Question

A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration. At one time it is rotating at 10 rev/s; 60 revolutions later, its angular speed is 15 rev/s. Calculate

(a) the angular acceleration,
(b) the time required to complete the 60 revolutions,
(c) the time required to reach the 10 rev/s angular speed, and
(d) the number of revolutions from rest until the time the disk reaches the 10 rev/s angular speed.

Answer 1

Step-by-step explanation:

Given:

`\omega_0` = 10 rev/s =
`20\pi\:\text{rad/s}`

`\omega` = 15 rev/s =
`30\pi\:\text{rad/s}`

`\theta` = 60 rev =
`120\pi\:\text{rads}`

a) the angular acceleration
`\alpha` is given by

`\alpha = (\omega^2 - \omega_0^2)/(2\theta)`

`\:\:\:\:\:\:\:=((30\pi)^2 - (20\pi)^2)/(240\pi) = 6.5\:\text{rad/s}^2`

b)
`t = (\omega - \omega_0)/(\alpha) = (30\pi - 20\pi)/(6.5) = 4.8\:\text{s}`

c)
`t = (\omega - \omega_0)/(\alpha)`

`=(20\pi - 0)/(6.5) = 9.7\:\text{s}`

d)
`\theta = (1)/(2)\alpha t^2`

`\:\:\:\:\:\:\:=(1)/(2)(6.5\:\text{rad/s}^2)(9.7\:\text{s})^2 = 305.8\:\text{rad}`

`\:\:\:\:\:\:\:= 48.7\:\text{revs}`