Angular momentum conservation explains how rotating objects keep or change their spin when external torque is absent. This reference helps students connect rotational motion to linear momentum ideas they already know. It is useful for solving problems about spinning disks, skaters, planets, collisions, and rotating systems.
The key goal is to identify the system, check external torque, and apply conservation correctly.
The central idea is that angular momentum is conserved when the net external torque is zero. For a rigid body rotating about a fixed axis, angular momentum is often written as , where is moment of inertia and is angular velocity. For a particle, angular momentum is with magnitude .
Changes in angular momentum follow , which is the rotational form of Newton's second law.
Key Facts
- Angular momentum for a particle is , and its magnitude is .
- For a rigid object rotating about a fixed axis, angular momentum is .
- Angular momentum is conserved when the net external torque is zero, so .
- For changing rotational inertia with no external torque, conservation gives .
- Net torque changes angular momentum according to .
- For constant torque over a time interval, angular impulse equals change in angular momentum: .
- Moment of inertia depends on mass distribution, so moving mass farther from the axis increases and usually decreases if is conserved.
- Angular momentum is a vector, and its direction is found using the right-hand rule.
Vocabulary
- Angular Momentum
- A vector measure of rotational motion, given by for a particle or for a rigid body about a fixed axis.
- Torque
- A rotational effect of a force, given by with magnitude .
- Moment of Inertia
- A measure of how difficult it is to change an object's rotational motion, depending on mass and how far the mass is from the axis.
- Conservation Law
- A rule stating that a quantity stays constant in a closed system when no external interaction changes it.
- External Torque
- A torque caused by forces from outside the chosen system, which can change the system's total angular momentum.
- Right-Hand Rule
- A method for finding the direction of angular velocity, torque, or angular momentum by curling the fingers of the right hand in the rotation direction.
Common Mistakes to Avoid
- Using for every situation is wrong because that form only works cleanly for a rigid body rotating about a fixed axis. For particles or orbital motion, use .
- Ignoring external torque is wrong because angular momentum is conserved only when . Always define the system and check whether outside forces create a torque.
- Treating angular momentum as a scalar is wrong because has direction. Opposite rotation directions must have opposite signs in one-dimensional rotational problems.
- Assuming angular speed stays constant when moment of inertia changes is wrong. If angular momentum is conserved, , so increasing decreases .
- Confusing torque with force is wrong because torque depends on lever arm and angle. Use , not just the size of the force.
Practice Questions
- 1 A skater has and spins at . If the skater pulls in their arms so , what is ?
- 2 A disk with rotates at . What is its angular momentum ?
- 3 A particle of mass moves at in a straight line that passes from a point. What is its angular momentum about that point if the velocity is perpendicular to the radius at closest approach?
- 4 A student says angular momentum is conserved for a spinning wheel because gravity acts on it. Explain why the correct decision depends on the chosen axis and whether gravity creates an external torque about that axis.
Understanding Angular Momentum Conservation Reference
Torque is not simply a turning force. Its effect depends on where the force acts and which way it acts. A door is easiest to rotate when you push near the handle, far from the hinges.
The same push near the hinges produces much less turning effect. A force aimed directly toward the axis produces no turning effect because its line of action passes through the axis.
This is why students should draw the axis first, then mark the point where each force acts. Gravity can create torque on a tilted object, but it creates no torque about an axis that passes through the object’s center of mass in the right arrangement.
Choosing the system is often the hardest part of a conservation problem. Consider a student standing on a low friction rotating platform while holding two masses. The student, masses, and platform should usually be treated as one system.
Forces between the student’s arms and the masses are internal forces. They can redistribute angular momentum within the system, but they do not change the total amount. Friction at the platform bearing may be small enough to ignore for a short time.
If a hand pushes on a wall, however, the wall is outside the chosen system and can provide an external torque. The conservation rule then may not apply to the student and platform alone.
Rotational collisions need extra care because angular momentum and kinetic energy follow different rules. When a lump of clay strikes a rotating disk and sticks, the collision is inelastic. The total angular momentum about a suitable axis can remain constant if outside torques are negligible during the short collision.
Some kinetic energy becomes heat, sound, and deformation. Students should not assume that rotational kinetic energy stays the same just because angular momentum does.
Pick an axis where unknown external forces have little or no lever arm. For example, a support force through a fixed axle has no torque about that axle, even though the force itself may be large.
Direction matters throughout rotational motion. A clockwise rotation seen from one side has the opposite angular momentum direction from the same rotation seen from the other side. Use the right hand rule consistently, with curled fingers following the rotation and the thumb giving the vector direction.
This matters when two rotating objects join, since their angular momenta may add or partly cancel. A spinning bicycle wheel can resist a change in its orientation because changing its direction requires torque. Planets provide a larger example.
Their orbital motion carries angular momentum, and gravity supplies a torque-free central force about the Sun. In problem solving, state the axis, list external torques about that axis, keep track of direction, then decide whether the situation involves conservation or a torque acting over time.