Question

A wheel with radius 32 cm is rotating at a rate of 14 rev/sA wheel with radius 32 cm is rotating at a rate of 14 rev/s(a) What is the angular speed in radians per second?(b) In a time interval of 5 s, what is the angle in radians through which the wheel rotates?(c) At t=10 s the angular speed begins to increase at a rate of 1.3 rad/s/s. At t=15 s, what is the angular speed in radians per second?

Answer 1

(a) 88.0 rad/s

The angular speed of the wheel is

$\omega = 14 rev/s$

Keeping in mind that

1 revolution = $2\pi$ radians

The angular speed in radians/second can be found by solving the proportion:

$14 rev/s : x = 1 rev : 2 \pi rad$

From which we find

$x=\frac{14\cdot 2 \pi}{1}=88.0 rad/s$

(b) 440 radians

Assuming the wheel is rotating at constant angular speed, the angular displacement of the wheel at time t is given by

$\theta= \omega t$

where

$\omega=88.0 rad/s$ is the angular speed

t is the time

Substituting

t = 5 s

we find the angle through which the wheel has rotated after 5 seconds:

$\theta=(88.0)(5)=440 rad$

(c) 94.5 rad/s

The angular speed after a time t is given by

$\omega(t) = \omega_o + \alpha t$

where

$\omega_0=88.0 rad/s$ is the angular speed at t=10 s, when the acceleration starts

$\alpha = 1.3 rad/s^2$ is the angular acceleration

The duration of the acceleration is

t = 15 s - 10 s = 5 s

So substituting this value into the equation, we find the new angular speed:

$\omega(15) = 88.0+(1.3)(5)=94.5 rad/s$