A bicycle wheel has a diameter of 64.6 cm and a mass of 1.87 kg. Assume that the wheel is a hoop with all of its mass concentrated on the outside radius. The bicycle is placed on a stationary stand on rollers 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 8.93 cm diameter sprocket if the wheel is to attain an acceleration of 4.60 rad/s2? N(b) What force is required if the chain shifts to a 5.55 cm diameter sprocket?
Question
A bicycle wheel has a diameter of 64.6 cm and a mass of 1.87 kg. Assume that the wheel is a hoop with all of its mass concentrated on the outside radius. The bicycle is placed on a stationary stand on rollers 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 8.93 cm diameter sprocket if the wheel is to attain an acceleration of 4.60 rad/s2? N(b) What force is required if the chain shifts to a 5.55 cm diameter sprocket?
Solution
To solve this problem, we need to use the formula for torque (τ) which is the force (F) times the radius (r), and the formula for the moment of inertia (I) of a hoop which is mass (m) times radius squared (r^2). We also need to use Newton's second law for rotation which states that torque equals moment of inertia times angular acceleration (α).
(a) First, we need to calculate the radius of the bicycle wheel and the sprocket. The radius of the wheel is half of the diameter, so it's 64.6 cm / 2 = 32.3 cm = 0.323 m. The radius of the sprocket is 8.93 cm / 2 = 4.465 cm = 0.04465 m.
The moment of inertia of the wheel is I = mr^2 = 1.87 kg * (0.323 m)^2 = 0.195 kgm^2.
The torque needed to attain the acceleration is τ = Iα = 0.195 kgm^2 * 4.60 rad/s^2 = 0.897 N*m.
The force applied by the chain is F = τ / r = 0.897 N*m / 0.04465 m = 20.1 N.
(b) If the chain shifts to a 5.55 cm diameter sprocket, the radius of the sprocket is 5.55 cm / 2 = 2.775 cm = 0.02775 m.
The force required is F = τ / r = 0.897 N*m / 0.02775 m = 32.3 N.
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