👋 Welcome to the World of Work and Energy in Mechanics!

Hello future Further Mathematicians! This chapter, "Work and Energy," is one of the most powerful tools in your Mechanics 2 toolkit. Why? Because instead of dealing purely with forces and acceleration (like in M1), we shift our focus to energy transfer. This often makes solving complex problems involving varying speeds and distances much, much easier!

We are moving from Newton's Laws (Force = mass × acceleration) to the language of energy (Work = Change in Energy). Don't worry if this seems tricky at first—we'll break down every concept step-by-step!

1. Work Done by a Constant Force

In Mechanics, "Work" has a very specific definition. It is a measure of the energy transferred when a force moves an object through a displacement.

1.1 The Definition and Formula

The simplest way to calculate the work done is when the force is acting in the same direction as the movement.

Key Definition: The Work Done (W) by a constant force is the product of the magnitude of the force and the distance moved in the direction of the force.

$$W = F d$$

  • F is the constant force (in Newtons, N).
  • d is the distance moved (in metres, m).
  • W (Work Done) is measured in Joules (J). (1 Joule = 1 Newton-metre).

Analogy: Imagine pushing a heavy box 5 meters across the floor using 10 N of force. The work done is \(10 \times 5 = 50 \text{ J}\).

1.2 When Force and Displacement are NOT Parallel

Often, a force is applied at an angle to the direction of motion (e.g., pulling a sledge with a rope). Only the component of the force parallel to the displacement does work.

If the force \(F\) makes an angle \(\alpha\) with the direction of motion, the formula becomes:

$$W = F d \cos \alpha$$

🧠 Memory Aid: The Cosine Rule

Remember that the component of the force acting along the displacement is \(F \cos \alpha\). You are just multiplying that effective force by the distance \(d\).

1.3 Positive, Negative, and Zero Work Done

Work done can be positive, negative, or zero, depending on the angle \(\alpha\):

  • Positive Work (\(\alpha < 90^\circ\)): The force aids the motion (e.g., a driving force). Energy is added to the system.
  • Negative Work (\(\alpha > 90^\circ\)): The force opposes the motion (e.g., friction or air resistance). Energy is removed from the system. This is sometimes called Work Done AGAINST the force.
  • Zero Work (\(\alpha = 90^\circ\)): The force is perpendicular to the motion (e.g., the normal reaction or gravity when moving horizontally). These forces do no work.

Common Mistake to Avoid: When calculating work done against resistance, remember that if the resistance \(R\) is 5 N, the work done by the resistance is \(-5d\). The work done against resistance is \((+5)d\). Be clear about what the question is asking!

Key Takeaway (Constant Force): Work is force times distance times the cosine of the angle. If the force acts perpendicular to the movement, the work done is zero.

2. Work Done by a Variable Force

In many realistic situations, the force applied or the resistance encountered changes depending on the object's position (for example, a spring force or air resistance depending on velocity, or a force deliberately varied by an engine).

2.1 Using Calculus to Find Work Done

When the force \(F\) is a function of the displacement \(x\), i.e., \(F = F(x)\), we must use integration to sum up the tiny amounts of work done (\(\delta W\)) over an interval.

The total work done by a variable force in moving a particle from position \(x_1\) to \(x_2\) is:

$$W = \int_{x_1}^{x_2} F(x) \, dx$$

Did you know? Geometrically, the work done is the area under the Force-Displacement graph. Integration is the mathematical way of finding this area.

Step-by-Step Process for Variable Force:

  1. Identify the function \(F(x)\) (the force as a function of position).
  2. Identify the initial position \(x_1\) and the final position \(x_2\).
  3. Set up the definite integral: \(\int_{x_1}^{x_2} F(x) \, dx\).
  4. Calculate the integral to find the total work \(W\).

Key Takeaway (Variable Force): When the force changes with position, work done is calculated using definite integration: \(\int F(x) \, dx\).

3. Energy Concepts

Work done is energy transfer. To understand energy transfers fully, we need to define the two main forms of mechanical energy studied in M2: Kinetic Energy and Gravitational Potential Energy.

3.1 Kinetic Energy (KE)

Kinetic Energy is the energy possessed by an object due to its motion. If an object is moving, it has KE.

$$KE = \frac{1}{2} m v^2$$

  • m is the mass (in kg).
  • v is the speed (in m s\(^{-1}\)).
  • KE is measured in Joules (J).

Analogy: A fast car has much more KE than a slow car of the same mass, because speed (\(v\)) is squared in the formula. Doubling the speed quadruples the KE!

3.2 Gravitational Potential Energy (GPE)

Gravitational Potential Energy is the energy stored in an object due to its position within a gravitational field, usually relating to its height above a defined level (a datum).

$$GPE = m g h$$

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

Important Note on Datum: The GPE value depends entirely on where you define \(h=0\). Always pick a convenient level (e.g., the bottom of the slope or the lowest point reached in the problem) and stick with it!

Worked Example: Work Done by Gravity

When an object of mass \(m\) is lifted vertically by a height \(h\), the force of gravity does negative work equal to \(-mgh\). The work done against gravity is \(+mgh\). This work done against gravity is the energy stored as GPE.

