A vacuum cleaner belt is looped over a shaft of radius 0.45 cm and a wheel of radius 1.80 cm. The arrangement of the belt, shaft, and wheel is similar to that of the chain and sprockets in Fig. Q9.4. The motor turns the shaft at 60.0 rev/s and the moving belt turns the wheel, which in turn is connected by another shaft to the roller that beats the dirt out of the rug being vacuumed. Assume that the belt doesn’t slip on either the shaft or the wheel. (a) What is the speed of a point on the belt? (b) What is the angular velocity of the wheel, in rad/s?

Solution 69P Step 1: Data given Radius of shaft R = 0.45 cm = 0.0045 m Radius of the wheel r = 1.80 cm = 0.018 m Angular velocity shaft = 60 rev/s × (2 rad/1rev) = 376.99 rad /s We need to find the speed of the belt It is obtained using v = R × Substituting values v = 0.0045 m × 376.99 rad /s v = 1.69 m /s This can be approximated to 1.70 m/s Thus we have the velocity of the belt as 1.70 m/s