Question

A bicycle wheel has a diameter of 63.0 cm and a mass of 1.74 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 117 N is applied tangent to the rim of the tire.

(a) What force must be applied by a chain passing over a 8.99-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 5.70-cm-diameter sprocket?
kN

Answer 1

A bicycle wheel has a diameter of 63 cm and a mass of 1.74 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 117 N is applied tangent to the rim of the tire.

(a) What force must be applied by a chain passing over a 8.99 cm diameter sprocket if the wheel is to attain an acceleration of 4.49 rad/s2?
in N

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

First of all lets find the inertia of the wheel.

hoop = mr² = 1.74 x 0.315² = 0.17265 kg-m²

Torque = 0.17265 x 4.49 = 0.7752 Nm

0.7752/radius = force
0.7752/0.04495 = 17.246 N

What force must be applied to attain acc of 4.49 rad/s²?

17.246 N

117 N x 0.315 m = 36.855 Nm

36.855/ 0.04495 = 819.911 N

Total = 17.246 + 819.91 = 837.157 N (answer)

(b) What force is required if the chain shifts to a 5.70 cm diameter sprocket?

Shifting to 5.70 cm gives the result

4.49 = (F*0.0285 - 117*0.315)/0.17265

F = 1320.3578 N