Key Takeaway (Energy): KE depends on motion (\(1/2 mv^2\)), and GPE depends on vertical height (\(mgh\)). Both are measured in Joules.

4. The Work-Energy Principle (WEP)

The Work-Energy Principle links the work done on an object to its change in kinetic energy. This is arguably the most important principle in this chapter!

4.1 The Core Relationship

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

$$W_{Total} = \Delta KE$$

$$\text{Total Work Done} = \text{Final } KE - \text{Initial } KE$$

$$W_{Total} = \frac{1}{2} m v_{final}^2 - \frac{1}{2} m u_{initial}^2$$

How to calculate \(W_{Total}\):

$$W_{Total} = \text{Work Done by Driving Forces} + \text{Work Done by Gravity} + \text{Work Done by Resistive Forces} + \dots$$

4.2 Applying the Work-Energy Principle

The WEP is extremely useful for problems involving distances and speeds, especially where acceleration might not be constant.

Example Scenario: A box is pushed up a rough incline.

The total work done (\(W_{Total}\)) includes:

  1. Work Done by Pushing Force: Positive work.
  2. Work Done by Friction/Resistance: Negative work (\(-R \times d\)).
  3. Work Done by Gravity: Negative work, related to the change in GPE (\(-mgh\)).

The sum of these three values must equal the change in \(\frac{1}{2} mv^2\).

🔥 Quick Review Box: WEP

If work is done ON the object (positive work), its KE increases. If work is done BY the object (negative work, like friction), its KE decreases.

5. Conservation of Mechanical Energy

Sometimes, the only forces doing work in a system are conservative forces.

5.1 Conservative Forces Explained

A force is conservative if the work done by it in moving a particle between two points is independent of the path taken. The primary conservative force we deal with in M2 is gravity.

(Non-conservative forces, like friction or air resistance, dissipate energy as heat, and the work done depends entirely on the path length.)

5.2 The Principle of Conservation of Mechanical Energy

If no non-conservative forces do work (or if their work is negligible), the total mechanical energy of the system remains constant.

$$\text{Total Energy (Initial)} = \text{Total Energy (Final)}$$

$$KE_1 + GPE_1 = KE_2 + GPE_2$$

$$\frac{1}{2} m u^2 + m g h_1 = \frac{1}{2} m v^2 + m g h_2$$

This principle is powerful because it simplifies problems significantly. You don't need to calculate the actual forces or accelerations—just the initial and final speeds and heights.

Example of Application: Roller Coasters

Ignoring friction, as a roller coaster car descends a hill (losing GPE), it gains exactly the same amount of KE, speeding up. As it climbs the next hill (gaining GPE), it loses KE, slowing down.

5.3 Dealing with Non-Conservative Forces

If non-conservative forces (like resistance \(R\)) ARE present and do work \(W_{NC}\), the total energy is NOT conserved. Instead, we adapt the equation:

$$\text{Initial Energy} + \text{Work In} = \text{Final Energy}$$

$$KE_1 + GPE_1 + W_{NC} = KE_2 + GPE_2$$

Where \(W_{NC}\) is the work done by the non-conservative forces (this is usually negative, representing energy lost due to resistance).

Key Takeaway (Conservation): If only gravity does work, the sum of KE and GPE is constant. If friction/resistance is involved, the work done by those forces must be included in the overall energy balance.

6. Power

Power is a measure of how quickly work is being done or how quickly energy is being transferred.

6.1 Definition and Units

Definition: Power is the rate at which work is done with respect to time.

$$P = \frac{dW}{dt}$$

  • Power (P) is measured in Watts (W). (1 Watt = 1 Joule per second, \(1 \text{ J s}^{-1}\)).

6.2 The Power Formula \(P = Fv\)

For an object moving at speed \(v\) under the action of a driving force \(F\), a very useful alternative definition of power is:

$$P = F v$$

This relationship is crucial, especially when dealing with motors, engines, or systems where the power output is constant but the force or speed varies.

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

We know \(P = \frac{dW}{dt}\). Since \(W = Fd\), then \(P = \frac{d(Fd)}{dt}\). If the force \(F\) is constant, \(P = F \frac{dd}{dt}\). Since \(\frac{dd}{dt}\) is the rate of change of displacement, or velocity \(v\), we get \(P = Fv\).

6.3 Applications of Constant Power

If an engine operates at constant power \(P\):

  • As the object speeds up, \(v\) increases, so the driving force \(F\) must decrease (since \(F = P/v\)).
  • At the maximum possible speed (terminal velocity) on a level road, the driving force \(F\) equals the total resistive forces \(R\). Therefore, the maximum speed \(v_{max}\) is found using \(P = R v_{max}\).

Key Takeaway (Power): Power is the rate of doing work, measured in Watts. The most important formula is \(P = Fv\), linking Power, Driving Force, and Velocity.

🎉 Conclusion and Next Steps

You have now grasped the foundational concepts of Work and Energy! These methods allow you to bypass complicated acceleration calculations and solve mechanical problems simply by accounting for energy transfer. Practice is key, especially mastering the application of the Work-Energy Principle when non-conservative forces are involved!

Keep up the great work!