Calculating Mass from Work Done Against Gravity

This problem asks us to find the mass of a person based on the work they did while carrying a box and moving upwards. The work done against gravity depends on the total mass being lifted, the acceleration due to gravity, and the vertical height moved.

Understanding Work Done Against Gravity

When an object is lifted vertically upwards, work is done against the force of gravity. The formula for this work done (\(W\)) is given by:

\(W = mgh\)

Where:

• \(m\) is the mass of the object being lifted (in kg).
• \(g\) is the acceleration due to gravity (in m/s\(\text{}^2\)).
• \(h\) is the vertical height the object is lifted (in m).

In this scenario, the person is carrying a box and moving upwards. Therefore, the total mass being lifted against gravity is the sum of the mass of the box and the mass of the person.

Problem Analysis and Given Values

We are given the following information:

• Mass of the box (\(m_b\)): 15 kg
• Vertical height (\(h\)): 5 m
• Work done by the person (\(W\)): 5000 J
• Acceleration due to gravity (\(g\)): 10 m/s\(\text{}^2\)

We need to find the mass of the person (\(m_p\)).

The total mass being lifted (\(M\)) is the sum of the mass of the box and the mass of the person:

\(M = m_b + m_p\)

Step-by-Step Mass Calculation

We can use the work done formula \(W = Mgh\), substituting \(M = m_b + m_p\):

\(W = (m_b + m_p)gh\)

Now, let's plug in the given values into the equation:

\(5000 \, J = (15 \, kg + m_p) \times (10 \, m/s^2) \times (5 \, m)\)

Simplify the right side of the equation:

\(5000 = (15 + m_p) \times 50\)

To isolate the term containing \(m_p\), divide both sides of the equation by 50:

\(\frac{5000}{50} = 15 + m_p\)

\(100 = 15 + m_p\)

Now, subtract 15 from both sides to find the value of \(m_p\):

\(m_p = 100 - 15\)

\(m_p = 85 \, kg\)

Therefore, the mass of the person is 85 kg.

Result Summary

Using the work-energy principle applied to work done against gravity, and considering the total mass being lifted (person + box), the calculated mass of the person is 85 kg.

Formula Used

\(W = (m_b + m_p)gh\)

Calculation Steps

1. Identify given values: \(W\), \(m_b\), \(h\), \(g\).
2. Identify the unknown: \(m_p\).
3. Write the formula for work done against gravity for the combined mass: \(W = (m_b + m_p)gh\).
4. Substitute known values into the formula: \(5000 = (15 + m_p) \times 10 \times 5\).
5. Solve the equation for \(m_p\): \(5000 = (15 + m_p) \times 50 \implies 100 = 15 + m_p \implies m_p = 85\).

Revision Table: Work Done and Mass Calculation

Additional Information: Potential Energy and Work

The work done against gravity when lifting an object is equal to the increase in its gravitational potential energy (\(PE\)).

\(PE = mgh\)

So, the work done \(W\) is equal to the change in potential energy (\(\Delta PE\)) if the object starts from a reference height (like the ground where \(h=0\)) and is lifted to height \(h\).

\(W = \Delta PE = PE_{final} - PE_{initial} = mgh - mg(0) = mgh\)

In this problem, the person lifts a combined mass of \(m_b + m_p\) through a height \(h\). The work done by the person against gravity increases the potential energy of the combined system by \( (m_b + m_p)gh \). The given work, 5000 J, is this increase in potential energy.

Understanding the relationship between work done against conservative forces (like gravity) and potential energy is crucial in physics problems.