A bicycle wheel has a diameter of 63.6 cm and a mass of 1.80 kg. Assume that the

Question

A bicycle wheel has a diameter of 63.6 cm and a mass of 1.80 kg. Assume that the wheel is a hoop with all of the mass concentrated on the outside radius. The bicycle is placed on a stationary stand and a resistive force of 120 N is applied tangent to the rim of the tire.

(a) What force must be applied by a chain passing over a 8.97-cm-diameter sprocket in order to give the wheel an acceleration of 4.49 rad/s2?
_______________ N

(b) What force is required if you shift to a 5.68-cm-diameter sprocket?
_______________N

Explanation / Answer

A bicycle wheel has a diameter of 63.2 cm and a mass of 1.72 kg. Assume that the wheel is a hoop with all of the mass concentrated on the outside radius. The bicycle is placed on a stationary stand and a resistive force of 123 N is applied tangent to the rim of the tire.

(a) What force must be applied by a chain passing over a 8.97 cm diameter sprocket if the wheel is to attain an acceleration of 4.50 rad/s2?
in N

Torque = Inertia x alpha (angular acceleration rad/s^2)
Torque also equals force x radius

First of all lets find the inertia of the wheel.

hoop = mr^2 = 1.80 x 0.318^2 = 0.1820 kg-m^2

Torque = 0.1820 x 4.49 = 0.8172 Nm

0.7729/radius = force
0.8172/0.04485 = 18.222 N

What force must be applied to attain acc of 4.49 rad/s^2?

18.222 N

120 N x 0.318 m = 38.16 Nm

38.16/ 0.04485 = 850.836 N

Total = 18.222 + 850.836 = 869.05 N (answer)

(b) What force is required if the chain shifts to a 5.68 cm diameter sprocket?

0.8172/radius = force

0.8172/0.0284=28.775 N

again

120 N x 0.318 m = 38.16 Nm

38.16/ 0.0284 = 1343.662 N

Total = 28.775  +1343.662 = 1372.437 N (answer)

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