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MTEL Physics: Practice & Study Guide21 chapters | 174 lessons

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Lesson Transcript

Instructor:
*Damien Howard*

Damien has a master's degree in physics and has taught physics lab to college students.

Learn what we mean when we say something experiences an angular impulse. Then discover the relationship between angular impulse and change in angular momentum.

When we work with forces in a physics course, we often think about them being applied to objects instantaneously. A golf club hits a golf ball and applies a force to it all at once, sending it flying down the fairway. Does that sound about right to you? Well as it turns out things are a little more complicated than that. Forces tend to be applied not instantaneously, but over some time period. Think about a nurse pushing a patient in a wheelchair down a hospital hallway. The nurse doesn't apply a force all at once, sending the patient zooming down the hallway, but instead applies it slowly over a period of time as they walk with the patient while pushing them.

Just like with the nurse, the golf club also exerts its force on the golf ball over the course of time. The difference is that the time interval here is a fraction of a second. In physics when we have a force applied to an object over a period of time, we say the object is experiencing an impulse. We write **impulse** as net force (*F*) multiplied by a change in time (*t*).

Now going back to our golf ball example, there is one more thing we've been ignoring up to this point. A ball hit by a club is often spinning while it flies through the air. In order to get the ball spinning, a torque must have been applied to it. **Torque** is the equivalent of force for rotational motion. So just like how net force multiplied by a change in time creates an impulse, a net torque (*tau*) multiplied by a change in time creates an **angular impulse**. In this lesson, we're going to look at angular impulse and see how it's related to a change in angular momentum.

In order to get the relationship between angular momentum and angular impulse, we first must look at Newton's 2nd Law. You may already be familiar with **Newton's 2nd law** as net force equals mass (*m*) times acceleration (*a*). What you might not know is that this is for translational motion, movement from one point to another. We can also get a version of Newton's 2nd law for rotational motion. To get this, we have to first define moment of inertia and angular acceleration.

**Moment of inertia** is a measurement of an object's ability to resist angular acceleration. An object's moment of inertia is dependent on its shape and how it is rotating. **Angular acceleration** is a change in angular velocity (*omega*) divided by a change in time. Using these three quantities, we get **Newton's 2nd Law for rotational motion** to be net torque equals moment of inertia (*I*) times angular acceleration (*alpha*).

To get this equation we replaced the three quantities in the translational motion formula for Newton's 2nd law with their rotational equivalents. So torque replaces force, moment of inertia replaces mass, and angular acceleration replaces acceleration.

You've probably spun a coin on its side. You know the faster you spin the coin, the longer it will stay up before falling over. **Angular momentum** is the measurement of an object's ability to keep spinning. So the faster you spin the coin, the more angular momentum it has, which keeps it wanting to spin longer.

All we need to see how angular momentum and angular impulse are related is some rearranging of the Newton's 2nd law for rotational motion formula. Let's go over step by step how Newton's 2nd law equation is rearranged:

Write Newton's 2nd law for rotational motion in terms of angular velocity. Remember, angular acceleration is a change in angular velocity divided by a change in time.

Write a change in angular velocity as final angular velocity minus the initial angular velocity.

Multiply both sides of the equation by change in time in order to move it from the right side of the equation to the left. Now we have the formula for angular impulse on the left of the equation.

Multiply the moment of inertia through the parenthesis to get moment of inertia times final angular velocity minus moment of inertia times initial angular velocity.

We've already seen that moment of inertia is the rotational equivalent of mass, and much like how momentum is mass times velocity, angular momentum (*L*) is moment of inertia times angular velocity.

Now let's look back at the formula we currently have for angular impulse. We can see it's equal to moment of inertia times final angular velocity minus moment of inertia times initial angular velocity.

Use the formula for angular momentum to get angular impulse equal to final angular momentum minus initial angular momentum. A final angular momentum minus an initial angular momentum is a change in momentum. This means we've found out that the relation between angular momentum and angular impulse is that an angular impulse is equal to a change in angular momentum.

Forces and torques applied to objects generally do not happen instantaneously, but over a period of time. A net force applied to an object over a period of time is called impulse, and a net torque applied to an object over a period of time is called **angular impulse**. We represent these mathematically as net force (*F*) times a change in time (*t*), and net torque (*tau*) times a change in time respectively.

It turns out that angular impulse has a unique relationship with **angular momentum**, which is the measurement of an object's ability to keep spinning. To see this relationship we need to look at **Newton's 2nd Law for rotational motion**, net torque equals moment of inertia (*I*) times angular acceleration (*alpha*).

If we rearrange the formula for Newton's 2nd law for rotational motion, we find that angular impulse is actually equal to a change in angular momentum.

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MTEL Physics: Practice & Study Guide21 chapters | 174 lessons

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