# Rotational Dynamics of a Bicycle wheel

## Homework Statement

A bicycle wheel has a diameter of 63.8 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 117 N is applied tangent to the rim of the tire.

(a) What force must be applied by a chain passing over a 9.01-cm-diameter sprocket in order to give the wheel an acceleration of 4.53 rad/s2?

(b) What force is required if you shift to a 5.60-cm-diameter sprocket?

R=wheel radius
R= .319m
m= wheel mass
m=1.79kg
F=Resistive force tangent to tire
F=117N
r=radius of sprocket
r=.0901m
α=angular acceleration of the wheel
α=4.53rad/s^2
τf= Torque from the bike wheel
τext=external torque for sprocket

## Homework Equations

Torque
τ=rF
Moment of Interia of a hoop
I=MR^2
Torque
τ=Iα

## The Attempt at a Solution

First calculate torque on the bike wheel

τf=.319m(117N)= 37.3Nm

Due to laws of rotational dynamics and why the wheel is spinning relate the torque of the wheel to torque of sprocket to moment of interia of the wheel

τext-τf=Iα

Break down each term into quantities we already know

τext=rF
τf=37.3Nm
I=MR^2
α=4.53rad/s^2

Thusly
rF-37.3Nm=MR^2(4.53Rad/s^2)

Rearrange and solve for F

F=MR^2(α)+τf/r

Input values
F=(1.79kg)(.319)^2(4.53rad/s^2)+37.3Nm/.0901m

F=423N

Maybe I messed up somewhere but I've tried twice already submitting the answer online and this is wrong but that was my shot, thank you for helping.

## Answers and Replies

I forgot to mention that to solve part b you would use the same equation just replace r with .0560m

You used the diameter of the sprocket where you needed the radius. Using your equations, I'm getting for part a:
F = 847 N

Oops, should have checked the date before replying.