Question

A bicycle wheel has a diameter of 63.4 cm and a mass of 1.83 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 121 N is applied tangent to the rim of the tire.

(a) What force must be applied by a chain passing over a 8.95-cm-diameter sprocket in order to give the wheel an acceleration of 4.51 rad/s²?
_____ N

(b) What force is required if you shift to a 5.65-cm-diameter sprocket?
____ kN

Answer 1

Given:

d = 63.4 cm

r = 63.4/2 cm=31.7cm =0.317m

m = 1.83 kg

F = 121 N

a)alpha = 4.51 rad/s^2

r' = 8.95 cm

r = 8.95/2 cm = 4.475cm =0.04475m

We know that

I = m r^2 = 1.83 x (0.317)^2 = 0.184 kg-m^2

net torque is:

Tnet = T - T'

F'r' - Fr = I alpha

F' = (I alpha + FR)/r'

F' = (0.184 x 4.51 + 121 x 0.317)/0.04475 =(0.82984+38.357)/0.04475 = 39.18684/0.04475 =875.6835754N

Hence, F =875.68 N (=876 N)

b)given

d=5.65

r' = 5.65/2 = 2.825cm = 0.02825m

F = (0.184 x 4.51 + 121 x 0.317)/0.02825 =(0.82984+38.357)/0.02825 = 39.18684/0.02825 = 1387.144 N

Hence, F =1387 N