Angular speed of 2 pulleys on a belt

[karush]

two pulleys connected by a belt have 15cm and 8cm radius

The larger pulley rotates $25$ times in $36$ sec,

Find the angular speed of each pulleey in radians per second.

the 15cm pulley has circumferce of $30\pi$ so

$\displaystyle\frac{25\text { rev}}{36 \text {sec}}
\cdot\frac{30\pi\text{ cm}}{ rev}
=\frac{750\text{ cm\pi}}{36\text {sec}}
=\frac{65.5\text{ cm}\text{ rad}}{\text{sec}}$

not sure how to get the v of the \(\displaystyle 8cm \) pulley

 

[MarkFL]

Re: angular speed of 2 pulleys on a belt

This is how I would work the first part:

\(\displaystyle \frac{25\text{ rev}}{36\text{ s}}\cdot\frac{2\pi\text{ rad}}{1\text{ rev}}=\frac{25}{18}\pi\frac{\text{rad}}{\text{s}}\)

Angular speed should have units of radians/time.

Since the pulleys are connected by a belt, then the linear velocity of the outer edge of each pulley will be the same:

\(\displaystyle v_2=v_1\)

Using, \(\displaystyle v=r\omega\), we may state:

\(\displaystyle r_2\omega_2=r_1\omega_1\)

Solve for \(\displaystyle \omega_2\):

\(\displaystyle \omega_2=\frac{r_1}{r_2}\omega_1\)

Now let \(\displaystyle r_1=15\text{ cm},\,r_2=8\text{ cm},\,\omega_1=\frac{25}{18}\pi\frac{ \text{rad}}{\text{s}}\)

What do you find?

 

[karush]

Re: angular speed of 2 pulleys on a belt

$\displaystyle\frac{15}{8}\cdot\frac{25}{18}\pi \text{ = } \frac{125}{48}\pi\ \frac{\text{rad}}{s}$