Work and Energy class 9 Notes, solved Numerical, worksheet pdf download

 

Work and Energy class 9 Notes, solved Numerical, worksheet pdf download

Work

Definition of Work

• Work is said to be done when a force applied on an object produces displacement in the direction of force.

Mathematical Definition:

$$\text{Work (W)}=\text{Force (F)}\times \text{Displacement (s)}$$

SI Unit:

• Joule (J)

• 1 Joule = Work done when a force of 1 N moves an object through 1 m.

Concept of Work (Important for Exams)

For work to be done:
✔ Force must act on the object
✔ Object must move (displacement)
✔ Displacement must be in the direction of force

Examples:

• Pushing a wall → No work (no displacement)

• Lifting a book → ✔ Work done

• Pulling a cart → ✔ Work done

Types of Work

(A) Positive Work

When force and displacement are in same direction.

Example:
A boy pulling a suitcase forward.

$$W = +Fs$$

(B) Negative Work

When force and displacement are in opposite directions.

Example:
Friction acting on a moving body.

$$W = -Fs$$

(C) Zero Work

When:

• No displacement OR
• Force is perpendicular to displacement

Examples:

• Carrying a load on a horizontal road
• Centripetal force in circular motion

Special Cases of Work

Work Done Against Gravity

$$W = mgh$$

Where:
m = mass (kg)
g = acceleration due to gravity (9.8 m/s²)
h = height (m)

Work Done by Gravity

Same formula, but work is positive when object moves downward.

Solved Numericals (VERY IMPORTANT)

Numerical 1 (Basic)

A force of 10 N moves an object by 4 m. Find work done.

$$W=F\times s=10\times \; 4\mathrm{<br>}=40J$$

Numerical 2 (Against Gravity)

Find work done in lifting a 3 kg body to a height of 10 m. (g = 10 m/s²)

$$W=mgh=3\times 10\times 10=300J$$

Numerical 3 (Zero Work)

A porter carries a load on his head while walking horizontally. Find work done.

Solution:
Displacement is horizontal, force is vertical.

$$W=0$$

Answer: 0 J

Numerical 4 (Negative Work)

Friction of 5 N acts on a body moving 3 m. Find work done by friction.

$$W=-Fs=-5\times 3=-15J$$

Energy

Definition of Energy

• Energy is the capacity of a body to do work.
• If a body can do work, it possesses energy.

SI Unit:

• Joule (J)
• 1 Joule = Energy required to do 1 Joule of work.

Different Forms of Energy

• Energy exists in various forms. Some important forms are:

(a) Mechanical Energy

(b) Heat Energy

(c) Light Energy

(d) Electrical Energy

(e) Chemical Energy

(f) Sound Energy

(g) Nuclear Energy

Among these, Mechanical Energy is most important for Class 9.

Mechanical Energy

• Mechanical energy is the energy possessed by an object due to its motion or position.

Types of Mechanical Energy:

1. Kinetic Energy

2. Potential Energy

$$\text{Mechanical Energy} = \text{Kinetic Energy} + \text{Potential Energy}$$

Kinetic Energy (KE)

Definition:

The energy possessed by a body due to its motion is called kinetic energy.

Examples:

• Moving car
• Flowing water
• Flying bird

Derivation of Kinetic Energy

Consider:

• Mass of body =        m
• Initial velocity =       u
• Final velocity =        v
• Acceleration =          a
• Distance travelled =  s

From Newton’s third equation of motion:

$$v^2 - u^2 = 2as$$From Newton's second law of motion:$$F = ma$$

Work done:

$$W=F\times s=ma\times s$$

Substitute value of s:

$$W=m\times \frac{(v^2-u^2)}{\mathrm{2}}$$

If body starts from rest, u = 0

$$W=\frac{1}{2}m{v}^2$$

This work done is stored as Kinetic Energy.

Formula:

$$\boxed{KE = \frac{1}{2} mv^2}$$

Numerical 2

A car of mass 2000 kg moves with a speed of 20 m/s. Find its kinetic energy.

$$KE = \frac{1}{2} × 1000 × 20^2$$
$$KE = 200000 J$$

​

Potential Energy (PE)

Definition:

The energy possessed by a body due to its position or configuration is called potential energy.

Examples:

• Water stored in a dam
• Stretched spring
• Raised stone

Derivation of Potential Energy

Consider:

• Mass of body = m
• Height raised = h
• Acceleration due to gravity = g

Force acting against gravity:

$$F=mg$$

Work done:

$$W=F\times h=mg\times h$$

This work done is stored as Potential Energy.

