A bicycle wheel has a diameter of 47.6 cm and a mass of 0.809 kg. The bicycle is placed on a stationary stand on rollers and a resistive force of 60.1 N is applied to the rim of the tire. Assume all the mass of the wheel is concentrated on the outside radius.
In order to give the wheel an acceleration of 3.18 rad/s2, what force must be applied by a chain passing over a sprocket with diameter 4.16 cm?
Answer in units of N.
I = M R ^2...moment of inertia of the wheel
Then the torque from resistive force = force * R
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To find the force required to accelerate the bicycle wheel, we can first calculate the moment of inertia of the wheel and then find the torque required to produce the desired acceleration.
1. Calculate the moment of inertia (I) of the wheel:
The moment of inertia of a solid disk rotating about its central axis is given by the formula: I = (1/2) * m * R^2, where m is the mass of the wheel and R is the radius of the wheel.
In this case, the mass of the wheel is given as 0.809 kg and the diameter is given as 47.6 cm. We need to convert the diameter to radius by dividing it by 2.
So, the radius of the wheel (R) = 47.6 cm / 2 = 23.8 cm = 0.238 m.
Now substitute the values into the formula: I = (1/2) * 0.809 kg * (0.238 m)^2.
2. Calculate the torque exerted by the resistive force:
The torque exerted by a force acting at a distance from the axis of rotation is given by the formula: torque = force * distance. In this case, the resistive force applied to the rim of the tire is given as 60.1 N. The distance from the axis of rotation can be taken as the radius of the wheel (R = 0.238 m).
So, the torque exerted by the resistive force = 60.1 N * 0.238 m.
3. Find the force required to accelerate the wheel:
The torque needed to produce an angular acceleration is given by the formula: torque = I * alpha, where alpha is the angular acceleration.
In this case, the angular acceleration (alpha) is given as 3.18 rad/s^2, and we have already calculated the moment of inertia (I).
So, we can equate the torque from the resistive force to the torque required to produce the acceleration: torque = force * R = I * alpha.
Now, solve for the force: force = (I * alpha) / R.
Substitute the values we have calculated into the equation to find the force required:
force = [(1/2) * 0.809 kg * (0.238 m)^2 * 3.18 rad/s^2] / 0.238 m.
Calculate the force using the given values and simplify the expression to get the final answer in N.