MHB Angular speed of 2 pulleys on a belt

[RobertoPink]

The two pulleys connected by a belt have a radii of 15 cm and 8 cm. The larger pulley rotates 24 times in 36 seconds.

a. Find the angular velocity of the small pulley in radians per second.

b. Find the linear velocity of a point on the belt that connects the two pulleys in centimeters per second.

 

[skeeter]

The two pulleys connected by a belt have a radii of 15 cm and 8 cm. The larger pulley rotates 24 times in 36 seconds.

a. Find the angular velocity of the small pulley in radians per second.

b. Find the linear velocity of a point on the belt that connects the two pulleys in centimeters per second.

(a) $r \omega_1 = R\omega_2$

$r = 8 \, cm$, $R = 15 \, cm$, $\omega_2 = \dfrac{2}{3} \, \text{rev per sec}$

you'll need to convert $\omega_2$ to radians per second

(b) note $v = r \omega_1 = R\omega_2$

 

[HOI]

The larger pulley has radius 15 cm so circumference $2\pi(15)= 30\pi$ cm. The larger pulley rotates 24 times in 36 seconds so at a rate of 24/36= 2/3 rotations per second. In one second, since the larger pulley has rotated 2/3 of a rotation, the belt has moved a distance or $(2/3)(30\pi)= 20\pi$ cm.

The smaller pulley has radius 8 cm so circumference $2\pi(8)= 16\pi$. When the belt moves $20\pi$ cm, it will have moved $\frac{16\pi}{20\pi}= \frac{4}{5}$ of one rotation of the smaller pulley. The smaller pulley is turning at 4/5 rotation per second. There are $2\pi$ radians in one rotation so that is a rate of $(4/5)(2\pi)= (8/5)\pi$ radians per second.

The linear velocity of the belt is the $20\pi$ cm/sec we got earlier.