Chapter 10.2, Problem 34SSC

(a)

To determine

To identify: If small or large gear should be chosen.

Explanation of Solution

Introduction:

The ideal mechanical advantage is equal to the displacement of the effort force divided by the displacement of the load. For the work to be done easily the IMA should be as high as possible.

The IMA for the rear wheel gear is equal to the radius of the rear gear divided by the rear wheel radius. That is,

IMA = $\frac{\text{rear}\text{gear}\text{radius}}{\text{rear}\text{wheel}\text{radius}}$  ......(1)

From this formula, one can see that more is the rear gear radius; more will be the mechanical advantage. That is, the force applied on the rear gear wheel increases proportionally with the size of the rear gear radius.

So, for exerting the greatest possible force, one needs to choose the largest possible radius of the gear which ultimately implies the largest rear gear.

Conclusion:

Hence, to start the bicycle, one needs to choose the large rear gear.

(b)

To determine

To identify: The gear that should be chosen if it is required to rotate the pedals as few times as possible.

Explanation of Solution

Introduction:

The ideal mechanical advantage is equal to the displacement of the effort force divided by the displacement of the load. For the work to be done easily the IMA should be as high as possible.

If it is required to rotate the pedals as few times as possible then it is needed to have minimum possible chain rotation to make the wheel rotate. Choosing the small gear helps in rotating the rear wheel in minimum possible pedaling.

So, to maintain the speed by rotating the pedals as few times as possible, one has to choose the small gear.

Conclusion:

Hence, one has to choose the small gear.

(c)

To determine

To identify: The gear that should be chosen if it is required to accelerate while climbing a hill.

Explanation of Solution

Introduction:

The ideal mechanical advantage is equal to the displacement of the effort force divided by the displacement of the load. For the work to be done easily, the IMA should be as high as possible.

The IMA for the front wheel pedal is equal to the radius of the pedal divided by the radius of the front gear. That is,

IMA = $\frac{\text{front}\text{pedal}\text{radius}}{\text{front}\text{gear}\text{radius}}$  ......(1)

From this formula, one can see that the smaller is the front gear radius; more will be the mechanical advantage. That is the force applied on the bicycle increases inversely with the size of the front gear radius.

So, for exerting the greatest possible force, one needs to choose the smallest possible radius of the front gear, which ultimately implies the smallest front gear.

Conclusion:

Hence, to accelerate the bicycle by creating more force while climbing the hill, one needs to choose the small front gear.