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Angular momentum and angular impulse review

Review how both rotating objects and objects with linear momentum can have angular momentum.  Recap how torque applied to and object over a time interval can change the angular momentum of an object.

Key terms

Term (symbol)Meaning
Angular momentum (L)Measure of how much rotational motion and rotational inertia an object has. Vector quantity with SI units of start fraction, start text, k, g, end text, dot, start text, m, end text, squared, divided by, start text, s, end text, end fraction.
Angular impulse (delta, L)Change in angular momentum. Vector quantity with SI units of start fraction, start text, k, g, end text, dot, start text, m, end text, squared, divided by, start text, s, end text, end fraction.

Equations

EquationSymbol breakdownMeaning in words
L, equals, I, omegaL is angular momentum, I is rotational inertia, and omega is angular velocity.Angular momentum of a spinning object without linear momentum is proportional to rotational inertia and angular velocity.
L, equals, m, v, r, start subscript, \perp, end subscriptL is angular momentum, m is mass, v is linear velocity, and r, start subscript, \perp, end subscript is the perpendicular radius from a chosen axis to the mass's line of motion.Angular momentum of an object with linear momentum is proportional to mass, linear velocity, and perpendicular radius from an axis to the line of the object's motion.
delta, L, equals, tau, delta, tdelta, L is change of angular momentum, tau is net torque, and delta, t is time interval.Change in angular momentum is proportional to average net torque and the time interval the torque is applied.

How to find the angular momentum of an object moving in a straight line

People forget that an object moving in a straight line (having linear momentum) can have angular momentum. For example, let’s say we throw a ball at one end of a stick (see Figure 1). The stick can pivot around point O. When the ball hits the stick, the stick rotates.
Figure 1. Collision between a ball and stick that transfers angular momentum.
If the system of the ball and stick has no net external torque, the only way the stick could get angular momentum is from the ball during the collision. Thus, the ball must initially have some angular momentum. The ball’s angular momentum about point O before the collision is
L, start subscript, start text, b, a, l, l, end text, end subscript, equals, m, v, r

Common mistakes and misconceptions

People mistakenly think any external force acting on a system will change angular momentum. Angular momentum is changed by a net external torque, but not all forces cause a torque. To produce a torque tau, a force F must have a lever arm r and a component perpendicular to the lever arm.
tau, equals, r, F, start subscript, \perp, end subscript
For more detail about torque, refer to our torque and equilibrium article.

Learn more

For deeper explanations of angular momentum concepts, see our video about angular momentum and impulse.
To check your understanding and work toward mastering these concepts, check out our exercises:

Want to join the conversation?

  • blobby green style avatar for user leonardo1452
    The equation for angular momentum is given as:
    L = Iw (where "w" is omega)
    The unit for I or rotational inertia is "kg.m^2".
    The unit for omega is "rad/sec"
    The unit for angular momentum L is "kg.m^2/sec"
    When we multiply I with w (omega) what happens to the radians?
    (4 votes)
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  • blobby green style avatar for user BrianHamm00
    Why does Linear Impulse (ΔP) get its own variable (J) whereas Angular Impulse (ΔL) is just known as ΔL ?? Or did I miss a variable somewhere?
    (1 vote)
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  • blobby green style avatar for user hola
    what does it mean if the unit was kg*m^2 *s^-1?
    (1 vote)
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  • female robot amelia style avatar for user migrosxd
    "If the system of the ball and stick has no net external torque, the only way the stick could get angular momentum is from the ball during the collision. Thus, the ball must initially have some angular momentum"

    The conclusion ("thus") does not seem to be true. It could also be, that the ball (or any object) has just linear momentum.

    As an exercise, when a ball with a known linear and angular momentum collides with a system that only allows angular momentum, we can solve (either via "transition of momentum" for lack of a better word, or "transition of energy") what the resulting momentum or energy is.
    (1 vote)
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    • aqualine ultimate style avatar for user Ricardo Z. G.
      I think the reason it says there is Angular Momentum is because the ball is hitting the stick in a point of it's radius (some distance away from it's center.

      P = m * V, this would work if you were hitting the object in its center of mass, but you are awat from, so you have to take that in account, so:

      L = P * r
      (1 vote)