Mathematics | Work and energy

Welcome to Work and Energy!

Hello future physicist! In Unit M1, you mastered forces, velocity, and acceleration. Now, in M2, we connect those concepts to the crucial ideas of Work and Energy. This chapter explains why things move and how much effort is required to change their motion. Understanding this allows us to solve complex problems without always relying on Newton's Second Law (\(F=ma\)), making problem-solving much faster!

Don't worry if this seems abstract at first. We will break down every concept into clear, manageable steps. Let's get started!

Section 1: Work Done by a Constant Force

1.1 Defining Work Done (\(W\))

In physics, Work Done is a measure of the energy transferred when a force moves an object through a displacement. It’s not just about effort; it's about effective effort.

Key Idea: Work is only done if the force causes movement in its direction (or opposite to its direction).

The unit for Work Done is the Joule (J), which is the standard unit for energy.

1.2 The Formula for Work Done

If a constant force \(F\) moves an object through a distance \(d\) in the direction of the force, the formula is simple:

$$W = Fd$$

However, force and displacement are not always parallel. If the force \(F\) acts at an angle \(\theta\) to the direction of displacement \(d\), we only use the component of the force that is parallel to the movement:

$$W = Fd \cos\theta$$

Where:

• \(F\) is the magnitude of the force (N).
• \(d\) is the distance displaced (m).
• \(\theta\) is the angle between the force and the displacement.

Memory Aid: Think of dragging a heavy suitcase. You pull the handle upwards at an angle (\(F\)), but the suitcase only moves horizontally (\(d\)). You are only doing work with the horizontal component of your pull (\(F \cos\theta\)).

1.3 Special Cases of Work Done

The angle \(\theta\) is crucial:

Case 1: Force is Parallel to Displacement

If the force pushes exactly in the direction of motion, \(\theta = 0^{\circ}\). Since \(\cos(0^{\circ}) = 1\):

$$W = Fd$$

Case 2: Force is Perpendicular to Displacement (Zero Work)

If the force is perpendicular to the motion (e.g., gravity acting on a object sliding perfectly horizontally), \(\theta = 90^{\circ}\). Since \(\cos(90^{\circ}) = 0\):

$$W = 0$$

Example: If you carry a heavy tray across a flat room, the upward force you exert (to support the tray against gravity) does zero work, because the displacement is horizontal.

Case 3: Work Done Against Motion (Negative Work)

If the force opposes the direction of motion (e.g., friction or air resistance), \(\theta = 180^{\circ}\). Since \(\cos(180^{\circ}) = -1\):

$$W = -Fd$$

Work done by forces like friction is usually negative, meaning they remove energy from the system.

Section 2: The Two Essential Forms of Mechanical Energy

In M2, we primarily deal with two types of mechanical energy: energy due to movement and energy due to position.

2.1 Kinetic Energy (KE)

Kinetic Energy (KE) is the energy a body possesses due to its motion. Everything that moves has KE. The unit is the Joule (J).

The Formula for Kinetic Energy

Kinetic Energy depends on the object's mass (\(m\)) and its speed (\(v\)):

$$KE = \frac{1}{2}mv^2$$

Important Points:

• Since \(v\) is squared, doubling the speed quadruples the KE (it has a much bigger impact than doubling the mass).
• KE is always non-negative (because \(m\) is positive and \(v^2\) is positive).

Did you know? Even small molecules moving at room temperature have KE!

2.2 Gravitational Potential Energy (GPE)

Gravitational Potential Energy (GPE) is the energy stored in an object due to its position within a gravitational field, specifically its height above a certain reference point.

The Formula for GPE

GPE depends on the mass (\(m\)), the acceleration due to gravity (\(g\)), and the height (\(h\)):

$$GPE = mgh$$

Where:

• \(m\) is the mass (kg).
• \(g\) is the acceleration due to gravity (usually \(9.8 \text{ m/s}^2\)).
• \(h\) is the vertical height above the reference level (m).

Choosing the Reference Level

The value of GPE is relative. We must always define a point where \(h=0\). This is often the ground, but you can choose any convenient point (like the lowest point in the problem).

