Work, energy and power

Welcome to Work, Energy, and Power!

In this chapter, we are going to look at the "currency" of the physical world: Energy. We'll explore how we spend it (Work), how we store it (Potential Energy), and how fast we can use it (Power). These concepts are the backbone of Further Mechanics 1 and will help you understand everything from how roller coasters work to why cars need powerful engines to go uphill.

Don't worry if these concepts feel a bit abstract at first. We'll break them down into simple pieces with plenty of real-world examples!

1. Work Done: The Effort of Moving

In physics, Work Done has a very specific meaning. You aren't "doing work" just by thinking hard; you do work when you apply a force to an object and move it over a distance.

The Formula

The work done by a constant force is defined as:
\(Work\ Done = Force \times Distance\ moved\ in\ the\ direction\ of\ the\ force\)

Mathematically, if a force \(F\) acts at an angle \(\theta\) to the direction of motion:
\(W = Fs \cos \theta\)

Where:
- \(W\) is Work Done (measured in Joules, J)
- \(F\) is the Force (in Newtons, N)
- \(s\) is the displacement (in metres, m)

Analogy: Pulling a Suitcase

Imagine you are pulling a suitcase on wheels at an airport. You pull the handle upwards at an angle, but the suitcase moves horizontally. Only the horizontal part (component) of your pull is actually doing the "work" of moving the suitcase forward. This is why we use \(\cos \theta\)!

Quick Review: Prerequisite Concept

Before moving on, remember how to resolve forces. If you are moving up an inclined plane (a slope), the component of weight acting down the slope is \(mg \sin \alpha\). If there is friction, remember that \(F = \mu R\), where \(R\) is the normal reaction force.

Key Takeaway: Work is only done when a force causes displacement. No movement = No work done!

2. Kinetic Energy (KE): The Energy of Motion

Anything that is moving has Kinetic Energy. The faster it moves or the heavier it is, the more kinetic energy it has.

The Formula

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

Where:
- \(m\) is mass (kg)
- \(v\) is speed (ms\(^{-1}\))

Important Point: Because the velocity is squared, doubling the speed of a car actually quadruples its kinetic energy! This is why high-speed crashes are so much more dangerous than low-speed ones.

Did you know? Energy is a scalar quantity. This means it doesn't have a direction—it’s just a "bucket" of capacity to do things.

3. Potential Energy (GPE): The Energy of Position

Gravitational Potential Energy (GPE) is the energy an object stores because of its height above the ground. You can think of it as "stored work."

The Formula

\(GPE = mgh\)

Where:
- \(g\) is the acceleration due to gravity (\(9.8\ ms^{-2}\))
- \(h\) is the vertical height (m)

The "Zero Level" Trick: You can choose anywhere to be your "zero height" (\(h = 0\)). Usually, we pick the lowest point in the problem. If a particle drops below that point, its GPE simply becomes negative!

Key Takeaway: GPE only depends on vertical height, not the path taken to get there. Whether you take the lift or the stairs, if you end up at the same floor, your change in GPE is the same.

4. The Work-Energy Principle

This is the most important tool in your kit. It connects the work you do to the energy the object gains or loses.

The Principle: The total work done by all forces (other than gravity) acting on a particle is equal to the change in its total mechanical energy.

\(Work\ Done\ by\ external\ forces = \Delta KE + \Delta GPE\)

In many exam problems, you will have a Driving Force (like an engine) and a Resistive Force (like friction or air resistance).
- Work done by Driving Force adds energy to the system.
- Work done against Resistance (Friction \(\times\) Distance) removes energy from the system (usually as heat).

Step-by-Step for Solving Problems:

1. Identify the Initial state (KE\(_i\) and GPE\(_i\)).
2. Identify the Final state (KE\(_f\) and GPE\(_f\)).
3. Identify External Work (Driving forces or Friction).
4. Plug them into: \(Initial\ Energy + Work\ Done\ by\ Driving\ Force - Work\ Done\ against\ Resistance = Final\ Energy\)

5. Conservation of Mechanical Energy

This is a special case of the Work-Energy Principle. If there are no external forces (like friction or engines) doing work, then the total mechanical energy stays the same!

\(KE + GPE = constant\)

Example: A bead sliding down a smooth (frictionless) wire. As it goes down, GPE turns into KE. As it goes up, KE turns back into GPE. The total never changes.

Common Mistake: Students often forget that "smooth" means no friction, but "rough" means you must include work done against friction in your equations.

6. Power: The Speed of Energy

Power is the rate at which work is done. If two people lift the same weight, but one does it faster, that person is more "powerful."

The Formulae

1. \(P = \frac{Work\ Done}{Time}\) (measured in Watts, W)
2. \(P = Fv\)

The second formula (\(P = Fv\)) is incredibly useful for car problems. It links the Driving Force (F) of the engine to the Velocity (v) of the car at any given moment.

Analogy: Think of a car climbing a hill. If the engine has a constant maximum power, the car must slow down (\(v\) decreases) if the force required to climb the hill (\(F\)) increases. This is why you change to a lower gear!

The "Variable Resistance" Scenario

Sometimes the resistance isn't constant; it might depend on the speed (e.g., \(R = kv\)). In these cases, you use \(P = Fv\) where the Driving Force \(F\) must equal the total resistance for the car to move at a constant speed.

Quick Review Box:
- 1 Watt = 1 Joule per second.
- Constant speed means Acceleration = 0, so Resultant Force = 0.
- Therefore, at constant speed: Driving Force = Total Resistances.

Final Summary Checklist

Check if you can:
- Calculate Work Done using \(Fs \cos \theta\).
- Find KE (\(\frac{1}{2}mv^2\)) and GPE (\(mgh\)).
- Set up a Work-Energy Principle equation for a particle on a slope.
- Use \(P = Fv\) to find the driving force or maximum speed of a vehicle.
- Handle friction by calculating work done against it (\(Friction \times Distance\)).

Remember, mechanics is all about the diagrams! Always start by drawing a clear sketch with all forces and heights labelled. You've got this!