Q. 117 - Precalculus Enhanced with Graphing Utilities

Question

Pulleys Two pulleys, one with radius $r_1$ and the other with radius $r_2$, are connected by a belt. The pulley with radius $r_1$rotates at $v_1$revolutions per minute, whereas the pulley with

radius $r_2$ rotates at $v_2$ revolutions per minute. Show that $\frac{r_1}{r_2}=\frac{{\omega }_1}{{\omega }_2}$

Step-by-Step Solution

Answer

The relationship between the radius and angular speed of two pulleys is $\frac{r_1}{r_2}=\frac{{\omega }_1}{{\omega }_2}$

Step 1: Understand the Belt-Pulley System

When two pulleys are connected by a belt, the linear speed at the rim of each pulley must be equal (since the belt moves at a constant speed).

Step 2: Express the Linear Speed

Linear speed \(v = r\omega\), where \(r\) is the radius and \(\omega\) is the angular speed.
For pulley 1: \(v_1 = r_1\omega_1\)
For pulley 2: \(v_2 = r_2\omega_2\)

Step 3: Derive the Relationship

Since \(v_1 = v_2\):
\(r_1\omega_1 = r_2\omega_2\)
Therefore: \(\frac{r_1}{r_2} = \frac{\omega_2}{\omega_1}\)