• If the object is above the reference level, \(h\) is positive, and GPE is positive.
• If the object is below the reference level, \(h\) is negative, and GPE is negative.

Section 3: The Work-Energy Principle

The Work-Energy Principle is one of the most fundamental ideas in mechanics. It provides the link between the work done on an object and its resulting change in energy.

3.1 The Principle Defined

The principle states that the net work done by all forces acting on a particle is equal to the change in its kinetic energy.

$$W_{\text{net}} = \Delta KE$$

Where \(\Delta KE = KE_{\text{final}} - KE_{\text{initial}} = \frac{1}{2}mv^2 - \frac{1}{2}mu^2\).

Analogy: If you spend £10 (net work input) on an object, its bank balance (KE) changes by £10. If friction removes £3 of energy, the net work is \(W_{\text{applied}} - W_{\text{friction}}\).

3.2 The Comprehensive Work-Energy Equation (W_{NC})

When solving complex M2 problems involving gravity and friction/resistance, it's often more practical to consider the work done by non-conservative forces (forces that depend on the path taken, like friction or air resistance).

Conservative Forces (like gravity) are those where the work done depends only on the start and end points (not the path). The energy associated with conservative forces is stored as Potential Energy (GPE).

The most useful equation in this chapter combines all elements:

$$W_{\text{NC}} = \Delta KE + \Delta GPE$$

This means:

Work done by resistance/driving force/etc. = (Change in KE) + (Change in GPE)

$$W_{\text{NC}} = \left(\frac{1}{2}mv_{\text{final}}^2 - \frac{1}{2}mu_{\text{initial}}^2\right) + \left(mgh_{\text{final}} - mgh_{\text{initial}}\right)$$

Step-by-Step Application

1. Identify State 1 (Initial) and State 2 (Final): Determine initial and final speeds (\(u, v\)) and heights (\(h_1, h_2\)).
2. Choose a Reference Level: Define where \(h=0\).
3. Calculate Changes: Find \(\Delta KE\) and \(\Delta GPE\). Remember, \(\Delta = \text{Final} - \text{Initial}\).
4. Calculate \(W_{\text{NC}}\): This is the work done by all forces *except* gravity. This typically includes a driving force (positive work) and resistance (negative work).
5. Solve the Equation: \(W_{\text{NC}} = \Delta KE + \Delta GPE\).

Section 4: The Principle of Conservation of Mechanical Energy

The Principle of Conservation of Energy states that energy cannot be created or destroyed, only transformed from one form to another.

4.1 When is Mechanical Energy Conserved?

Mechanical Energy is the sum of Kinetic Energy and Potential Energy (\(E = KE + GPE\)).

This total energy is conserved (remains constant) only if no non-conservative forces are doing work. In practical terms, this means:

• There is no air resistance.
• There is no friction.
• There is no external driving or braking force.

If these conditions are met, then \(W_{\text{NC}} = 0\).

4.2 The Conservation Equation

If Mechanical Energy is conserved, the total energy at the initial position (1) must equal the total energy at the final position (2):

$$E_1 = E_2$$

$$KE_1 + GPE_1 = KE_2 + GPE_2$$

$$\frac{1}{2}mu^2 + mgh_1 = \frac{1}{2}mv^2 + mgh_2$$

Example: Think of a frictionless roller coaster. As it goes down a hill, GPE is converted into KE, increasing speed. As it goes up the next hill, KE is converted back into GPE, slowing it down. The total energy remains constant.

4.3 Transforming Between Energy Principles

It is important to recognize that the Conservation of Mechanical Energy equation is simply a special case of the Work-Energy Principle:

Start with the Work-Energy Principle:

$$W_{\text{NC}} = \Delta KE + \Delta GPE$$

If there are no non-conservative forces, \(W_{\text{NC}} = 0\):

$$0 = (KE_2 - KE_1) + (GPE_2 - GPE_1)$$

Rearranging the terms gives the conservation equation:

$$KE_1 + GPE_1 = KE_2 + GPE_2$$

Choose the conservation method if the system is ideal (no resistance). Choose the work-energy method if non-conservative forces (like friction or a driving force) are involved.