Problem 3 - Applied Mathematics

Question

The potential energy of a body of mass \(m\) is \(m g h\) where \(h\) is: (a) the distance from a chosen level, (b) the height above the ground, (c) the height above a chosen level, (d) the vertical distance moved.

Step-by-Step Solution

Answer

Option (c): the height above a chosen level.

Step 1: Identify the given equation

The potential energy (PE) of a body is given by the equation \(PE = mgh\), where \(m\) is the mass, \(g\) is the acceleration due to gravity, and \(h\) is the height.

Step 2: Understand the components

In the equation \(mgh\), \(h\) represents the height. It is important to determine what this height refers to in the context of potential energy.

Step 3: Analyze each option

Option (a): The distance from a chosen level. Option (b): The height above the ground. Option (c): The height above a chosen level. Option (d): The vertical distance moved.

Step 4: Match the correct definition

Potential energy is dependent on the height relative to a reference point. This can be a chosen level or a datum line from which the height is measured.

Step 5: Select the best option

Among the given choices, option (c) 'the height above a chosen level' best matches the definition of height \(h\) in the potential energy equation \(mgh\).

Key Concepts

Gravitational Potential Energy

Gravitational potential energy represents the energy that a body possesses due to its position in a gravitational field. When a body is lifted to a certain height, work is done against the force of gravity. This work is stored as potential energy in the body. The equation for gravitational potential energy is \( PE = mgh \) Here’s what each term means:

• m: the mass of the body, measured in kilograms (kg)
• g: the acceleration due to gravity, typically \(9.81 m/s^2\) on the surface of the Earth
• h: the height or vertical distance from a reference point, measured in meters (m)

This means that the higher a body is lifted in a gravitational field, and the more massive it is, the more potential energy it has. This can be illustrated with simple real-world examples: a book held above the ground has more gravitational potential energy than the same book placed on the floor.

Acceleration Due to Gravity

Acceleration due to gravity, denoted by \(g\), is the rate at which an object accelerates when falling freely near the Earth's surface. This is approximately \(9.81 m/s^2\). This value may slightly vary depending on your location on Earth, but for most practical calculations, \(9.81 m/s^2\) is used. Gravitational acceleration is what makes objects fall towards the Earth when dropped. It’s a constant force that acts on all objects, regardless of their mass. Hence, in the formula for potential energy, \(g\) remains constant. Here are some key points to understand about \(g\):

• It is the reason why objects speed up as they fall.
• It is always directed towards the center of the Earth.
• It is a key factor in calculating both gravitational potential energy and the weight of an object (\(F = mg\)).

Understanding \(g\) helps you understand why even lightweight objects can have considerable potential energy when at a high elevation, owing to this constant pull of gravity.

Reference Point in Physics

A reference point is a point against which position and potential energy are measured. In the context of gravitational potential energy, the reference point is crucial to determine the 'height' or distance in the equation \(PE = mgh\).
Choosing a reference point often involves selecting a level ground or specific datum line from which to measure the height. For example:

• It could be the ground level in a building.
• A specific floor in a multi-storey structure.
• A sea level when talking about geographical elevations.

In our exercise, option (c) 'the height above a chosen level' is correct. This is because gravitational potential energy is relative. The chosen level acts as the zero point or baseline from which height ( \(h\)) is measured. This reference point ensures that potential energy can be quantified meaningfully and comparably in different scenarios.
For instance, if you were lifting a box to a height of 2 meters above the second floor of a building, the second floor would be your reference point. The potential energy can then be measured from this chosen level.