Calculating Height from Energy and Mass

This physics problem involves calculating the height a mass is raised when a specific amount of energy is spent. The energy spent in raising an object against gravity is converted into its gravitational potential energy.

Understanding Potential Energy and Work Done

When a force acts over a distance, work is done. In this case, the force is applied to lift the mass against the force of gravity. The work done against gravity is stored as gravitational potential energy (PE) in the mass. The formula for gravitational potential energy is:

\( \text{PE} = mgh \)

Where:

• \( \text{PE} \) is the potential energy gained (in Joules, J)
• \( m \) is the mass of the object (in kilograms, kg)
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \) on Earth)
• \( h \) is the height the object is raised (in meters, m)

The problem states that 9800 Joules of energy was spent to raise the mass. This energy is equal to the potential energy gained by the mass.

Step-by-Step Calculation of Height

We are given the following values:

• Energy spent (\( \text{PE} \)) = 9800 J
• Mass (\( m \)) = 80 kg
• Acceleration due to gravity (\( g \)) = \( 9.8 \, \text{m/s}^2 \) (standard value unless otherwise specified)

We need to find the height (\( h \)). We can rearrange the potential energy formula to solve for \( h \):

\( h = \frac{\text{PE}}{mg} \)

Now, substitute the given values into the formula:

\( h = \frac{9800 \, \text{J}}{80 \, \text{kg} \times 9.8 \, \text{m/s}^2} \)

First, calculate the denominator:

\( 80 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 784 \, \text{J/m} \)

Now, divide the energy by the result:

\( h = \frac{9800 \, \text{J}}{784 \, \text{J/m}} \)

\( h = 12.5 \, \text{m} \)

So, the mass was raised to a height of 12.5 meters.

Comparing Calculated Height with Options

Let's compare our calculated height of 12.5 m with the given options:

• Option 1: 10.5 m
• Option 2: 15.0 m
• Option 3: 12.5 m
• Option 4: 22.5 m

Our calculated height matches Option 3.

Revision Table: Energy, Mass, Height Calculation

Additional Information: Work-Energy Principle

The principle used here is a specific application of the work-energy principle, which states that the net work done on an object is equal to its change in kinetic energy. In this case, if the mass starts and ends at rest (or is raised slowly at a constant velocity), the change in kinetic energy is zero. Therefore, the work done against gravity (which is the energy spent) is equal to the change in potential energy.

If there were other forces involved, like friction or air resistance, the total energy spent might be higher than the potential energy gained, with the extra energy being converted into heat or sound.

The value of \( g \) can vary slightly depending on location on Earth, but \( 9.8 \, \text{m/s}^2 \) is a widely accepted standard value for calculations at the Earth's surface.