Mathematics (9660) | Work and energy

M2.6: Comprehensive Study Notes – Work and Energy

Welcome to the Work and Energy chapter! This topic is absolutely essential in Mechanics (M2) because it gives you a powerful, alternative way to solve problems involving motion, especially those where forces or speeds are changing. Instead of constantly relying on Newton’s Laws and the SUVAT equations (which can get complicated), we use the concepts of energy and work, which often make the math much simpler!

Don't worry if this seems tricky at first; we will break down each concept step-by-step, using clear analogies.

1. Work Done by a Constant Force

In physics, "work" has a very specific meaning. You only do work if you apply a force and the object moves (is displaced) in the direction of that force.

Key Formula: Work Done (W)

The work done by a constant force \(F\) causing a displacement \(d\) is given by:

\(W = Fd \cos \theta\)

• \(F\) is the magnitude of the constant force (in Newtons, N).
• \(d\) is the magnitude of the displacement (in metres, m).
• \(\theta\) is the angle between the direction of the force and the direction of the displacement.
• The unit of work is the Joule (J). \(1 \text{ J} = 1 \text{ N m}\).

Understanding the Angle (\(\theta\))

The \(\cos \theta\) part is vital! It means only the component of the force that is parallel to the direction of motion does the work.

1. Force parallel to motion (\(\theta = 0^\circ\)):
   \(W = Fd \cos(0^\circ) = Fd\). This gives maximum positive work. Example: Pushing a trolley forwards.
2. Force opposite to motion (\(\theta = 180^\circ\)):
   \(W = Fd \cos(180^\circ) = -Fd\). This gives negative work, meaning the force is removing energy from the system. Example: Work done by friction or air resistance.
3. Force perpendicular to motion (\(\theta = 90^\circ\)):
   \(W = Fd \cos(90^\circ) = 0\). Zero work is done. Example: The normal reaction force or gravity when an object is sliding horizontally.

Quick Tip: If a force is not causing the object to speed up or slow down along its path, that force is probably doing zero work!

2. Energy Forms in Mechanics

Energy is the capacity to do work. In M2, we primarily focus on two types of mechanical energy:

2.1. Kinetic Energy (KE)

This is the energy an object possesses due to its motion.

• \(KE = \frac{1}{2} mv^2\)
• \(m\) is the mass (kg).
• \(v\) is the speed (m s\(^{-1}\)).

Did you know? Because velocity is squared, doubling the speed of a car increases its kinetic energy by a factor of four!

2.2. Gravitational Potential Energy (GPE)

This is the energy stored in an object due to its position in a gravitational field (i.e., its height).

• \(GPE = mgh\)
• \(m\) is the mass (kg).
• \(g\) is the acceleration due to gravity (\(9.8 \text{ m s}^{-2}\)).
• \(h\) is the vertical height above a defined zero level (m).

Important Note: We get to choose where \(h=0\). Usually, this reference level is the ground, or the lowest point in the problem. The absolute value of GPE doesn't matter, only the change in GPE matters.

3. The Work-Energy Principle

The work-energy principle links the concepts of force, displacement, and energy change. It is one of the most powerful ideas in mechanics.

The Principle

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

\(W_{\text{net}} = \Delta KE = KE_{\text{final}} - KE_{\text{initial}}\)

This means if you calculate the work done by *every* force (gravity, tension, friction, engine force, etc.), the resulting value tells you exactly how much the object's movement energy (KE) has changed.

Step-by-step process for using the Work-Energy Principle:

1. Identify the start point (initial state) and end point (final state).
2. Calculate the initial and final Kinetic Energy (\(KE_i\) and \(KE_f\)).
3. Calculate the work done by all individual forces (\(W_F\), \(W_{\text{friction}}\), \(W_{\text{gravity}}\), etc.) along the displacement. (Remember that forces perpendicular to motion do zero work!)
4. Sum the work done: \(W_{\text{net}} = W_1 + W_2 + W_3 + \dots\)
5. Set \(W_{\text{net}} = KE_f - KE_i\) and solve for the unknown quantity (often the final speed, \(v_f\)).

4. Conservation of Mechanical Energy

This is a specific case of the work-energy principle where things get even simpler!

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

When is Mechanical Energy Conserved?

Mechanical energy is conserved (i.e., remains constant) if and only if no work is done by non-conservative forces, such as friction or air resistance. Only conservative forces (like gravity or tension in a smooth pulley) must be acting.

If energy is conserved, the total mechanical energy at the start equals the total mechanical energy at the end:

\(KE_{\text{initial}} + GPE_{\text{initial}} = KE_{\text{final}} + GPE_{\text{final}}\)

or

\(\frac{1}{2} m v_i^2 + mgh_i = \frac{1}{2} m v_f^2 + mgh_f\)

What if Non-Conservative Forces are Present?

If forces like friction *are* present, they convert mechanical energy into other forms (like heat and sound). The energy is still conserved overall (the total energy of the universe is conserved), but the mechanical energy is not.

In this common scenario, the work done by the non-conservative forces (\(W_{\text{NC}}\)) is equal to the total change in mechanical energy:

\(W_{\text{NC}} = (KE_f + GPE_f) - (KE_i + GPE_i)\)

Since forces like friction always do negative work, the final energy will be less than the initial energy.

5. Power

Power measures how quickly work is being done or how quickly energy is being transferred.

Definition and Basic Formula

Power is the rate at which a force does work.

\(P = \frac{W}{t}\)

• \(P\) is power (in Watts, W).
• \(W\) is work done (in Joules, J).
• \(t\) is time taken (in seconds, s).

\(1 \text{ Watt} = 1 \text{ Joule per second} (1 \text{ J s}^{-1})\).

Power in terms of Force and Velocity

For a constant force \(F\) acting on a particle moving with constant velocity \(v\) in the direction of the force, the power output is simply:

\(P = Fv\)

How we get this: We know \(P = W/t\). We also know \(W = Fd\). Substituting \(W\), we get \(P = Fd/t\). Since distance over time is velocity (\(v = d/t\)), we arrive at \(P = Fv\).

This formula is incredibly useful, especially when dealing with constant engine power or drag forces. If a vehicle maintains a constant power output, then as its velocity \(v\) increases, the driving force \(F\) must decrease, because their product \(Fv\) must stay constant.