# FOUNDATION Physics Work and Energy

## Work and Energy Synopsis

**Synopsis**

**Work & Work – Energy Theorem **

- Work is any physical or mental activity that one does in performing daily tasks. However, in scientific parlance, work is done when a force produces motion in an object.
- When a boy pushes a small toy, he is able to move it. Thus, work is said to be done on the toy.
- However, few humans pushing a huge stone cannot move it in spite of putting in a lot of effort. Thus, work is not done on the stone as it does not move.
- Thus, work depends on two factors:

o Magnitude of force applied to an object.

o Distance/displacement through which the object moves. - Work done in moving a body is equal to the product of force exerted on the body and the distance moved by the body in the direction of force.

Work = Force × Distance

W = F × s - If a retarding force is applied to a body moving in a particular direction, then the work done is negative.

W = F × -s OR W = -F ×s - Thus, work can be positive, negative or zero.
- Work is a scalar quantity.
- The SI unit of work is
**newton metre (N m)**or**joule (J)**.

**Expression of Work**

- The displacement of a body need not always be in the direction of force.
- Consider the following case.
- Displacement is AC which is at an angle θ with the horizontal.
- Thus, the displacement along the direction of force is given by the horizontal component of displacement.
- Hence, work done is W = F × AB
- From right angled triangle ABC, we have
- Thus, the expression of work is

W = Fs Cosθ - Thus, we can conclude that, the amount of work done is the product of force, displacement and the cosine of angle between the force and displacement.

**CASES:**

- If displacement is in the direction of force, that is, θ = 0°.

Hence, the work done is, W = F × s

This work is maximum and positive. - If displacement is normal to the direction of force, that is, θ = 90°.

Hence, the work done is, W = 0.

Thus, no work is done.

Also, if displacement is zero, then the work done is zero. This is the case when a body is performing circular motion. - If displacement is in the direction opposite to that of force, that is, θ = 180°.

Hence, the work done is, W = -F × s - This work is minimum and negative.

**Work Done By Gravity**

- If a body of mass m moves down from a height h, the force of gravity or weight acts on the body through a displacement h.
- Thus, work done by the force of gravity is

W = Force × displacement

= Weight × height

= mg × h - Similarly, if the body is thrown up to a height h, the work done by gravity is

W = -mg × h

**Work Energy Theorem **

**Statement**

Work done on a body by a resultant force acting on it is equal to the change in the kinetic energy of the body

∴ work done, W = ∆ KE

where, ∆KE is the change in kinetic energy.

**Relation between kinetic energy and momentum: - **

K.E. of body is given as,

K = ½ mv^{2} = ½ pv = ½ (p^{2}/2m)

**Kinetic Energy and Potential Energy**

**Kinetic Energy**

- A moving object can do work, and a fast-moving object can do more work.
- The energy possessed by an object on account of its motion is known as
**kinetic energy**. - Consider an object of mass m, moving with initial velocity u. Let it be displaced through a distance s when a constant force F acts on it in the direction of its displacement. The work done is

W = F × s - The work done on the object will cause a change in its velocity. Let its velocity change from u to v. Let ‘a’ be the acceleration produced.
- We know that

V^{2}- U^{2}= 2as - We also know that F = ma
- Thus, the work done is
- So, we can say that work done is equal to the change in kinetic energy of the object.
- If the object starts from rest, then u = 0. Thus, we have
- Hence, the kinetic energy of a body is

o Directly proportional to the mass of the body.

o Directly proportional to the square of the velocity of the body.

**Potential Energy**

- If work is done on an object and it is not used in changing the velocity of the object, then that work is stored inside the object as potential energy.
**Work is done in pulling the spring. This is stored as potential energy.** - The energy possessed by an object on account of its position or configuration is known as
**kinetic energy**.

**Potential Energy of an object at a height**

- To raise a body to a certain height, work is done against gravity.
- The gravitational potential energy of an object at a point above the ground is defined as the work done in raising it from the ground to that point against gravity.
- The force required to raise the object is equal to the weight F = mg of the object. If it is raised to a height h, then the work done is W = mg × h
- This work is stored in the body as potential energy.
- Work done by gravity depends on the initial and final positions and not on the path taken to raise it to the specified height.
- The potential energy is the same when the object is lifted from A to B along path 1 and path 2.

**Conservative and non – conservative Forces**

**Transformation and Conservation of Energy**

- According to the law of conservation of energy, energy can neither be created nor can it be destroyed. It only changes from one form to another.
**Conservation of mechanical energy:**The total mechanical energy of an isolated system at any instant is equal to the sum of its kinetic energy and the potential energy.- According to the law of conservation of mechanical energy, whenever there is an interchange between the potential energy and the kinetic energy, the total mechanical energy remains constant, i.e., K + U remains constant when there are no frictional forces.
- Consider a body of mass m freely falling under gravity from a height h.

**At position A:**

Initial velocity of body = 0

Thus, kinetic energy = K =0

Potential energy = U = mgh

Hence, total energy = K + U = mgh

**At position B:**

Let v

_{1}be the velocity acquired by the body after falling through distance x.

Then, u = 0, S = x, a = g

From equation v

^{2}= u

^{2}+ 2aS, we have

Thus, kinetic energy is

Now at B, height of body above the ground = h – x

Thus, potential energy is = mg (h - x)

Hence, total energy = K + U = mgx + mg(h - x) = mgh

**At position C:**

Let v be the velocity acquired by the body on reaching the ground.

Then, u = 0, S = h, a = g

From equation v

^{2}= u

^{2}+ 2aS, we have

^{2}= 0 + 2gh = 2gh

Thus, kinetic energy is

Now at C, height of body above the ground = 0

Thus, potential energy is 0.

Hence, total energy = K + U = mgh + 0 = mgh

• Thus, the mechanical energy always remains constant. Hence, mechanical energy is conserved.

**Power**

- Power is rate of doing work.
- Some machines do work at a faster rate, whereas some do at a slower rate. A stronger person can do a work in relatively less time.
- Power is defined as the rate of doing work or the rate of transfer of energy.
- Power is a scalar quantity.
- Power can also be written as

Here, v is the average speed of the body. - If displacement is at an angle θ, then the power is
- Its SI unit is
**watt (W)**or**joule per second****(J/s)**and its C.G.S. unit is**erg per second (erg/s).** - Another unit of power is kilowatts (kW).

o1 kW = 1000 W

o1 MW = 10^{6}W - Another unit of power is the horse power which is generally used in mechanical engineering. 1 horse power = 746 W = 0.746 kW.

## Videos

## FOUNDATION Class 9 Tests

### Kindly Sign up for a personalised experience

- Ask Study Doubts
- Sample Papers
- Past Year Papers
- Textbook Solutions

#### Sign Up

#### Verify mobile number

Enter the OTP sent to your number

Change