Question

A bug traveling with tangential velocity `v` sits halfway between the axis and the edge of a phonograph record. What will happen to its tangential speed if the RPM rate is tripled? What will happen to its tangential speed if it crawls out to the edge?

Answer 1

1) Tangential velocity becomes three times, 2) Tangential velocity becomes two times

Explanation:

The tangential velocity `v` of bug sitting at radial distance `r` from axis of rotation on the phonograph record which rotates at `N\ \text{rpm}` is given as

`v=\frac{2\pi Nr}{60}`

`v\prop N`

`v_2/v_1=N_2/N_1`

Keeping `r` constant when speed is tripled i.e. `N_2=3N_1` then tangential velocity `v_2` is given as

`v_2/v={3N_1}/N_1`

`v_2=3v`

The tangential speed `v_2` becomes three times the initial tangential velocity `v`

Now, again from above equation

`v=\frac{2\pi Nr}{60}`

`v\prop r`

`\frac{v_2}{v_1}=\frac{r_2}{r_1}`

when bug travels from half way `r=R/2` to the edge `r=R` the tangential velocity `v_2` is given as

`\frac{v_2}{v}=\frac{R}{R/2}`

`v_2=2v`

The tangential velocity becomes two times the original velocity `v`