momentum

Angular momentum of a body's center of mass about an external point.

Moment arm of momentum

The perpendicular distance from the origin to a particle's line of motion.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.