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 rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together
and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec
differentiate above equation w.r.t
time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find
* The angular acceleration
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
the velocity of a boat related to water is 3i+4j and that of water related to earth is i-3j. what is the velocity of the boat relative to earth.If unit vector i and j represent 1km/hour east and north respectively