Work, energy and power

Welcome to Work, Energy, and Power!

In this chapter, we are going to explore the "currency" of the universe: Energy. We’ll look at how we measure the effort we put into moving things (Work), how fast we can get things done (Power), and the amazing rule that energy never actually disappears—it just changes its "outfit."

Don't worry if these terms sound like things you use in everyday speech; in Physics, they have very specific meanings, and we’re going to break them down step-by-step.

1. Work Done: The Result of Your Effort

In Physics, you can push a brick wall until you are exhausted, but if the wall doesn't move, you haven't done any work! For work to be done, a force must move an object over a displacement.

The Formula

Work done is defined as the product of the force and the displacement in the direction of the force:
\( W = Fs \)

Where:
\( W \) = Work done (measured in Joules, J)
\( F \) = Force (measured in Newtons, N)
\( s \) = Displacement (measured in meters, m)

Important Condition: Direction Matters!

The force and the displacement must be in the same direction. If you pull a suitcase at an angle, only the part of the force pulling forward does work.

Common Mistake to Avoid: If you carry a heavy box while walking horizontally at a constant speed, you are technically doing zero work on the box because your lifting force is upwards, but the movement is sideways!

Quick Review: Work

• 1 Joule is the work done when a force of 1 Newton moves an object 1 meter.
• If there is no movement, there is no work.
• Key Takeaway: Work is energy being transferred by a force.

2. The Law of Conservation of Energy

This is one of the most important rules in all of science. It states that:
Energy cannot be created or destroyed. It can only be transformed from one form to another.

Analogy: Think of energy like money. You can have it in cash (Kinetic Energy), in a savings account (Potential Energy), or spend it on something that generates heat (Work against friction). You haven't "destroyed" the money; it's just in a different place or form.

Did you know? When a car brakes, its motion energy isn't "lost"—it's mostly turned into heat energy in the brake discs!

3. Efficiency: How Good is the System?

In the real world, no machine is perfect. Some energy is always "wasted" (usually as heat or sound).

The Formula

\( \text{Efficiency} = \frac{\text{Useful energy output}}{\text{Total energy input}} \times 100\% \)

Or using power:
\( \text{Efficiency} = \frac{\text{Useful power output}}{\text{Total power input}} \times 100\% \)

Quick Tip: Efficiency is always less than 1 (or 100%). If your calculation gives you 120%, go back and check your numbers—you might have swapped the input and output!

Key Takeaway: Energy Conservation

• Total Energy In = Useful Energy Out + Wasted Energy Out.
• Efficiency tells us what percentage of energy actually does what we want it to do.

4. Gravitational Potential Energy (\( E_P \))

This is the energy an object has because of its position in a gravitational field. Basically, the higher you lift something, the more "stored" energy it has.

Deriving the Formula

Don't worry, this derivation is quite simple!
1. We know Work \( W = F \times s \).
2. To lift an object, the Force \( F \) must equal its weight, which is \( mg \).
3. The displacement \( s \) is the height \( h \).
4. So, \( W = mg \times h \).

Therefore: \( \Delta E_P = mg\Delta h \)

Where:
\( m \) = mass (kg)
\( g \) = acceleration of free fall (\( 9.81 \, \text{m s}^{-2} \))
\( h \) = change in height (m)

5. Kinetic Energy (\( E_K \))

This is the energy an object possesses because of its motion. If it's moving, it has \( E_K \).

The Formula

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

Deriving the Formula (The Step-by-Step Way)

We use one of our equations of motion: \( v^2 = u^2 + 2as \).
1. Assume the object starts from rest, so \( u = 0 \). This gives us \( v^2 = 2as \).
2. Rearrange to find acceleration: \( a = \frac{v^2}{2s} \).
3. We know Force \( F = ma \). Substitute our \( a \) into this: \( F = m(\frac{v^2}{2s}) \).
4. Work done \( W = Fs \). Substitute our \( F \) into this: \( W = (m \frac{v^2}{2s}) \times s \).
5. The \( s \) cancels out, leaving us with: \( W = \frac{1}{2}mv^2 \).

Since the work done to accelerate the object equals its kinetic energy, \( E_K = \frac{1}{2}mv^2 \).

Key Takeaway: Potential vs. Kinetic

• \( E_P \) is about height; \( E_K \) is about speed.
• In a falling object (with no air resistance), the loss in \( E_P \) equals the gain in \( E_K \).

6. Power: Speeding Things Up

Power is the rate at which work is done. It's not about how much work you do, but how fast you do it.

Example: Two people climb the same stairs. They both do the same amount of work (lifting their weight to the same height). But the person who runs up has a higher power because they did the work in less time.

The Formulas

1. General formula: \( P = \frac{W}{t} \)
(Measured in Watts, W. 1 Watt = 1 Joule per second)

2. For a moving object: \( P = Fv \)

How do we get \( P = Fv \)?

1. Start with \( P = \frac{W}{t} \).
2. Since \( W = Fs \), we get \( P = \frac{Fs}{t} \).
3. We know that velocity \( v = \frac{s}{t} \).
4. Substitute \( v \) into the equation: \( P = F \times v \).

This formula is very useful for vehicles moving at a constant speed where the engine force equals the resistive forces (like air resistance).

Quick Review: Power

• Power is Work divided by Time.
• Unit: Watt (W).
• Memory Aid: "Power is how fast you spend your energy."

Summary Checklist

Before you finish this chapter, make sure you can:
• Calculate Work using \( W = Fs \).
• Explain why energy is always conserved.
• Calculate Efficiency as a percentage.
• Use \( \Delta E_P = mg\Delta h \) and \( E_K = \frac{1}{2}mv^2 \).
• Use \( P = \frac{W}{t} \) and \( P = Fv \) to solve problems.

You've got this! Practice a few calculation questions, and these formulas will become second nature.