Customers are complaining that it is too hard to pedal the X

Customers are complaining that it is too hard to pedal the X-20 model bike up a moderate hill, and your boss tells you to solve the problem. Since the rear sprocket would be the easiest to replace in the field, you decide to modify the rear sprocket\'s size to correct the problem. The known parameters are: ? The back wheel\'s diameter is 700 mm. ? The front sprocket radius is 120 mm. ? The chain connects the front sprocket to the rear wheel. What size should the rear sprocket be so that the rider will be pedaling at 60 rev/min when moving at 7 m/s?

Back Wheel Front Sprocket r = 120 mm = 350 Reah Sprocket 60 rev/min (!), =

Solution

What is known:

The back wheel\'s diameter is 700 mm.

The front sprocket radius is 120 mm.

The chain connects the front sprocket to the rear wheel.

Approach:

Notice that both sprockets and the back wheel are rotating around their respective center axes,

Separate the two sprockets and the back wheel into three separate objects.

Relate the angular velocity of each object to the tangential velocity of its surface.

w1r1=w2r2

w2=(60*120)/350

w2=20.57 rev/min

the angular velocity of the rear sprocket and back wheel is equal

w3=w2= 20.57 rev/min= 6.85 rad/s

v3=w3*r3

r3 = v3/w3

r3 = 7/6.85

r3= 1.02 mm