# Angular Momentum

### Also known as Moment of Momentum and Rotational Momentum

### Written by tutor Bibhaw P.

Often referred to as rotational equivalent of Linear Momentum. Angular momentum is a vector quantity. For an object rotating about an axis, angular momentum (L) is represented by a the product of the object’s Rotational Inertia (I) and its Angular Velocity (ω).

**L=Iω ⇒ I**

The definition could be extended to a point mass (an object whose radius is much smaller as compared to its distance from the axis of rotation). Angular momentum in this case is equal to the cross product of an objects’ linear momentum (mv) and its distance from the axis of rotation (r).

**L=r x mv (where m and v are mass and velocity of the object) ⇒ II**

The unit of Angular Momentum is Kg*m^{2}/s^{2}

### Angular Momentum Example 1

What is the angular momentum of a thin hoop of radius 2 m and mass 1 kg that is rotating at a velocity of 4 rad/s?

### Solution

Given: r= 2m , m=1 Kg and ω= 4 rad/s

Formula to use : L = Iω

Recall Moment of inertia of a loop is just mr^{2}, then

L= mr^{2} ω = 1.2^{2}. 4 = 16 Kg m^{2}/s^{2}

### Test Your Understanding

What is the angular momentum of a solid cylindrical object with mass 5 Kg and radius 6 inches rotating with a velocity of 14 rpm about a vertical axis thru the center.

(Hint: convert all the units to the standard units i.e distance in meters , time in seconds, angular velocity in rad/sec. NOTE: rpm= revolution per min and 1 rev= 2 π rad)

(Scroll to the end of the page for the answer.)

## Conservation of Angular Momentum

A very important concept in Physics. Ever wonder why an iceskater folds his/her hands when he/she is trying to spin faster? Ever wonder why comets gains speed as they approach the Sun? And ever wonder why when you add an extra mass in a spinning record disc it rotates slowly? etc. Well all these scenarios could be answered using the concept of Conservation of Angular Momentum, which states that:

"If all the external torques acting on an object is negligible, then the angular momentum acting on an object is always conserved/constant."

ΣT_{torque} = __Change in Momentum__ = __dL__

Change in time dt

If T_{torque} = 0, then, __dL__ = 0

dt

Or in other words momentum is constant. We can also represent it as

**I _{i}ω_{i} = I_{f}ω_{f} ⇒ III**

Where I_{i} and ω_{i} are initial Moment of Inertia and Angular Velocity while I_{f} and ω_{f}
are final Moment of Inertia and Angular Velocity respectively.

### Angular Momentum Example 2

A disk is spinning at a rate of 10 rad/s. A second disk of the same mass and shape, with no spin, is placed on top of the first disk. Friction acts between the two disks until both are eventually traveling at the same speed. What is the final angular velocity of the two disks?

### Solution

Recall, this is different from a problem we did in the last section. Instead of one object rotating, now we have two objects interacting in the same motion. That should always be a hint as to one must use momentum conservation (specifically angular momentum) to solve this problem.

L_{i} = L_{f}

**I _{i}ω_{i} = I_{f}ω_{f}**

Recall the moment if inertia of a disc rotating about an axis about its center is ^{1}/_{2} mr^{2}. Since both of those disc are identical, they would
have same moment of inertia. In that case the final moment of inertia when both of those disc are rotating together is sum of individual
moment of inertias, which would basically be equal to 2(^{1}/_{2}mr^{2}). We could now easily solve the problem as

. 10 = 2^{1}/_{2}mr^{2}~~(~~. ω^{1}/_{2}mr^{2})_{f}

ω_{f} = 5 rad/s

The combination of two disc would be rotating slower, as we predicted to conserve the momentum.

### Test Your Understanding

A particle attached to a string of length 2 m is given an initial velocity of 6 m/s. The string is attached to a peg and, as the particle rotates about the peg, the string winds around the peg. What length of string has wound around the peg when the velocity of the particle is 20 m/s?

Scroll to the bottom to see the answer.

## Conceptual Problems

In an isolated system the moment of inertia of a rotating object is doubled. What happens to angular velocity of the object?

**A.**Doubled

**B.**Halved

**C.**Tripled

**D.**Remains the same

**B**.

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?

**A.**It takes less time to stop the ping pong

**B.**It takes the same time to stop both balls

**C.**It takes more time to stop the ping pong ball

**D.**It is impossible to calculate

**B**.

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the distances needed to stop them compare?

**A.**It takes a shorter distance to stop the ping pong ball

**B.**It takes a longer distance to stop the ping pong ball

**C.**It takes the same distance to stop the ping pong ball

**D.**It is impossible to calculate

**B**.

A cart moving at speed v collides with an identical stationary cart on an air track, and the two stick together after the collision. What is their velocity after colliding?

**A.**v

**B.**-v

**C.**0.5v

**D.**2v

**C**.

Answers to the Test Your Understanding problems:

1. 0.085 kg*m^{2}/s^{2}

2. 0.6 m

References

1. SparkNote on Angular Momentum.” SparkNotes.com. SparkNotes LLC. n.d.. Web. 29 Apr. 2013

2. MITOpenCourseware. ocw.mit.edu.n.d.web.29 Apr.2013.