Work, Energy and Power

Work done

Work done by a force is the product of the forces and the distance moved in the direction of the force.
SI unit: joule (J) – the SI unit of energy.
1 J = 1 N m

\begin{aligned} W=F\times s \end{aligned}

Work done must be calculated by using the component of the force that is parallel to the direction of the displacement. If the two directions are the same, work is done on the object and the objects gains energy. If the two directions are opposite, then energy is lost through work done(typically as friction or air resistance)

Work done must be calculated by using the component of the force that is parallel to the direction of the displacement.
If the two directions are the same, work is done on the object and the objects gains energy.
If the two directions are opposite, then energy is lost through work done(typically as friction or air resistance)

Work done by a gas

work on gas

Work done by a gas equals to the pressure it exerts over the cross-sectional area and the displacement the piston moves.

\begin{aligned} F&=p\times A\\W&=p\times A\times s\\&=p\times \Delta V \end{aligned}

Gravitational potential energy

Work is done by you during lifting. Object gains height when work is done against gravity. Gravitational potential energy increases as an object moves to a higher ground.

gpe

Other forms of potential energy

There are many forms of potential energy. Some examples are below:

  1. Electrical potential energy – capacitor. A capacitor stores charge which provides the capacitor the ability to produce current. A battery does not store electrical potential energy. Instead, it is stored as chemical energy.
  2. Elastic potential energy – spring
  3. Chemical potential energy – food, battery. Usually, chemical energy allows the substance to burn for a period of time, such as most organic compounds.
  4. Nuclear energy – radioactive nuclei

Kinetic energy

Kinetic energy is the energy due to motion. A moving object contains kinetic energy. To derive the equation of kinetic energy, we first assume an object is at rest. A constant force does work on the object, causing it to accelerate.

\begin{aligned} v^2&=u^2+2as\\&=2as \text{ since the object is initially at rest} \end{aligned}

Multiplying \frac{1}{2}m on both sides,

\begin{aligned} \frac{1}{2}mv^2&=mas\\&=Fs \end{aligned}

Hence, work done is converted into kinetic energy of \frac{1}{2}mv^2 .

Gravitational PE and Kinetic energy Transformation

When a high object falls to a lower height, the GPE is converted to KE. The increase in KE cause the object to increase in speed. The decrease in height should always be calculated based on the drop in vertical height.

Efficiency

Efficiency is the ratio of the useful output energy to the total input energy expressed as a percentage.

\begin{aligned} \text{efficiency}=\frac{\text{useful output energy}}{\text{total input energy}} \times 100\% \end{aligned}

It is important for you to determine what makes the useful output energy in an energy transformation situation. Below are some situations of energy transformations. Suggest the useful output energy and the energy/energies that are lost.

  1. A luggage being delivered from the ground floor to the first floor.
  2. A man turning a generator to produce electricity.
  3. Water flowing through a hydroelectric damp.
  4. Wind flowing through a windmill.

Coulomb’s Law

Coulomb’s force is the electrical force between two charges. Since there are positive and negative charges, coulomb’s force can be attractive or repulsive. However, the magnitude of the force produced by a point charge is similar to gravitational force:

\begin{aligned} F=\pm \frac{1}{4\pi \epsilon _0} \frac{Q_1Q_2}{r^2} \end{aligned}

\epsilon_o is the permittivity of free space. It is a description of how well electric field lines permeates in vacuum. It has a value of 8.85\times 10^{-12} \text{m}^{-3}\text{ kg}^{-1}\text{s}^4\text{A}^2

Electric Field

An electric field is a region which a charge experiences an electrical force.

Similar to gravitational field strength, the electric field strength at a point is the coulomb force per unit charge placed at that point.

\begin{aligned} E=\pm \frac{1}{4\pi \epsilon _0}\frac{Q}{r^2} \end{aligned}

Properties of electric field

  • Field lines are represented by arrows, starting from positive and ends at the negative.
  • Field lines do not cross.
  • Field lines always emerge or enters a conducting surface(s.g. metal) at a perpendicular direction.
  • There are no electric field in a conductor.

The direction of the field line represents the direction of electric force acting on a positive charge when it is placed in the field. As such, the direction of electric force on a negative charge is opposite to the direction of the field lines.

