Question

A solid disk, spinning counter-clockwise, has a mass of `4 kg` and a radius of `3 m`. If a point on the edge of the disk is moving at `8 m/s` in the direction perpendicular to the disk's radius, what is the disk's angular momentum and velocity?

Answer 1

1. `omega=8/3 1/s`
2. `L=96pi (kg*m^2)/s`

Explanation:

For this equation, we need to find two things: 1. the angular velocity of the disk and 2. the angular momentum of the disk

Angular velocity, `omega`, is the rate of change of angular displacement, and it is the angular analogue to linear velocity. Likewise, these two share a relationship of `omega=v/r`
If we substitute our linear velocity of `8m/s` and our radius of `3m`into the equation:

`omega=v/r`

`omega=(8cancel(m)/s)/(3cancelm)`

`omega=8/3 1/s`

We now have the answer to part 1.

Part 2. is asking about angular momentum. One equation used to calculate this is `L=Iomega` where `L` is angular momentum, `I` is the moment of inertia, and `omega` is angular velocity.

The moment of inertia is an objects resistance to angular acceleration, and its calculation changes on the shape of the object. For a solid disk, the equation is
`I=1/2 Mr^2` where `M` is the object's mass and `r` is the radius of the disk. If we substitute in for those values, we see that

`I=1/2*4kg*3m^2`

`I=18kg*m^2`

If we substitute in this value into our first equation, we see that

`L=Iomega`

`L=18kg*m^2*(16pi)/3 1/s`

`L=96pi (kg*m^2)/s`