Question

Consider a person climbing and descending stairs. Construct a problem in which you calculate the long-term rate at which stairs can be climbed considering the mass of the person, his ability to generate power with his legs, and the height of a single stair step. Also consider why the same person can descend stairs at a faster rate for a nearly unlimited time in spite of the fact that very similar forces are exerted going down as going up. (This points to a fundamentally different process for descending versus climbing stairs.)

Short answer

The gravitational potential energy gained by the climber is \(9187.5{\rm{ J}}\). The person takes \(13.4{\rm{ s}}\) to get to the roof if he can climb 116 steps in a minute. During climbing stairs, the work is done against the gravity makes it harder than descending stairs

Step-by-step solution

Step 1: Definition of Concepts 

Power: Power is defined as the rate at which energy is being consumed.Mathematically,

\(P = \frac{E}{T}\)

Here, \(E\) is the energy consumption, and \(T\) is the time.

Construction of problem

A person of mass \(75{\rm{ kg}}\) is climbing the stairs of his house to get to the roof. He has to climb \(50\) stairs each of height \(0.25{\rm{ m}}\) to get to the roof. Find the gravitational potential energy gained. How long does he take to to get to the roof if he can climb 116 steps in a minute. Explain why it is easier to descend stairs than climbing the stairs

Find the gravitational potential energy

The height of the roof is,

\(h = 50{h_s}\)

Here,\({h_s}\)is the height of each stair\(\left( {{h_s} = 0.25{\rm{ m}}} \right)\).

Putting all known values,

\(\begin{array}{c}h = 50 \times \left( {0.25{\rm{ m}}} \right)\\ = 12.5{\rm{ m}}\end{array}\)

The gravitational potential energy gained is,

\(E = mgh\)

Here,\(m\)is the mass of the person\(\left( {m = 75{\rm{ kg}}} \right)\),\(g\)is the acceleration due to gravity\(\left( {g = 9.8{\rm{ m}}/{{\rm{s}}^2}} \right)\), and\(h\)is the height of the roof\(\left( {h = 12.5{\rm{ m}}} \right)\).

Putting all known values,

\(\begin{array}{c}E = \left( {75{\rm{ kg}}} \right) \times \left( {9.8{\rm{ m}}/{{\rm{s}}^2}} \right) \times \left( {12.5{\rm{ m}}} \right)\\ = 9187.5{\rm{ J}}\end{array}\)

Therefore, the required gravitational potential energy gained by the climber is \(9187.5{\rm{ J}}\).

Calculate the time required to get to the roof

The time taken to get to the roof is,

\(t = \frac{E}{P}\)

Here,\(E\)is the gravitational potential energy gained\(\left( {E = 9187.5{\rm{ J}}} \right)\), and\(P\)is the power required to climb stairs at the rate of\(116\)stairs in a minute\(\left( {P = 685{\rm{ W}}} \right)\).

Putting all known values,

\(\begin{array}{c}t = \frac{{9187.5{\rm{ J}}}}{{685{\rm{ W}}}}\\ = 13.4{\rm{ s}}\end{array}\)

Therefore, the person takes \(13.4{\rm{ s}}\) to get to the roof if he can climb \(116\) steps in a minute

Explain why it is easier to descend stairs than climbing the stairs

During climbing the climber must lift his body mass against the gravity, he has to overcome the force of gravity which makes it difficult than descending down the stairs.