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.

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.