Work, energy and power

Welcome to Work, Energy, and Power!

Ever wondered why pushing a heavy shopping cart across a flat floor feels different from lifting it into the boot of a car? Or why a small engine takes longer to get a car up to speed than a powerful one? In this chapter, we explore the "currency" of the universe: Energy. We will look at how we spend it (Work) and how fast we spend it (Power).

This chapter is part of the Understanding processes: space, time and motion section. It builds on what you know about forces and motion, but gives us a much simpler way to solve problems without always needing to worry about complex accelerations.

1. Work Done: Spending Energy

In physics, "Work" has a very specific meaning. You might spend all day "working" on a difficult math problem, but if you haven't moved an object, you haven't done any Work in the eyes of a physicist!

Definition: Work is done whenever a force moves an object through a distance. Doing work is simply the process of transferring energy from one form to another.

The Formula

The energy transferred (Work Done) is calculated as:

\(\Delta E = F \Delta s\)

Where:
\(\Delta E\) = Work done (or change in energy) measured in Joules (J)
\(F\) = Force applied in Newtons (N)
\(\Delta s\) = Displacement (distance moved) in metres (m)

What if the force is at an angle?

Imagine pulling a sledge with a rope. You are pulling upwards and forwards, but the sledge only moves forward. Only the part of the force acting in the direction of movement does work.

The formula becomes:

\(\Delta E = F \Delta s \cos(\theta)\)

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

Quick Tip: If the force is perpendicular (at \(90^{\circ}\)) to the motion, no work is done. This is why a waiter carrying a tray at a constant height across a room does no work against gravity—the upward force on the tray is at right angles to the horizontal movement!

Key Takeaway: Work is done only when a force causes displacement in the direction of that force. No movement = No work.

2. The Law of Conservation of Energy

This is one of the most important rules in all of science. It’s the "Gold Standard" for checking if your physics calculations make sense.

The Rule: Energy cannot be created or destroyed. It can only be transferred from one store to another or dissipated (spread out) into the surroundings.

In our section on motion, this usually means that the Total Energy at the start of a process must equal the Total Energy at the end. If a ball rolls down a hill, its stored "height energy" turns into "movement energy."

Note: In the real world, some energy usually turns into heat due to friction. We say this energy is "dissipated," but it isn't gone—it's just no longer useful.

3. Kinetic Energy (\(E_k\))

Kinetic Energy is the energy an object possesses because it is moving. Any object with mass that has velocity has kinetic energy.

The Formula

\(E_k = \frac{1}{2} m v^2\)

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

Don't forget the square! Because the velocity is squared, doubling your speed actually quadruples (\(2^2 = 4\)) your kinetic energy. This is why car crashes at high speeds are so much more dangerous than at low speeds.

Quick Review: To find the speed of an object if you know its kinetic energy, rearrange the formula: \(v = \sqrt{\frac{2 E_k}{m}}\).

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

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

The Formula

\(E_p = m g h\)

Where:
\(m\) = mass in kg
\(g\) = gravitational field strength (on Earth, this is always \(9.81 N kg^{-1}\))
\(h\) = vertical height change in metres (m)

Analogy: Think of GPE like a savings account. You "deposit" energy by lifting the object up, and you can "withdraw" it (turn it into kinetic energy) by letting it fall.

Key Takeaway: In many exam questions, you can assume \(E_p\) lost = \(E_k\) gained (if friction is ignored). This means \(m g h = \frac{1}{2} m v^2\). Notice how the mass \(m\) cancels out? This tells us that, in a vacuum, all objects fall at the same rate regardless of their weight!

5. Power: The Rate of Doing Work

Imagine two people climbing the same flight of stairs. They both have the same mass. They both do the same amount of work (\(m g h\)). However, one person sprints up in 5 seconds, while the other takes 20 seconds. The sprinter is more powerful.

Definition: Power is the rate at which work is done or the rate at which energy is transferred.

Formula 1: The Basic Definition

\(P = \frac{\Delta E}{t}\)

Where:
\(P\) = Power in Watts (W)
\(\Delta E\) = Energy transferred in Joules (J)
\(t\) = Time taken in seconds (s)
Note: \(1 Watt = 1 Joule \ per \ second\).

Formula 2: Moving Objects

If an object is moving at a constant velocity \(v\) against a constant resistive force \(F\) (like a car driving down a highway), the power required is:

\(P = F v\)

Did you know? This formula comes from combining \(Work = Force \times Distance\) and \(Speed = Distance / Time\). Since \(P = \frac{F \times s}{t}\) and \(\frac{s}{t} = v\), we get \(P = F v\)!

6. Practical Skills & Modeling

In this module, you might be asked to model motion in small discrete time steps. This is a fancy way of saying "breaking a journey down into tiny chunks."

• In each tiny chunk of time (\(\Delta t\)), we assume the force is constant.
• We calculate the work done in that tiny step.
• We update the energy and velocity for the next step.

Common Practical: Measuring the "acceleration of free fall" (\(g\)). You might drop a ball through light gates to measure its velocity at different heights and use the conservation of energy (\(m g h = \frac{1}{2} m v^2\)) to calculate \(g\).

Summary Checklist

Before you move on to Newton's Laws and Momentum, make sure you can:

• Define Work Done and use \(\Delta E = F \Delta s \cos(\theta)\).
• State the Principle of Conservation of Energy.
• Calculate Kinetic Energy (\(\frac{1}{2} m v^2\)) and GPE (\(m g h\)).
• Calculate Power using both \(\frac{\Delta E}{t}\) and \(F v\).
• Explain why no work is done if the force and motion are perpendicular.

Final Tip for the Exam: Always check your units! Mass must be in kg (not grams), and distances must be in metres (not cm or mm). A common trick is to give you power in kilowatts (\(kW\))—multiply by \(1,000\) to get Watts!