Work, Energy and Power

Welcome to Work, Energy, and Power!

Hi there! Welcome to one of the most practical chapters in Mechanics. Have you ever wondered why it's harder to pull a sledge at an angle than straight ahead, or how engineers calculate the top speed of a supercar? This chapter is all about the "currency" of the universe: Energy. We will look at how we spend it (Work), how we store it (Potential Energy), and how fast we use it (Power). Don't worry if these seem like big ideas—we’ll break them down into bite-sized pieces together!

1. Work Done: The Cost of Movement

In physics, Work Done isn't just about effort; it’s about results. You can push against a brick wall all day until you're exhausted, but if the wall doesn't move, you haven't done any "Work" in the mechanical sense!

Work Done by a Constant Force

If a constant force \(F\) moves an object a distance \(d\) in the direction of the force, the work done is:
\(W = F \times d\)

What if the force is at an angle?
Imagine pulling a suitcase on wheels. You pull up and along, but the suitcase only moves along the floor. Only the part (component) of the force in the direction of motion counts. If the force is at an angle \(\theta\) to the direction of motion, we use:
\(W = Fd \cos(\theta)\)

Work Done as a Scalar Product (Stage 2)

If you are given the force and displacement as vectors, finding the work is even easier! It is the scalar product (dot product) of the force vector \(\mathbf{F}\) and the displacement vector \(\mathbf{x}\):
\(W = \mathbf{F} \cdot \mathbf{x}\)

Work Done by a Variable Force

Sometimes a force changes as you move (like a gusty wind). To find the work done by a variable force \(F(x)\) moving from position \(a\) to \(b\), we use integration:
\(W = \int_{a}^{b} F(x) dx\)

Quick Review:
• Work is measured in Joules (J).
• Work is a scalar quantity (it has size but no direction).
• Common Mistake: Forgetting to use \(\cos(\theta)\) when the force and motion aren't in the same line.

Key Takeaway: Work is done only when a force causes displacement. If there is no movement, or if the force is perpendicular to the motion (like gravity acting on a car driving on a flat road), the work done by that force is zero!

2. The Three Types of Mechanical Energy

Energy is the capacity to do work. In this course, we focus on three main "storage heaters" for energy.

A. Kinetic Energy (KE) - The Energy of Motion

Anything that moves has KE. The faster it goes or the heavier it is, the more KE it has.
\(KE = \frac{1}{2}mv^2\)
Stage 2 Tip: You can also express this using the scalar product of velocity: \(KE = \frac{1}{2}m(\mathbf{v} \cdot \mathbf{v})\).

B. Gravitational Potential Energy (GPE) - The Energy of Height

When you lift something up, you do work against gravity. That work is "stored" as GPE.
\(GPE = mgh\)
(Where \(m\) is mass, \(g\) is acceleration due to gravity \(9.8 \, ms^{-2}\), and \(h\) is the vertical height gained).

C. Elastic Potential Energy (EPE) - The Energy of Stretch

Before we calculate EPE, we need Hooke’s Law. It describes the tension (\(T\)) in a spring or string with natural length \(l\) and extension \(x\):
\(T = \frac{\lambda x}{l}\)
Here, \(\lambda\) is the modulus of elasticity. Think of it as a measure of how "stiff" the spring is.

The energy stored in that stretched spring (EPE) is:
\(EPE = \frac{\lambda x^2}{2l}\)

Memory Aid:
• KE is for Kicking (movement).
• GPE is for Going up.
• EPE is for Elastic/stretchy stuff.

Key Takeaway: Total Mechanical Energy is the sum of these three: \(ME = KE + GPE + EPE\).

3. The Work-Energy Principle

This is the "Golden Rule" for solving complex mechanics problems without needing to know the acceleration at every second. It links the work you do to the energy change you see.

The Principle:
The Work Done by external forces (like an engine or friction) equals the total change in Mechanical Energy.

In Equation form:
\(Work \, Done \, by \, Driving \, Force - Work \, Done \, against \, Resistance = \Delta KE + \Delta GPE + \Delta EPE\)

Conservation of Mechanical Energy

If there is no friction and no engine (no external work), then the total mechanical energy stays constant!
\(Initial \, (KE + GPE + EPE) = Final \, (KE + GPE + EPE)\)

Did you know?
A rollercoaster is a perfect example of this. At the top of the first hill, it has max GPE. As it drops, GPE turns into KE (speed!). If we ignore friction, the energy just keeps swapping forms back and forth.

Step-by-Step for Problems:
1. Identify the "Start" and "Finish" points.
2. List the KE, GPE, and EPE at both points.
3. Identify any Work Done by engines or friction.
4. Set up the equation: \(Energy_{Start} + Work_{In} - Work_{Out} = Energy_{Finish}\).

4. Power: How Fast are you Working?

Power is the rate of doing work. Two people might lift the same weight to the same height (same Work Done), but the person who does it faster is more Powerful.

Average Power

\(Power = \frac{Work \, Done}{Time \, Taken}\)
Power is measured in Watts (W), where \(1 \, W = 1 \, J/s\).

Power and Velocity

For a moving object (like a car) being pushed by a constant tractive force \(F\) at a velocity \(v\):
\(P = F \times v\)

Stage 2 Vector Form:
If force and velocity are vectors: \(P = \mathbf{F} \cdot \mathbf{v}\).

Maximum Speed

A car reaches its maximum speed when the driving force from the engine is exactly balanced by the resistance forces (like air resistance). At this point, the acceleration is zero.
To find max speed:
1. Use \(P = Fv\) to find the driving force.
2. Set driving force \(F\) = resistance forces.
3. Solve for \(v\).

Quick Review:
• Power is Work divided by Time.
• Use \(P = Fv\) for vehicles on roads or slopes.
• Encouraging Phrase: If you get stuck on a slope problem, always start by resolving forces parallel to the slope to find the driving force \(F\) first!

Key Takeaway: Power tells us how much energy is being converted every second. In car engines, we often look for the "Tractive Force" to find the Power.

Final Summary Table

Work Done: \(Fd\cos(\theta)\) or \(\mathbf{F} \cdot \mathbf{x}\) (Joules)
Kinetic Energy: \(\frac{1}{2}mv^2\) (Joules)
G. Potential Energy: \(mgh\) (Joules)
Hooke's Law: \(T = \frac{\lambda x}{l}\) (Newtons)
E. Potential Energy: \(\frac{\lambda x^2}{2l}\) (Joules)
Power: \(\frac{Work}{Time}\) or \(Fv\) (Watts)

Don't worry if this seems tricky at first—Mechanics is all about practice! Try drawing a clear diagram for every energy problem, and you'll be a pro in no time.