Challenge 1

Can you suggest why the electric field lines emerging or entering a conducting surface is perpendicular?

Electric Potential

Electric potential of a point is the amount of work done in bringing a unit charge from infinity to that point.

Similar to gravitational potential,

\begin{aligned} \phi=\pm \frac{1}{4 \pi \epsilon_o}\frac{Q}{r} \end{aligned}

The electric potential energy is the amount of work done in bring a charge from infinity to that point.

The difference between electric potential and potential energy is that for electric potential, it is the energy per unit charge while in potential energy, we are considering the entire charge, which may be more than 1 coulomb.

Relationship between electric field and potential

It is important to remember that the field strength is the negative of the potential gradient. Potential gradient means that gradient of the potential-distance graph.

The negative gradient of this graph gives the value of the electric field strength.
The negative gradient of this graph gives the value of the electric field strength.

Uniform Electric Field

A uniform electric field can be produced by a pair of parallel plates. A charge placed anywhere inside a uniform field experiences the same force, regardless of whether it is nearer to the positive or negative plate.

You can think that a positive charge, if placed near the negative plate, experience more attraction by the negative plate and less repulsion by the positive plate. The same charge placed near the positive plate experiences more repulsion from the positive plate and less attraction by the negative plate. Hence the charge experiences the same force anywhere in this uniform field.

Energy of a Simple Harmonic Oscillator

Energy changes in according to displacement

The kinetic energy of a simple harmonic oscillator is

\begin{aligned} E_\text{kinetic} &= \frac{1}{2}m \omega^2(x_o^2 - x^2) \end{aligned}

In a SHM, the oscillator’s kinetic energy and potential energy always changes from maximum to zero throughout the oscillations. However, at all time, the total energy of the oscillator is constant. This value can be obtained by calculating the maximum kinetic energy of the system:

\begin{aligned} E_\text{total} &= \frac{1}{2}mv_o ^2\\&=\frac{1}{2}m \omega^2 x_o^2 \end{aligned}

To find out the potential energy of a simple harmonic oscillator,

\begin{aligned} E_\text{potential} &= E_\text{total} - E_\text{kinetic}\\&=\frac{1}{2}m \omega^2x_o^2 - \frac{1}{2}m \omega^2(x_o^2 - x^2)\\&=\frac{1}{2}m\omega^2 x^2\end{aligned}

The total energy is constant, but the kinetic and potential energy changes throughout the oscillator.
The total energy is constant, but the kinetic and potential energy changes throughout the oscillator.

The energy changes of an SHM oscillator changes in a sinusoidal pattern.

Energy changes according to time

From the velocity equation, the kinetic energy is

\begin{aligned} E_\text{kinetic}&=\frac{1}{2}mv^2\\&=\frac{1}{2}mv_o ^2 \cos^2{\omega t}\end{aligned}

The potential energy is the difference between the total energy and the kinetic energy,

\begin{aligned} E_\text{potential}&=\frac{1}{2}mv_o ^2 - \frac{1}{2}mv_o ^2 \cos^2{\omega t}\\&=\frac{1}{2}mv_o^2(1-\cos^2{\omega t})\\&=\frac{1}{2}mv_o \sin^2{\omega t} \end{aligned}

Uniform Gravitational Force

 

Uniform gravitational force

The gravitational force acting on a mass is called weight. Weight always acts vertically down and its magnitude is directly proportional to the mass.

\begin{aligned} W=mg \end{aligned}

where g is the gravitational field strength on the Earth’s surface. Gravitational field strength of a point is the gravitational force acting on a unit mass at that point.

\begin{aligned} g=\frac{F}{m} \end{aligned}

The SI unit of gravitational field strength is \text{N kg}^{-1} .

On the Earth’s surface, the gravitational field is uniform and its direction is vertically down. A uniform field is represented by parallel field lines. A mass placed anywhere in a uniform field experiences the same gravitational force.

The gravitational field on Earth's surface is pointing vertically down
The gravitational field on Earth’s surface is pointing vertically down

The direction of the gravitational field indicates the direction of the force acting on a mass placed in the field.

When a mass is thrown vertically upward, its initial velocity is v. At its maximum height, its velocity is 0\text{ m s}^{-1}. It then falls and its velocity reaches v in the downward direction.

forces

Challenge 1

What are the values of acceleration at point A, B and C?