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23 March, 06:38

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.

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  1. 23 March, 06:40
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    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.
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