A bicycle wheel has a diameter of 57.7 cm

and a mass of 2.66 kg. The bicycle is placed
on a stationary stand and a resistive force of
156 N is applied tangent to the rim of the tire.
Assume that the wheel is a hoop with all of
the mass concentrated on the outside radius.
In order to give the wheel an acceleration of
4.67 rad/s
2
, what force must be applied by a
chain passing over a 3 cm diameter sprocket?
Answer in units of N

To find the force required to give the bicycle wheel an acceleration of 4.67 rad/s^2, we can use Newton's second law, which states that force (F) is equal to mass (m) multiplied by acceleration (a).

First, let's calculate the moment of inertia (I) of the bicycle wheel, assuming it is a thin hoop with all the mass concentrated on the outside radius. The moment of inertia of a hoop is given by the formula I = m * r^2, where m is the mass and r is the radius.

Given:
Diameter of the bicycle wheel = 57.7 cm
Radius of the bicycle wheel (r) = diameter / 2 = 57.7 cm / 2 = 28.85 cm = 0.2885 m
Mass of the bicycle wheel (m) = 2.66 kg

Calculating the moment of inertia:
I = m * r^2 = 2.66 kg * (0.2885 m)^2

Next, we can calculate the torque (τ) required to produce the desired angular acceleration (α) using the formula τ = I * α, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

Given:
Angular acceleration (α) = 4.67 rad/s^2

Calculating the torque:
τ = (2.66 kg * (0.2885 m)^2) * 4.67 rad/s^2

Finally, we can find the force (F) required by the chain passing over the sprocket using the relationship between force and torque. As the chain is passing over a sprocket with a small diameter, the force required to produce the torque is given by F = τ / (radius of the sprocket).

Given:
Diameter of the sprocket = 3 cm
Radius of the sprocket = diameter / 2 = 3 cm / 2 = 1.5 cm = 0.015 m

Calculating the force:
F = (2.66 kg * (0.2885 m)^2 * 4.67 rad/s^2) / 0.015 m

Simplifying the equation and calculating:
F ≈ 208.5 N

Therefore, to give the bicycle wheel an acceleration of 4.67 rad/s^2, a force of approximately 208.5 N must be applied by the chain passing over the 3 cm diameter sprocket.