Work, energy, power and efficiency

Introduction to Work, Energy, and Power

Welcome to one of the most important chapters in your Physics journey! In the previous chapters, we looked at how objects move (kinematics) and why they move (forces). Now, we are going to look at the "currency" of the universe: Energy. Whether it's a car accelerating down a highway or a lightbulb glowing, energy is being transferred or transformed. Understanding how to calculate Work, Energy, and Power allows us to predict exactly how systems will behave and how "efficient" our machines really are.

Don't worry if these terms sound similar right now; by the end of these notes, you’ll be able to distinguish between them with ease!

1. Work Done

In Physics, "Work" has a very specific meaning. You might feel like you're doing "work" by sitting still and studying, but in Physics, Work Done only happens when a force moves an object through a distance.

What is Work Done?

Work is defined as the product of the force and the displacement in the direction of that force. It is measured in Joules (\(J\)).

The basic formula is:
\(W = F \Delta s\)

Where:
\(W\) = Work done (\(J\))
\(F\) = Constant force (\(N\))
\(\Delta s\) = Displacement (\(m\))

Forces Not Acting Along the Line of Motion

Sometimes, we pull an object at an angle. For example, if you are pulling a suitcase with a handle, you are pulling upwards and forwards, but the suitcase only moves horizontally. In this case, only the component of the force acting in the direction of motion does work.

The formula becomes:
\(W = F \Delta s \cos(\theta)\)

Where \(\theta\) is the angle between the force and the direction of motion.

Quick Tip: If the force is perpendicular (\(90^{\circ}\)) to the motion, no work is done by that force! This is why the Earth's gravity does no work on a satellite in a perfectly circular orbit.

Quick Review:
- Work is energy transferred by a force.
- Unit: Joules (\(J\)).
- Only the force component parallel to the displacement counts.

2. Kinetic Energy (\(E_k\))

Kinetic Energy is the energy an object possesses because it is moving. If an object is at rest, its kinetic energy is zero.

The Formula

The amount of kinetic energy depends on the object's mass and its velocity:
\(E_k = \frac{1}{2} m v^2\)

Where:
\(m\) = mass (\(kg\))
\(v\) = velocity (\(m s^{-1}\))

Did you know? Because the velocity is squared, if you double the speed of a car, it actually has four times the kinetic energy (\(2^2 = 4\)). This is why high-speed crashes are so much more dangerous!

3. Gravitational Potential Energy (\(E_{grav}\))

Gravitational Potential Energy (GPE) is the energy an object has because of its position in a gravitational field. When you lift an object up, you are doing work against gravity, and that work is "stored" as GPE.

The Formula

Near the Earth’s surface, we use:
\(\Delta E_{grav} = m g \Delta h\)

Where:
\(m\) = mass (\(kg\))
\(g\) = gravitational field strength (\(9.81 \, N kg^{-1}\))
\(\Delta h\) = change in height (\(m\))

Note: You will always be given the value of \(g = 9.81 \, m s^{-2}\) on your data sheet. Always use this value unless the question tells you otherwise!

4. Conservation of Energy

This is one of the most fundamental laws in Physics. The Principle of Conservation of Energy states that energy cannot be created or destroyed; it can only be transferred from one form to another.

In a closed system:
Total Energy Before = Total Energy After

Energy Transfers in Mechanics

In many Unit 1 problems, you will see energy swapping between GPE and Kinetic Energy.
Example: A ball dropped from a height \(h\).
At the top: All energy is GPE.
As it falls: GPE is converted into Kinetic Energy.
At the bottom (just before hitting): All energy is Kinetic Energy.

Mathematically, if there is no air resistance:
\(m g \Delta h = \frac{1}{2} m v^2\)

Common Mistake: In the real world, some energy is usually "lost" to the surroundings as heat due to air resistance or friction. The energy isn't gone; it's just no longer "useful" mechanical energy.

5. Power

Power is the rate at which work is done or energy is transferred. If two people climb the same set of stairs, they do the same amount of work. However, the person who runs up the stairs has a higher Power because they did the work faster.

The Formula

\(P = \frac{W}{t}\)

Where:
\(P\) = Power (\(Watts, W\))
\(W\) = Work done (\(J\))
\(t\) = Time taken (\(s\))

Another very useful version of this formula for moving objects is:
\(P = F v\)
(Power = Force \(\times\) Velocity)

Unit Check: \(1 \, Watt = 1 \, Joule \, per \, second\).

6. Efficiency

No machine is perfect. Some energy is always transferred into non-useful forms (like heat or sound). Efficiency tells us how much of the "input" energy actually ends up as "useful output" energy.

The Formulas

Efficiency can be calculated using either energy or power:
\(\text{Efficiency} = \frac{\text{Useful energy output}}{\text{Total energy input}}\)

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

Important Points:
- Efficiency is usually expressed as a decimal (e.g., \(0.75\)) or a percentage (e.g., \(75\%\)).
- Efficiency can never be greater than \(1\) (or \(100\%\)). If your calculation gives you \(120\%\), go back and check your numbers—you’ve likely swapped the input and output!

Summary Table for Quick Revision

Work Done: \(W = F \Delta s \cos(\theta)\) (Unit: \(J\))
Kinetic Energy: \(E_k = \frac{1}{2} m v^2\) (Unit: \(J\))
Gravitational Potential Energy: \(\Delta E_{grav} = m g \Delta h\) (Unit: \(J\))
Power: \(P = \frac{W}{t}\) or \(P = F v\) (Unit: \(W\))
Efficiency: \(\frac{\text{Useful}}{\text{Total}}\) (No Unit)

Top Exam Tips:

1. Check your units: Always convert mass to \(kg\), distance to \(m\), and time to \(s\) before calculating.
2. Vector Directions: Remember that displacement is a vector. For Work Done, the force and displacement must be in the same line of action.
3. Energy Conservation: If a question asks for the speed of a falling object and doesn't mention time, try using the conservation of energy (\(GPE \to KE\)) instead of SUVAT equations. It's often much faster! (See the chapter on Motion and Kinematics Graphs for a refresher on SUVAT).