Work, energy and power

Introduction to Work, Energy, and Power

Welcome to one of the most important chapters in your A-level Physics journey! In the "Mechanics and materials" section, we’ve already looked at how forces cause motion. Now, we are going to look at the "currency" of the universe: Energy. Whether it’s a car accelerating, a lightbulb shining, or a person climbing stairs, energy is being transferred. Understanding how to calculate these transfers using Work and Power is essential for mastering Physics.

Don't worry if these terms feel familiar from GCSE; we are going to add the precision and mathematical depth you need for the AQA 7408 specification. Let's dive in!

Note: This chapter connects closely to "Newton’s laws and momentum" and "Motion along a straight line."

1. Work Done: Energy in Action

In Physics, "Work" isn't just something you do at a desk. Work is done whenever a force moves an object over a distance. When work is done, energy is transferred from one form to another.

The Mathematical Definition

If a constant force \(F\) acts on an object and moves it a distance \(s\) in the direction of the force, the work done \(W\) is:

\(W = Fs\)

However, life isn't always that simple! Sometimes the force is at an angle to the direction of motion (like pulling a suitcase on wheels). In that case, we only care about the component of the force acting along the line of movement:

\(W = Fs \cos(\theta)\)

Where:
- \(W\) is Work Done measured in Joules (J).
- \(F\) is the Force in Newtons (N).
- \(s\) is the Displacement in metres (m).
- \(\theta\) is the angle between the force and the direction of motion.

Important Tip: The "No Work" Scenarios

If you hold a heavy box perfectly still, you might feel tired, but in Physics terms, you are doing zero work because the displacement \(s\) is zero. Similarly, if you carry a box horizontally at a constant speed, the upward force you provide is at \(90^{\circ}\) to the motion. Since \(\cos(90^{\circ}) = 0\), you are technically doing no work on the box in the direction of travel!

Quick Review: Work is energy transferred by a force. Always check if the force and the distance are in the same line!

2. Conservation of Energy

This is a fundamental law of the universe. The Principle of Conservation of Energy states that energy cannot be created or destroyed, only transferred from one form to another.

In A-level problems, we usually focus on Mechanical Energy, which is the sum of Kinetic Energy (\(E_k\)) and Potential Energy (\(E_p\)).

Kinetic Energy (\(E_k\))

This is the energy an object has due to its motion:

\(E_k = \frac{1}{2}mv^2\)

Gravitational Potential Energy (\(\Delta E_p\))

This is the energy an object gains when it is lifted in a uniform gravitational field:

\(\Delta E_p = mg\Delta h\)

Applying the Principle

In a perfect world (where there is no air resistance or friction), the energy at the start of a process equals the energy at the end. For a falling object:
Loss in \(E_p\) = Gain in \(E_k\)
\(mg\Delta h = \frac{1}{2}mv^2\)

Common Mistake to Avoid: In the real world, some energy is always transferred to the surroundings as heat due to friction or air resistance. In these cases, the equation becomes:
Total Energy at Start = Total Energy at End + Work done against friction.

3. Power: The Speed of Energy Transfer

If two people climb the same set of stairs, they both do the same amount of Work (because they lifted the same weight the same distance). However, the person who runs up the stairs is more Powerful because they did the work faster.

Power is defined as the rate of doing work (or the rate of energy transfer).

The Formulas for Power

1. The general definition:
\(P = \frac{\Delta W}{\Delta t}\)

2. For a moving object (like a car traveling at a constant speed):
\(P = Fv\)

Where:
- \(P\) is Power measured in Watts (W). Note: \(1 W = 1 J/s\).
- \(\Delta W\) is the Work Done (or energy transferred).
- \(\Delta t\) is the Time taken in seconds.
- \(F\) is the Motive Force.
- \(v\) is the Velocity.

Did you know? The formula \(P = Fv\) is derived from \(P = \frac{Fs}{t}\). Since velocity \(v = \frac{s}{t}\), we can substitute it directly!

4. Efficiency

No machine is 100% efficient. Some energy is always "wasted"—usually spreading out into the environment as thermal energy (heat). Efficiency is a measure of how much of the energy we put in actually does the job we want.

Calculating Efficiency

Efficiency can be calculated using either energy or power. It is usually expressed as a percentage or a decimal (always less than 1 or 100%).

\(\text{Efficiency} = \frac{\text{Useful output power}}{\text{Total input power}}\)

or

\(\text{Efficiency} = \frac{\text{Useful energy output}}{\text{Total energy input}}\)

Example: A Car Engine

If a car engine takes in \(100,000 J\) of chemical energy from fuel, but only \(25,000 J\) is used to move the car forward (the rest being lost as heat and sound), the efficiency is:
\(\frac{25,000}{100,000} = 0.25\) or 25%.

Summary and Key Takeaways

To succeed in exam questions on this topic, remember these three core pillars:

1. Work Done: Always look for the force acting in the direction of the displacement. Use \(W = Fs \cos(\theta)\) if there's an angle involved.

2. Conservation: Energy cannot vanish. If a calculation doesn't balance, look for "work done against friction" or "thermal energy loss."

3. Power: Power is just "Work per second." If a question gives you a velocity and a force, use \(P = Fv\) for a quick solution.

Quick Review Box:
- Work Done \(W\): Joules (\(J\))
- Power \(P\): Watts (\(W\))
- Efficiency: No units (it's a ratio!)
- Gravity \(g\): Always use \(9.81 m/s^2\) as per the AQA data sheet.