Since
$\overrightarrow{r}$ is constant, the term
$\overrightarrow{\theta}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\frac{d\overrightarrow{r}}{dt}=0$ . Since
$\overrightarrow{v}=\frac{d\overrightarrow{s}}{dt}$ is the tangential velocity and
$\overrightarrow{\omega}=\frac{d\overrightarrow{\theta}}{dt}$ is the angular velocity, we have
That is, the tangential velocity is the cross product of the angular velocity and the position vector, as shown in
[link] . From part (a) of this figure, we see that with the angular velocity in the positive
z -direction, the rotation in the
xy -plane is counterclockwise. In part (b), the angular velocity is in the negative
z- direction, giving a clockwise rotation in the
xy- plane.
Rotation of a flywheel
A flywheel rotates such that it sweeps out an angle at the rate of
$\theta =\omega t=(45.0\phantom{\rule{0.2em}{0ex}}\text{rad}\text{/}\text{s)}t$ radians. The wheel rotates counterclockwise when viewed in the plane of the page. (a) What is the angular velocity of the flywheel? (b) What direction is the angular velocity? (c) How many radians does the flywheel rotate through in 30 s? (d) What is the tangential speed of a point on the flywheel 10 cm from the axis of rotation?
Strategy
The functional form of the angular position of the flywheel is given in the problem as
$\theta (t)=\omega t$ , so by taking the derivative with respect to time, we can find the angular velocity. We use the right-hand rule to find the angular velocity. To find the angular displacement of the flywheel during 30 s, we seek the angular displacement
$\text{\Delta}\theta $ , where the change in angular position is between 0 and 30 s. To find the tangential speed of a point at a distance from the axis of rotation, we multiply its distance times the angular velocity of the flywheel.
Solution
$\omega =\frac{d\theta}{dt}=45\phantom{\rule{0.2em}{0ex}}\text{rad}\text{/}\text{s}$ . We see that the angular velocity is a constant.
By the right-hand rule, we curl the fingers in the direction of rotation, which is counterclockwise in the plane of the page, and the thumb points in the direction of the angular velocity, which is out of the page.
In 30 s, the flywheel has rotated through quite a number of revolutions, about 215 if we divide the angular displacement by
$2\pi $ . A massive flywheel can be used to store energy in this way, if the losses due to friction are minimal. Recent research has considered superconducting bearings on which the flywheel rests, with zero energy loss due to friction.
We have just discussed angular velocity for uniform circular motion, but not all motion is uniform. Envision an ice skater spinning with his arms outstretched—when he pulls his arms inward, his angular velocity increases. Or think about a computer’s hard disk slowing to a halt as the angular velocity decreases. We will explore these situations later, but we can already see a need to define an
angular acceleration for describing situations where
$\omega $ changes. The faster the change in
$\omega $ , the greater the angular acceleration. We define the
instantaneous angular acceleration$\alpha $ as the derivative of angular velocity with respect to time:
Questions & Answers
a length of copper wire was measured to be 50m with an uncertainty of 1cm, the thickness of the wire was measured to be 1mm with an uncertainty of 0.01mm, using a micrometer screw gauge, calculate the of copper wire used
When paddling a canoe upstream, it is wisest to travel as near to the shore as possible. When canoeing downstream, it may be best to stay near the middle. Explain why?
one ship sailing east with a speed of 7.5m/s passes a certain point at 8am and a second ship sailing north at the same speed passed the same point at 9.30am at what distance are they closet together and what is the distance between them then