Problem Diagram

First, break the problem into three parts that can be related to one another.

  1. Front Sprocket, r1
  2. Rear Sprocket, r2
  3. Back Wheel, r3

Next analyze each part and relate their tangent and angular velocities. The angular velocity for the back sprocket and back wheel must be equal (assuming no slipping) and the tangent velocity of the two sprockets must be equal.


Back Sprocket (disk 3)

Back Wheel

  Part a) Back Wheel


Since the back wheel does not slip, the point of contact with the ground has zero velocity for that instant. The wheel is essentially rotating about the contact point, and the angular velocity ω3 can be calculated as

     ω3 = v3/r3 = 7/0.350 = 20 rad/s

It should be noted the center velocity v3 is relative to the ground.

The angular velocity of the back wheel is the same as the velocity of the rear sprocket since they are fixed together.

     ω2 = ω3                                                     (1)

    Part b) Front Sprocket

Front Sprocket (disk 1)

Tangential Velocity of
Both Disks are Equal


The velocity of the outside edge of the front sprocket can be obtained by using the basic equation for rotation about a fixed axis.

     v1 = ω1 r1

But first, the given angular velocity needs to be converted to standard units.

     ω1 = 60 rev/min = 6.28 rad/s

Substituting into the previous equation,

     v1 = (6.28 rad/s)(120 mm) = 754 mm/s

The velocity v1 is relative to the center of the sprocket, which is moving with the bike. This will not affect the final answer since the rear sprocket is also moving with the bike.

Since the rear sprocket and front sprocket are connected by a chain, the tangential velocity of their edges will be the same

     v1 = v2                                                        (2)


  Part c) Rear Sprocket

Rear Sprocket (disk 2)

Finally, the rear sprocket size can now be calculated since its angular velocity ω2 and tangential velocity v2 are known. The radius for the rear sprocket is

     r2 = v2 / ω2

        = (754 mm/s)/(20 rad/s)

        = 37.7 mm

It should be noted, that this exact size may not be available since the gear has teeth that come in increment sizes.

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