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The Relationship Between Angular Momentum & Torque

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  • 0:00 What Is Rotational Motion?
  • 1:10 Angular Momentum
  • 1:57 Torque And Angular Momentum
  • 3:46 The Mathematical Relationship
  • 4:56 Lesson Summary
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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.

In this lesson, we'll explore how torque and angular momentum affect objects in rotational motion. In doing this, we'll also discover the important relationship between torque and angular momentum.

What Is Rotational Motion?

In a physics course, we often deal with an object moving from one place to another. You might see problems that involve situations such as a ball being dropped from a roof or a book sliding down a ramp. These are examples of translational motion, meaning an object travels from one point to another. However, this is not the only type of motion an object can experience.

Another common type of motion we see in physics is rotational motion. Rotational motion happens when an object spins around an axis. You can see rotational motion in objects such as a bicycle wheel spinning around its center point or the Earth's rotation around its poles.

When we compare translational and rotational motion, we find that the two have similar physical quantities. For example, translational motion has linear velocity, force, and linear momentum. Correspondingly, rotational motion has angular velocity, torque, and angular momentum. In this lesson, we're going to focus on angular momentum and its relationship with torque.

Angular Momentum

As a child, have you ever played with a spinning top? You might remember that the faster you spin it, the longer it takes to stop. This is due to angular momentum. The more angular momentum an object has, the more it wants to keep rotating.

You might be familiar with the formula for linear momentum (p) as mass (m) times velocity (v). Angular momentum works similarly but uses physical quantities specific to rotational momentum. Angular momentum (L) is defined as moment of inertia (I) times angular velocity (omega). An object's moment of inertia is a measurement of its ability to resist angular acceleration. In this way it works similarly to mass for linear momentum.

momentum equations

Torque and Angular Momentum

Now that we know what angular momentum is, let's start looking at how torque affects it. As we saw in the beginning of this lesson, torque is the equivalent of force for rotational motion. In fact torque can be described as a twisting force that causes rotation. This is a little misleading, however, as torque isn't exactly a force. Torque (tau) is a force (F) multiplied by a radius (r) multiplied by the sine of the angle (theta) at which the force is applied.

Torque equation
torque equation

So how are torque and angular momentum related? To see this, we need to think about how objects in rotational motion get moving in the first place. Have you ever thought about what gets a wind turbine spinning? You know it's the wind, but what exactly is the wind doing? It's pushing on the wind turbine's blade, applying a force to the blade at some angle and radius from the turbine's axis of rotation. In other words, the wind is applying a torque to the wind turbine.

Torque is what gets rotatable objects spinning when they're standing still. Also, if a torque is applied to an already spinning object in the direction it's spinning, it increases its angular velocity. If it's applied in the opposite direction the object is spinning, it decreases its angular velocity. So torque is directly affecting the angular velocity of a spinning object.

Remember that our angular momentum equation tells us that angular momentum is moment of inertia multiplied by angular velocity. Since torque can change angular velocity, and the amount of angular momentum an object has depends on its angular velocity, it makes sense that torque can change angular momentum. This is how the two are related.

Mathematical Relation for Torque and Angular Momentum

Finally, we can take what we just learned to relate angular momentum and torque mathematically as well. We know torque can change angular momentum, but in what way? Let's think about a merry-go-round like you might see at an older playground. At the start, the merry-go-round is standing still; it has zero angular momentum.

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