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### Course: Mechanics (Essentials) - Class 11th>Unit 12

Lesson 6: Angular momentum

# 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 $\frac{\text{kg}\cdot {\text{m}}^{2}}{\text{s}}$.
Angular impulse ($\mathrm{\Delta }L$)Change in angular momentum. Vector quantity with SI units of $\frac{\text{kg}\cdot {\text{m}}^{2}}{\text{s}}$.

## Equations

EquationSymbol breakdownMeaning in words
$L=I\omega$$L$ is angular momentum, $I$ is rotational inertia, and $\omega$ is angular velocity.Angular momentum of a spinning object without linear momentum is proportional to rotational inertia and angular velocity.
$L=mv{r}_{\perp }$$L$ is angular momentum, $m$ is mass, $v$ is linear velocity, and ${r}_{\perp }$ 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.
$\mathrm{\Delta }L=\tau \mathrm{\Delta }t$$\mathrm{\Delta }L$ is change of angular momentum, $\tau$ is net torque, and $\mathrm{\Delta }t$ 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.
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
${L}_{\text{ball}}=mvr$

## 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.
$\tau =r{F}_{\perp }$
For more detail about torque, refer to our torque and equilibrium article.

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?

• 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?
• Radians have no units and act as a placeholder. In the case you specified we can simply remove them.
• "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.
• 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)
• 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)
• You didn't miss a variable anywhere. There just isn't any commonly used variable for Angular Impulse in physics.
(1 vote)
• what does it mean if the unit was kg*m^2 *s^-1?
(1 vote)
• A negative exponent means "the inverse of", so s^-1 is the same as 1/s.

The unit above is equivalent to (kg*m^2)/s
(1 vote)