Question 29: (II) A 66.5-kg hiker starts at an elevation of 1270 m and climbs to the top of a peak 2660 m high. (a) What is the hiker’s change in potential energy? (b) What is the minimum work required of the hiker? (c) Can the actual work done be greater than this? Explain.

The energy possessed by an object due to its position above the surface of the earth is referred to as the gravitational potential energy of the object.

When an object of mass m is raised by a height h, the gravitational potential energy of the object at this height is given as:

\(P{E_{\rm{G}}} = m{\rm{g}}h\)

Here, g is the acceleration due to gravity on the surface of the earth.

The mass of the hiker is m = 66.5 kg.

The hiker moves from an elevation of 1270 m to a height of 2660 m. Thus, the total height covered by the hiker is:

\(\begin{array}{c}h = \left( {2660 - 1270} \right)\;{\rm{m}}\\ = 1390\;{\rm{m}}\end{array}\)

Suppose the elevation of 1270 m is taken as the reference level. In that case, the change in the gravitational potential energy of the hiker will be equal to his gravitational potential energy at the top of the peak with respect to the reference level. Therefore,

\(\begin{array}{c}\Delta P{E_{\rm{G}}} = m{\rm{g}}h\\ = \left( {66.5\;{\rm{kg}}} \right) \times \left( {9.8\;{\rm{m/}}{{\rm{s}}^2}} \right) \times \left( {1390\;{\rm{m}}} \right)\\ = 905,863\;{\rm{J}}\\ = 9.06 \times {\rm{1}}{{\rm{0}}^5}\;{\rm{J}}\end{array}\)

Thus, the change in the gravitational potential energy of the hiker is \(9.06 \times {\rm{1}}{{\rm{0}}^5}\;{\rm{J}}\).

The potential energy of the hiker changes when he does some work to change his position. Therefore, the minimum work required of the hiker to climb a height of 2660 m from the elevation of 1270 m is equal to the change in the potential energy of the hiker in going through this height. So,

\(\begin{array}{c}W = \Delta P{E_{\rm{G}}}\\ = \;9.06 \times {\rm{1}}{{\rm{0}}^5}\;{\rm{J}}\end{array}\)

Thus, the minimum work required of the hiker to climb to the top of the peak is \(9.06 \times {\rm{1}}{{\rm{0}}^5}\;{\rm{J}}\).

The actual work done by the hiker can be greater than the minimum work required of the hiker as energy other than mechanical energy can also be present. Some amount of work also gets lost in the surroundings in the form of heat. When the hiker climbs, some work must also be done to overcome the friction between his joints of arms and legs.

Therefore, it is clear that the actual work done can be greater than the minimum work required of the hiker to climb to the top of the peak.