Formula:

$$\boxed{{\displaystyle PE=mg\mathrm{h}}}$$

Solved Numericals on Potential Energy

Numerical 1

Find potential energy of a 6 kg object kept at a height of 10 m. (g = 10 m/s²)

$$PE=6\times 10\times 10=600J$$

Mechanical Energy (Numerical)

Numerical 1

A body of mass 2 kg is at a height of 5 m and moving with velocity 4 m/s. Find total mechanical energy.

Kinetic Energy:

$$KE=\frac{1}{2}\times 2\times 4^2=16J$$

Potential Energy:

$$PE=2\times 10\times 5=100J$$

Total Mechanical Energy:

$$ME = KE + PE = 58 J$$

LAW OF CONSERVATION OF ENERGY

Statement of the Law

"Energy can neither be created nor destroyed. It can only be transformed from one form to another."

• The total energy of an isolated system remains constant.
• This law is universal and applies to all physical and chemical processes.

Concept of Law of Conservation of Energy

• Energy does not disappear.
• Energy does not appear from nothing.
• Only conversion of energy takes place.

Examples:

• Electric bulb: Electrical → Light + Heat
• Falling object: Potential → Kinetic
• Electric fan: Electrical → Mechanical

Mathematical Derivation (VERY IMPORTANT)

Consider:

• Mass of body = m
• Height = h
• Velocity at bottom = v
• Acceleration due to gravity = g

At Height h (Top Position)

Velocity = 0

$$KE = 0$$
$$PE = mgh$$

Total Energy:

$$E = PE + KE = mgh$$

At Ground (Bottom Position)

Height = 0

$$PE = 0$$

Using equation of motion:

$$v^2 = 2gh$$
$$KE=\frac{1}{2}m{v}^2$$

Substitute value:

$$KE = \frac{1}{2}m(2gh) = mgh$$

Total Energy:

$$E = KE + PE = mgh$$

Conclusion:

$$\text{Total Energy at top}=\text{Total Energy at bottom}$$

Hence, energy is conserved.

Solved Numericals

Numerical 1

A body of mass 3 kg is raised to a height of 10 m. Find total energy. (g = 10 m/s²)

$$PE=mgh=3\times 10\times 10=300J$$

Numerical 2

A stone of mass 1 kg falls freely from a height of 20 m. Find its velocity just before hitting the ground.

Using conservation of energy:

$$mgh = \frac{1}{2}mv^2$$
$$v^2 = 2gh = 2 × 10 × 20 = 400$$
$$v = 20 m/s$$

Numerical 3

A ball is dropped from a height of 40 m. Find its kinetic energy at a height of 10 m. (g = 10 m/s²)

Initial total energy:

$$E = mgh = m × 10 × 40 = 400m$$

PE at 10 m:

$$PE=m\times 10\times 10=100m$$

KE at 10 m:

$$KE = 400m - 100m = 300m$$

Numerical 4

A body of mass 7 kg is thrown upward. Find its maximum height if initial velocity is 20 m/s.

Using energy conservation:

$$\frac{1}{2}m{v}^2=mgh$$$$h = \frac{v^2}{2g} = \frac{400}{20} = 20 m$$

POWER

Definition of Power

• Power is the rate at which work is done or energy is transferred.
• It tells us how fast a work is done.

Concept of Power (Very Important)

Two persons may do the same amount of work, but the one who does it in less time is said to have more power.

Example:

• Person Y lifts a load in 10 s

• Person Z lifts the same load in 5 s

✔ Both do same work
✔ Person Z is more powerful

Mathematical Expression (Formula)

$$\text{Power (P)} = \frac{\text{Work (W)}}{\text{Time (t)}}$$

SI Unit of Power

• Watt (W)

Definition of 1 Watt:

If 1 Joule of work is done in 1 second, the power is 1 watt.

Commercial Unit (Important)

• kilowatt-hour (kWh)

• 1 kWh = 1 unit of electrical energy

Power in Terms of Force and Velocity

Since:

$$W = F × s$$
$$P=\frac{W}{t}=\frac{F\times s}{\mathrm{t}}$$

But:

$$\frac{s}{t} = v$$

So,

$$\boxed{{\displaystyle P=F\times \mathrm{v}}}$$

​Solved Numericals on Power