# Angular velocity and rotation

• Dec 30th 2006, 05:07 AM
totalnewbie
Angular velocity and rotation
Before braking angular velocity was 300 rad per second. It took 10 seconds to stop. Braking was steady. How many rotations did wheel ?
• Dec 30th 2006, 06:08 AM
topsquark
Quote:

Originally Posted by totalnewbie
Before braking angular velocity was 300 rad per second. It took 10 seconds to stop. Braking was steady. How many rotations did wheel ?

$\omega_0 = 300 \, rad/s$ and the wheel stopped in $t = 10 \, s$, so $\omega = 0 \, rad/s$. The braking was "steady" which I will take to mean that the angular acceleration $\alpha$ was constant. We wish to know what angle the wheel rotated through during this time period, so I'm going to set $\theta _0 = 0 \, rad$.

One way to do this is to use:
$\omega = \omega _0 + \alpha t$
to find the $\alpha$ then use:
$\theta = \theta _0 + \omega _0 t + \frac{1}{2} \alpha t^2$
to find $\theta$.

The more direct way is to use:
$\theta = \theta _0 + \frac{1}{2}( \omega _0 + \omega )t$

$\theta = \frac{1}{2} \omega _0 t$

$\theta = \frac{1}{2} (300 \, rad/s) (10 \,s ) = 1500 \, rad$

Now we need to find out how many revolutions this is. This is a simple unit conversion: there are $2 \pi \, rad$ for every revolution. Thus:
$\frac{1500 \, rad}{1} \cdot \frac{1 rev}{2 \pi \, rad} \approx 238.732 rev$