Question

An object whose mass is $400\mathrm{kg}$ is located at an elevation of $25\mathrm{m}$ above the surface of the earth. For $g=9.78\mathrm{m}/{\mathrm{s}}^2$, determine the gravitational potential energy of the object, in $\mathrm{kJ}$, relative to the surface of the earth.

Short answer

The gravitational potential energy is 97.8 kJ.

Step-by-step solution

- Understand the formula

Gravitational potential energy (GPE) is calculated using the formula: $$GPE=mgh$$ where $m$ is mass, $g$ is the acceleration due to gravity, and $h$ is the height above the reference point.

- Identify given values

From the problem, the given values are: Mass $m=400$ kg Height $h=25$ m Acceleration due to gravity $g=9.78$ m/s\textsuperscript{2}

- Substitute the values into the formula

Plug the given values into the formula: $$GPE=400\times 9.78\times 25$$

- Calculate the result

Multiply the values: $$GPE=400\times 9.78\times 25=97800\text{J}$$

- Convert joules to kilojoules

Since 1 kJ = 1000 J, convert the result: $$97800\text{ J}=\frac{97800}{1000}\text{ kJ}=97.8\text{ kJ}$$

Key concepts

Understanding Mass

Mass is a measure of the amount of matter in an object. It is commonly measured in kilograms (kg). The mass of an object is a scalar quantity, meaning it only has magnitude and no direction.
In the problem, we are given a mass of 400 kg. This value represents the quantity of matter in the object, and it remains constant regardless of its location in the universe.
Understanding mass is important, as it directly affects the gravitational potential energy of the object. The larger the mass, the greater the potential energy when positioned at a certain height above the Earth's surface.
Key points about mass:

• Measured in kilograms (kg).
• Scalar quantity (only magnitude).
• Directly influences gravitational potential energy.

Acceleration Due to Gravity

Acceleration due to gravity, denoted as $g$, is the rate at which an object accelerates when it is in free fall towards the Earth. Its value varies slightly depending on location but is generally approximated as $9.81{\text{ m/s}}^2$. In our problem, $g$ is specifically given as $9.78{\text{ m/s}}^2$.
Gravitational acceleration is a vector quantity, meaning it has both magnitude and direction. It always points towards the center of the Earth.
Key points about acceleration due to gravity:

• Measured in meters per second squared (${\text{ m/s}}^2$).
• Near Earth's surface, typically around $9.81{\text{ m/s}}^2$.
• Vector quantity (magnitude and direction).

Gravitational acceleration determines how quickly an object speeds up as it falls, influencing the potential energy stored due to its position.

Importance of Height

Height, denoted as $h$, is the distance above a reference point, typically the Earth's surface in these problems. It is measured in meters (m). In the problem at hand, the object is 25 meters above the Earth's surface. Height is directly proportional to gravitational potential energy.
More height means more potential energy for the same mass and gravitational acceleration. This is because energy is stored by lifting the object against Earth's gravity.
Key points about height:

• Measured in meters (m).
• Scalar quantity (only magnitude).
• Directly influences gravitational potential energy.

By understanding the height component, we can accurately determine how much energy is stored due to the object's elevated position.

Gravitational Potential Energy and Energy Conversion

Gravitational Potential Energy (GPE) is the energy stored in an object due to its position in a gravitational field. It is given by the formula: $$GPE=mgh$$
This means GPE depends on the mass ($m$) of the object, gravitational acceleration ($g$), and height ($h$).
In our problem, we calculated the GPE of an object to be 97.8 kJ, using: $m=400\text{ kg}$, $g=9.78{\text{ m/s}}^2$, and $h=25\text{ m}$.
Energy conversion is a key concept here. GPE can convert into other energy forms, such as kinetic energy when the object falls. Hence, the higher the GPE, the more energy is available to convert as the object moves.
Key points about energy conversion:

• GPE depends on mass, gravitational acceleration, and height.
• Measured in Joules (J) or kilojoules (kJ).
• Can convert into other energy forms, such as kinetic energy.

Understanding GPE helps in comprehending energy conversion processes in various physical contexts.