## SAT Physics Subject Test

## Chapter 6 Curved and Rotational Motion

### ANGULAR MOMENTUM

So far we”ve developed rotational analogs for displacement, velocity, acceleration, and force. We will finish by developing a rotational analog for linear momentum; it”s called **angular momentum**.

Consider a small point mass *m* at distance *r* from the axis of rotation, moving with velocity **v** and acted upon by a tangential force **F**.

Then, by Newton”s second law

If we multiply both sides of this equation by *r* and notice that *r***F** = τ, we get

Therefore, to form the analog of the law **F** = ∆*p*/∆*t* (force equals the rate-of-change of linear momentum), we say that torque equals the rate-of-change of angular momentum, and the angular momentum (denoted by *L*) of the point mass *m* is defined by the equation

*L* = *rm*v

If the point mass *m* does not move in a circular path, we can still define its angular momentum relative to any reference point.

If **r** is the vector from the reference point to the mass, then the angular momentum is

*L* = *rm*v_{⊥}

where **v**_{⊥} is the component of the velocity that”s perpendicular to **r**.

For a rotating object, the angular momentum equals the sum of the angular momentum of each individual particle. This can be written as *L*=*Iω*, where *I* is the object”s **moment of inertia** and *ω* is the angular velocity (to be discussed later). *I* is basically a measure of how difficult it is to start an object rotating (analogous to mass in the translational world). *I* increases with mass and average radius from the axis of rotation.