**Energy **of a system is defined as its capacity to do work. (Note: Work is covered in the next sub-topic)

- SI unit : joules (J)
- Scalar Quantity; Energy has magnitude only.

**The Principle of conservation of energy states that energy cannot be created nor destroyed in any process.**

- Total amount of energy of a closed system remains
**constant.** - Energy can be converted/transformed from one form to another. (
**Note:**You can think of the different forms of energy as some form of “currencies”, whereby the exchange rates among the different “currencies” are 1:1.)**E.g. 1:**A television converts electrical energy(electricity) into light, sound and thermal energies.**E.g. 2:**Burning of fuels (wood) converts stored chemical energy into heat and light energies.

- It can also be transferred from one body to another through work done and/or heat exchanges.

### Forms of Energy:

- Potential Energy (Elastic, gravitational and chemical)
- Kinetic Energy (or mechanical energy)
- Electrical Energy
- Thermal Energy (or heat)
- Light
- Nuclear Energy

**Kinetic Energy**, $E_{k}$ is the energy a body possessed by virtue of its motion.

- Moving objects have kinetic energy.
- Kinetic energy can be used to do work.
- $E_{k} = \frac{1}{2} m v^{2}$
- where m = Mass (in kg), v = Velocity (in $\text{m s}^{-1}$)

### Worked Example 1: Speeding Projectile

A projectile of mass 0.02 kg travels at a speed of $1200 \text{ m s}^{-1}$. Calculate its kinetic energy.

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$$\begin{aligned} E_{k} &= \frac{1}{2} mv^{2} \\ &= \frac{1}{2} (0.02)(1200)^{2} \\ &= 14400 \, \text{J} \end{aligned}$$

**Potential Energy** is the stored energy in a system.

- Example of chemical potential energy: Wood; When you burn wood, the chemical potential energy in wood is converted into thermal energy (heat) and light.
- Example of elastic potential energy: Rubber band; When you stretch a rubber band, elastic potential energy is stored in the stretched rubber band.

**Gravitational Potential Energy** is defined as the amount of work done in order to raise the body to the height *h* from a reference level.

- G.P.E.= mgh, where m = mass (in kg), g = acceleration due to gravity (in $\text{m s}^{-2}$), h = height (in m)
- An object at X m above the reference level (commonly taken to be the ground level) will have gravitational potential energy of mgX. When the object is released from the height (X m), the object will have all its gravitational potential energy gradually converted into kinetic energy, just before it hits the ground. (Assuming that there is no air resistance)
- From the diagram above, the conversion of kinetic energy to gravitational potential energy, and back to kinetic energy is shown.

### Worked Example 2: Lifting Object

An object with a mass of 5 kg is lifted vertically through a distance of 10 m at a constant speed. What is the gravitational potential energy gained by the object?

(Take the acceleration due to gravity to be $10 \text{ m s}^{-2}$)

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$$\begin{aligned} E_{gpe} &= mgh \\ &= (5)(10)(10) \\ &= 500 \, \text{J} \end{aligned}$$

### Worked Example 3: Nail In A Plank

A nail is being hammered into a plank.

- What energy does a raised hammer possess?
- When it falls, what energy will the energy in part a. convert into?
- What is the subsequent energy used for?
- Are there any other forms of energy produced? If so, name them.

### Click here to show/hide answer

- Gravitational potential energy
- Kinetic energy
- It is used to drive the nail into the plank
- Some of the gravitational potential energy is converted into sound and thermal energy

The case studies below highlights the conversion of kinetic energy into gravitational potential energy and vice versa. They will help to solidify your understanding towards the concept.

### Case Study 1: Ideal Pendulum

Consider an ideal pendulum (as shown in the diagram below). Note that ideal pendulum means that there is no energy lost to overcome air resistance and friction during oscillation.

When a pendulum is displaced to one side (Point A), it gains gravitational potential energy. The amount of gravitational potential energy gained will be mgh, where h is the height difference of point A and point B.

When the raised pendulum is released from A, it will swing towards the equilibrium position (Point B). During this movement, the gravitational potential energy is converted into kinetic energy. Hence, at A, gravitational potential energy is at minimum, while kinetic energy is at maximum.

From Point B, the pendulum will swing upwards to the other side (Point D). During this movement, the pendulum will lose kinetic energy and gain gravitational potential energy. Since this is an ideal pendulum, the pendulum will swing up to Point D, where Point D is at the same height as Point A.

In summary, the relevant energy conservation equations are:

From Point A to Point B,

$$\text{Initial G.P.E. of pendulum at Point A} = \text{Final K.E. of pendulum at Point B}$$

From Point B to Point D,

$$\text{Initial K.E. of pendulum at Point B} = \text{Final G.P.E. of pendulum at Point D}$$

#### Energy Loss Due To Resistive Forces

In the real world, resistive forces like frictional and drag forces, air resistance, convert some of the total energy of the swinging pendulum into thermal energy. This thermal energy will be dissipated into the surroundings and cannot be converted back into the kinetic or gravitational potential energy of the pendulum.

From this, it can be seen that the pendulum system will lose energy (due to air resistance) and the pendulum bob will reach a lower height with each successive swing.

However, please note that the total energy is still conserved. (Total energy referred to in this statement is the total energy of the Universe)

#### Non-Ideal Pendulum

In summary, for a non-ideal pendulum (accounting for the dissipative effects of friction and air resistance), the relevant energy conservation equations are:

From Point A to Point B,

$$\text{Initial G.P.E. of pendulum at Point A} = \text{Final K.E. of pendulum at Point B} + \text{Energy lost due to friction and air resistance}$$

From Point B to Point D,

$$\text{Initial K.E. of pendulum at Point B} = \text{Final G.P.E. of pendulum at Point D} + \text{Energy lost due to friction and air resistance}$$

If you still require more examples to grasp the concept, please click the navigational buttons below to go to the next page.

Gaga Godfrey John scribbled

Adjust your way of writing the formulars

Farhan Johnson scribbled

But no one is changing it, why?

Nwaniji Samuel scribbled

I think there is an error here: “When the raised pendulum is released from A, it will swing towards the equilibrium position (Point B). During this movement, the gravitational potential energy is converted into kinetic energy. Hence, at A, gravitational potential energy is at minimum, while kinetic energy is at maximum.” The sentence- “Hence, at A,” should be hence at B,

Saad Chaudhary scribbled

Yes, there is an error…

Fring Maustar scribbled

Yes it is written wrong here