Question

A \(60 \mathrm{~kg}\) JATAN with \(10 \mathrm{~kg}\) load on his head climbs 25 steps of \(0.20 \mathrm{~m}\) height each. What is the work done in climbing ? \(\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) (A) \(5 \mathrm{~J}\) (B) \(350 \mathrm{~J}\) (C) \(100 \mathrm{~J}\) (D) \(3500 \mathrm{~J}\)

Short answer

The work done in climbing is 3500 J. The correct answer is (D) \(3500 \mathrm{~J}\).

Step-by-step solution

1. Identify the given variables.

We are given the following variables: - Mass of JATAN (m1) = 60 kg - Mass of load (m2) = 10 kg - Number of steps (n) = 25 - Height of each step (h) = 0.20 m - Gravitational acceleration (g) = 10 m/s²

2. Calculate the total mass being lifted.

The total mass being carried by the person as he climbs the stairs is the sum of the JATAN's own mass and the mass of the load. Therefore, we have: Total mass (m) = m1 + m2 m = 60 kg + 10 kg = 70 kg

3. Determine the total height climbed by the person.

To find the total height climbed by the person, multiply the number of steps by the height of each step: Total height (H) = n * h H = 25 * 0.20 m = 5 m

4. Calculate the change in potential energy.

The change in potential energy (ΔU) can be calculated using the formula: ΔU = m * g * H ΔU = 70 kg * 10 m/s² * 5 m = 3500 J

5. Determine the work done in climbing.

According to the work-energy principle, the work done (W) is equal to the change in potential energy: W = ΔU W = 3500 J So the work done in climbing is 3500 J. The correct answer is (D) \(3500 \mathrm{~J}\).

Key concepts

Potential Energy

When discussing work and energy, potential energy is a fundamental concept to understand. Potential energy is the energy stored within a system due to the position of objects within it. It is called "potential" because it has the potential to do work.

In the context of our problem, JATAN and the load experience potential energy as they are lifted higher against gravitational force. The formula to calculate gravitational potential energy is: \[ U = m \cdot g \cdot h \] where:

• \( U \) is the potential energy
• \( m \) is the total mass
• \( g \) is the gravitational acceleration
• \( h \) is the height climbed

The potential energy increases as the height increases, and since JATAN is climbing, the increase in potential energy directly relates to the work done against the gravitational force.

Gravitational Acceleration

Gravitational acceleration is a critical component of the equations related to work done against gravity. It is the rate at which objects accelerate towards Earth when in free fall, typically denoted as \( g \). On Earth, this value is approximately \( 9.81 \mathrm{~m/s^2}\), but it is often rounded in problems for simplicity, as in this case to \( 10 \mathrm{~m/s^2}\).

Gravitational acceleration is constant, meaning all objects, regardless of their mass, experience this acceleration equally when falling freely. This universal force not only keeps us grounded but also allows us to calculate how the potential energy changes with height. In our exercise, gravitational acceleration is crucial in determining the potential energy increase as JATAN moves up the steps. Without this acceleration, we wouldn't be able to calculate the work done effectively, as it directly multiplies the mass and height in the potential energy formula.

Mass and Weight

Understanding the difference between mass and weight is essential when solving problems related to work and energy. Mass is a measure of how much matter an object contains and is measured in kilograms, not dependent on local gravity. On the other hand, weight is the force that gravity exerts on an object's mass, mathematically defined as: \[ W = m \cdot g \] where:

• \(W\) is the weight
• \(m\) is the mass
• \(g\) is the gravitational acceleration

In the exercise, the total mass carried by JATAN includes his own body weight and the additional 10 kg load, combining to a total mass of 70 kg. The weight of this total mass acts against the force of gravity, and this force is what JATAN overcomes while climbing, which is why mass is crucial to calculate the work done.