NCERT Physics Class 11 Example 5.8 & 5.9 Solutions | Spring Compression Explained

 - Dr.Sanjaykumar Pawar 

A moving car compresses a spring, demonstrating conservation of mechanical energy and work-energy theorem.

Internal Links

• NCERT Class 11 Physics Chapter 5 Notes
• Work, Energy and Power Formulas
• Conservation of Mechanical Energy
• Work-Energy Theorem Explained
• Hooke's Law and Spring Force
• Potential Energy and Kinetic Energy
• NCERT Physics Example 5.1–5.7 Solutions
• NCERT Physics Chapter 5 Exercise Solutions
• JEE Physics Work Energy Questions
• NEET Physics Chapter-wise MCQs
• Energy Conservation Numerical Problems
• Class 11 Physics Formula Sheet

NCERT Physics Class 11

Chapter 5 - Work, Energy and Power

Example 5.8 & Example 5.9

Question 1

A car of mass 1000 kg is moving with a speed of 18 km h^-1 on a smooth horizontal road and collides with a spring of spring constant 5.25 × 10³ N m^-1. Find the maximum compression of the spring.

Answer

Given

• Mass (m) = 1000 kg
• Speed (v) = 18 km h^-1 = 5 m s^-1
• Spring constant (k) = 5.25 × 10³ N m^-1

Formula

At maximum compression,

Kinetic Energy = Spring Potential Energy

½mv² = ½kx²

Step 1 : Calculate Kinetic Energy

K = ½ × 1000 × 5²

K = 12500 J

Step 2 : Calculate Compression

12500 = ½ × 5.25 × 10³ × x²

12500 = 2625x²

x² = 12500 / 2625

x² = 4.76

x = √4.76

x ≈ 2.18 m

Maximum Compression = 2.0 m (Approx.)

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Question 2

Using Example 5.8, if the coefficient of friction is 0.5, calculate the maximum compression of the spring.

Answer

Given

• Mass = 1000 kg
• Speed = 5 m s^-1
• Spring constant = 5.25 × 10³ N m^-1
• Coefficient of friction = 0.5
• g = 10 m s^-2

Formula

Work-Energy Theorem

ΔK = W

½mv² = ½kx² + μmgx

Step 1

12500 = 2625x² + 5000x

Step 2

2625x² + 5000x − 12500 = 0

Step 3

Using quadratic formula,

x = 1.35 m

Maximum Compression = 1.35 m

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Practice Questions

Practice Question 1

A spring of spring constant 400 N m^-1 is compressed by 0.5 m. Calculate the elastic potential energy stored.

Formula

U = ½kx²

U = ½ × 400 × (0.5)²

U = 50 J

Answer = 50 J

Practice Question 2

A body of mass 2 kg moves with speed 10 m s^-1. Find its kinetic energy.

Formula

K = ½mv²

K = ½ × 2 × 10²

K = 100 J

Answer = 100 J

Practice Question 3

A spring stores 100 J of energy. Its spring constant is 500 N m^-1. Find the compression.

100 = ½ × 500 × x²

100 = 250x²

x² = 0.4

x = 0.63 m

Answer = 0.63 m

Practice Question 4

A 500 kg car moving at 10 m s^-1 hits a spring of spring constant 10000 N m^-1. Find the maximum compression.

K = ½mv²

K = ½ × 500 × 10²

K = 25000 J

25000 = ½ × 10000 × x²

25000 = 5000x²

x² = 5

x = 2.24 m

Answer = 2.24 m

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Important Formulae

Formula	Expression
Kinetic Energy	K = ½mv²
Spring Potential Energy	U = ½kx²
Work-Energy Theorem	ΔK = W
Mechanical Energy	KE + PE = Constant (Without Friction)

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Important Viva Questions

1. Why does the car stop at maximum compression?
   Because all kinetic energy is converted into spring potential energy.


2. Why is conservation of mechanical energy not applicable when friction is present?
   Because friction is a non-conservative force and converts mechanical energy into heat.


