Question

A bicyclist notes that the pedal sprocket has a radius of rp = 9.5 cm while the wheel sprocket has a radius of rw = 6.5 cm. The two sprockets are connected by a chain which rotates without slipping. The bicycle wheel has a radius R = 65 cm. When pedaling the cyclist notes that the pedal rotates at one revolution every t = 1.1 s. When pedaling, the wheel sprocket and the wheel move at the same angular speed.

Answer 1

Explanation:

Since the chain isn't slipping, the pedal sprocket and wheel sprocket have the same linear velocity.

First, we find the angular velocity of the pedal sprocket:

1.1 rev/s × 2π rad/rev = 2.2π rad/s ≈ 6.9 rad/s

Next, we find the linear velocity of the pedal sprocket:

v = ωr = (6.9 rad/s)(9.5 cm) = 66 cm/s

Now we find the angular velocity of the wheel sprocket:

ω = v/r = (66 m/s) / (6.5 cm) = 10. rad/s

So the linear velocity of the bike is:

v = ωr = (10. rad/s) (0.65 m) = 6.6 m/s

The linear velocity of the bike is proportional to the angular velocity of the pedal.  So if the bicyclist wanted to move at a different speed, say 5.5 m/s, we could find the new angular velocity of the pedal by writing a proportion:

1.1 rev/s / 6.6 m/s = x / 5.5 m/s

x = 0.92 rev/s

Or, 1/x = 1.1 seconds per revolution.