Physics (9630) | Work, energy and power

Physics 9630 Study Notes: Work, Energy, and Power

Hey there! Welcome to one of the most fundamental and useful chapters in Physics: Work, Energy, and Power.
This trio explains how things move, why they stop, and how quickly we can get things done. Understanding these concepts is essential because energy is the currency of the universe—it's what makes everything happen, from lifting a pen to launching a rocket!

Don't worry if the formulas seem tricky; we will break them down step-by-step and use real-life examples to make them stick. Let's dive in!

1. Work Done (\(W\))

In everyday life, 'work' means effort. In Physics, it has a very specific definition.

1.1 Definition of Work Done

Work Done is the energy transferred when a force causes an object to move through a distance. If you push on a wall all day and it doesn't move, you might be tired, but physically, you've done zero work on the wall!

The unit of work done is the Joule (J). (1 Joule is the work done when a force of 1 Newton moves an object 1 metre).

1.2 The Work Done Formula

The energy transferred (Work Done) by a constant force \(F\) moving an object a distance \(s\) is given by:

\(W = Fs \cos \theta\)

• \(F\) is the magnitude of the force (in Newtons, N).
• \(s\) is the displacement (distance moved) (in metres, m).
• \(\theta\) (theta) is the angle between the direction of the force and the direction of the displacement.

Key Point: The only part of the force that does work is the component parallel to the direction of movement. This is what the \(\cos \theta\) term accounts for.

Example Scenarios for \(\cos \theta\):

1. Pulling horizontally: If you pull a trolley straight forward, the force \(F\) and displacement \(s\) are in the same direction. \(\theta = 0^{\circ}\), and \(\cos 0^{\circ} = 1\).
   Therefore, \(W = Fs\).
2. Pulling a suitcase handle: If you pull a suitcase handle at an angle to the ground, only the horizontal component of your force does useful work moving it forward. \(\theta\) is the angle between the handle and the ground.
3. Force perpendicular to motion: If a force acts at a right angle to the displacement (e.g., gravity acting on a trolley moving perfectly horizontally), \(\theta = 90^{\circ}\), and \(\cos 90^{\circ} = 0\).
   Therefore, \(W = 0\).

Common Mistake Alert: Students often forget \(\cos \theta\). Remember that work is only done by the force in the direction of motion.

1.3 Work from Graphs

If the force changes while the object moves, we can't just use \(W = Fs\). However, the work done is represented by the area under a force-displacement graph.

(This is useful for variable forces, like stretching a spring or air resistance changes.)

2. Energy and Conservation

Energy is the ability to do work. In this chapter, we focus on three main forms of mechanical energy and how they interact.

2.1 Kinetic Energy (\(E_k\))

Kinetic Energy is the energy possessed by an object due to its motion.

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

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

Did you know? Because velocity is squared, doubling the speed quadruples the kinetic energy!

2.2 Gravitational Potential Energy (\(\Delta E_p\))

Gravitational Potential Energy (GPE) is the energy stored in an object because of its position in a gravitational field (i.e., its height above a certain point). We are usually interested in the change in GPE, \(\Delta E_p\).

\(\Delta E_p = mgh\)

• \(m\) is the mass (in kg).
• \(g\) is the acceleration due to gravity (approximately 9.81 N kg\(^{-1}\) or m s\(^{-2}\)).
• \(h\) is the change in vertical height (in m).

We treat GPE as zero at an arbitrary reference level (e.g., the ground).

2.3 Elastic Potential Energy

While the full treatment is in the next syllabus section (3.2.9), we need to know that Elastic Potential Energy is energy stored in objects like springs or stretched materials when they are deformed (e.g., compressed or extended). This energy can be converted into KE or GPE.

2.4 The Principle of Conservation of Energy

This is one of the most powerful laws in all of science:

Energy cannot be created or destroyed; it can only be transferred from one form to another.

Applying Energy Conservation

In many mechanics problems, especially those involving vertical motion, we apply this principle quantitatively:

Total Initial Energy = Total Final Energy + Energy Lost to Resistive Forces

If there is no air resistance or friction (ideal case):

\((E_k + E_p)_{\text{initial}} = (E_k + E_p)_{\text{final}}\)

If resistive forces (like friction or drag) are present, they do work against the motion, turning mechanical energy into heat (thermal energy):

\((E_k + E_p)_{\text{initial}} = (E_k + E_p)_{\text{final}} + W_{\text{resistive}}\)

Analogy: Think of a pendulum. At the highest point, it has maximum GPE (zero KE). As it swings down, GPE turns into KE. At the bottom, KE is maximum (zero GPE). If you ignore air resistance, it will always return to the same height!

3. Power (\(P\)) and Efficiency

We've looked at the energy transferred (Work, \(W\)). Now we look at the rate at which that transfer happens—that’s Power.

3.1 Definition of Power

Power is defined as the rate of doing work or the rate of energy transfer. It tells us how fast a machine or person can convert or use energy.

$$P = \frac{\text{Work Done}}{\text{time taken}} = \frac{\Delta W}{\Delta t}$$

Since work done is equal to the change in energy, we also write:

$$P = \frac{\text{Energy Transferred}}{\text{time taken}} = \frac{\Delta E}{\Delta t}$$

The SI unit for Power is the Watt (W). (1 Watt is equivalent to 1 Joule per second, 1 J s\(^{-1}\)).

3.2 Power and Velocity (The \(P=Fv\) Equation)

There is a crucial relationship linking power, force, and velocity, particularly useful when analyzing cars, aircraft, or motors operating at a constant speed (or terminal speed).

Since \(W = Fs\) (assuming force is parallel to displacement) and \(P = \frac{W}{t}\):

$$P = \frac{Fs}{t}$$

Since velocity \(v = \frac{s}{t}\):

$$P = Fv$$

Application: If a car engine is producing a constant power output \(P\), and the driving force \(F\) remains the same, as the velocity \(v\) increases, the engine must supply less force to maintain that velocity, or the power output must increase to maintain the force. At constant speed (cruising), the driving force \(F\) equals the resistive forces (drag).

3.3 Efficiency

No machine is perfect! Some energy is always wasted (usually as heat or sound). Efficiency measures how much of the energy input is converted into useful output energy or power.

$$\text{Efficiency} = \frac{\text{useful output power}}{\text{input power}}$$

Efficiency is often expressed as a percentage:

$$\text{Efficiency } (\%) = \frac{\text{useful output power}}{\text{input power}} \times 100\%$$

A perfect machine would have 100% efficiency, but this is never achieved in reality due to energy losses (like friction or electrical resistance).