Problem 13.

A cyclist intends to cycle up a 8 degrees hill whose vertical height is 150 m. If each complete revolution of the pedals moves the bike 6 m   along its path, calculate the average force that must be exerted on the pedals tangent to their circular path. Neglect work done by friction and other losses. The pedals turn in a circle of diameter 30 cm. The total mass of the cyclist and his bike is 100 kg.

Solution:

In this problem we need to use generalized work-energy theorem: work done by an external force is equal to the change of the net mechanical energy of the system:

We assume that the cyclist moves with constant speed. Then the initial and the final kinetic energies are the same. Therefore in the above expression we need to take into account only the gravitational potential energy:

where is the total mass of the cyclist and bike.

Then

Now we need to define the initial and the final states of the system. We introduce the final state as the state after one complete revolution (relative to the initial state).

We know that after one complete revolution the cyclist moves 6 meters (as show in the figure). Then the change in the height of the cyclist is

Therefore the change in the gravitation potential energy of the cyclist+bike is

The work done by the force is

During one revolution the pedal travels a distance of

(circumference of a circle with diameter ). Where is the diameter of the circle. Then the work done by the force is

This work is equal to the change in gravitational potential energy:

From this expression we can find the force: