A bicycle wheel has a diameter of 64.2 cm and a mass of 1.79 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 115 N is applied tangent to the rim of the tire.
A.What force must be applied by a chain passing over a 9.06 cm diameter sprocket if the wheel is to attain an acceleration of 4.41 rad/s2? (in Newtons)
B.What force is required if the chain shifts to a 5.54 cm diameter sprocket? (in Newtons)
To solve this problem, we will use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
A. To find the force required for a 9.06 cm diameter sprocket, we first need to determine the moment of inertia of the wheel. The moment of inertia for a hoop is given by I = m * r^2, where m is the mass and r is the radius.
Given:
Diameter of the wheel = 64.2 cm
Radius of the wheel (r) = 64.2 cm / 2 = 32.1 cm = 0.321 m
Mass of the wheel (m) = 1.79 kg
Using the formula for moment of inertia, we can calculate:
I = m * r^2
I = 1.79 kg * (0.321 m)^2
I = 0.183 kg·m^2
Next, we can use Newton's second law to find the force. The net force is given by F = I * α, where α is the angular acceleration.
Given:
Angular acceleration (α) = 4.41 rad/s^2
Using the formula for net force, we can calculate:
F = I * α
F = 0.183 kg·m^2 * 4.41 rad/s^2
F ≈ 0.807 N
Therefore, the force required for the chain passing over a 9.06 cm diameter sprocket is approximately 0.807 Newtons.
B. To find the force required for a 5.54 cm diameter sprocket, we can repeat the same steps as above.
Given:
Radius of the wheel (r) = 5.54 cm / 2 = 2.77 cm = 0.0277 m
Using the formula for moment of inertia, we can calculate:
I = m * r^2
I = 1.79 kg * (0.0277 m)^2
I = 0.0136 kg·m^2
Using the formula for net force, we can calculate:
F = I * α
F = 0.0136 kg·m^2 * 4.41 rad/s^2
F ≈ 0.060 N
Therefore, the force required for the chain passing over a 5.54 cm diameter sprocket is approximately 0.060 Newtons.