Question

A gear with a diameter of 12.0 inches makes 35 revolutions every three minutes. Find the linear and angular velocities of a point on the outer edge of the gear. Give your answers in both exact and approximate forms.

Answer 1

Angular velocity:
`\omega = (7\pi)/(18) \,(rad)/(s)` (
`1.222\,(rad)/(s)`)

Linear velocity:
`v = (14\pi)/(3)\,(m)/(s)` (
`14.661\,(m)/(s)`)

Explanation:

The gear experiments a pure rotation with axis passing through its center, the angular (
`\omega`), in radians per second, and linear velocities (
`v`), in inches per second, of a point on the outer edge of the element are, respectively:

`\omega = (2\pi)/(60)\cdot \dot n` (1)

`v = R\cdot \omega` (2)

Where:

`\dot n` - Rotation rate, in revolutions per minute.

`R` - Radius of the gear, in inches.

If we know that
`\dot n = (35)/(3)\,(rev)/(min)` and
`R = 12\,in`, then the linear and angular velocities of the gear are, respectively:

`\omega = (2\pi)/(60)\cdot \dot n`

`\omega = (7\pi)/(18) \,(rad)/(s)` (
`1.222\,(rad)/(s)`)

`v = R\cdot \omega`

`v = (14\pi)/(3)\,(m)/(s)` (
`14.661\,(m)/(s)`)