3. What is the formula of spring potential energy?
   U = ½kx²


4. What is the formula of kinetic energy?
   K = ½mv²

NEET Physics Notes

Chapter: Work, Energy and Power

Topics Covered

• Spring Force (Hooke's Law)
• Spring Potential Energy
• Kinetic Energy
• Conservation of Mechanical Energy
• Work-Energy Theorem
• Friction
• NCERT Example 5.8
• NCERT Example 5.9
• Important Formulae
• NEET MCQs

1. Spring Force (Hooke's Law)

When a spring is stretched or compressed, it tries to return to its original length. This restoring force is called Spring Force.

Formula

F = -kx

Symbol	Meaning
F	Spring Force (N)
k	Spring Constant (N/m)
x	Compression or Extension (m)

Remember

• Negative sign shows restoring force.
• Force acts opposite to displacement.
• Larger k means stronger spring.
• Smaller k means softer spring.

2. Spring Potential Energy

Energy stored inside a compressed or stretched spring.

Formula

U = ½ kx²

Symbol	Meaning
U	Potential Energy
k	Spring Constant
x	Compression

3. Kinetic Energy

Energy possessed by a moving object.

Formula

K = ½ mv²

Symbol	Meaning
m	Mass
v	Velocity

4. Conservation of Mechanical Energy

When there is no friction, the total mechanical energy remains constant.

½mv² = ½kx²

Used to calculate maximum compression of a spring.

Example 5.8

To simulate car accidents, auto manufacturers study the collisions of moving cars with mounted springs and other constraints. Consider a typical simulation with a car of mass 1000 kg moving at a speed of 18.0 km/h on a smooth road and colliding with a horizontal mounted spring of spring constant 5.25 × 10³ N m⁻¹. What is the maximum compression of the spring?

Example 5.9

Consider Example 5.8 taking the coefficient of friction, μ = 0.5, and calculate the maximum compression of the spring.

NCERT Example 5.8

Question

Mass = 1000 kg
Speed = 18 km/h
Spring Constant = 5.25 × 10³ N/m
Find the maximum compression.

Solution

Step 1 18 km/h = 5 m/s

Kinetic Energy

K = ½ ×1000×5²

K = 12500 J

Using Conservation of Energy

½kx² = 12500

x = 2.0 m

Answer: Maximum Compression = 2.0 m

5. Work-Energy Theorem

The work done by all forces acting on a body equals the change in kinetic energy.

Formula

W = ΔK

6. Friction

Friction always opposes motion.

Formula

F = μN

On horizontal surface,

N = mg

Therefore,

F = μmg

NCERT Example 5.9

Question

Mass = 1000 kg
Speed = 5 m/s
k = 5.25 ×10³ N/m
μ = 0.5

Solution

Apply Work-Energy Theorem

−½mv² = −½kx² − μmgx

Rearranging,

kx² + 2μmgx − mv² = 0

x = 1.35 m

Answer: Maximum Compression = 1.35 m

Important Formula Sheet

Formula	Equation
Kinetic Energy	½mv²
Spring Force	F = -kx
Spring Potential Energy	½kx²
Friction	μmg
Work-Energy Theorem	W = ΔK
Conservation of Energy	½mv² = ½kx²

NEET Important Points

• Hooke's law is valid only within elastic limit.
• Spring force is a restoring force.
• Potential energy is always positive.
• Friction is a non-conservative force.
• Mechanical energy is conserved only without friction.
• Work done by friction is negative.
• Spring energy depends on x².
• Kinetic energy depends on velocity².

NEET Practice MCQs

Q1. The force exerted by a spring is

• A. Constant
• B. Proportional to displacement
• C. Inversely proportional
• D. Zero

Answer: B

Q2. Potential energy stored in a spring is

• A. kx
• B. kx²
• C. ½kx²
• D. ½kx

Answer: C

Q3. Which force is non-conservative?

• A. Gravity
• B. Spring
• C. Electrostatic
• D. Friction

Answer: D

Q4. Mechanical energy remains constant when

• A. Friction acts
• B. Air resistance acts
• C. Only conservative forces act
• D. External work is done

Answer: C

Quick Revision

• Hooke's Law → F = -kx
• Spring Energy → ½kx²
• Kinetic Energy → ½mv²
• Work-Energy Theorem → W = ΔK
• Without friction → Energy conserved.
• With friction → Use Work-Energy Theorem.
• Maximum compression occurs when kinetic energy becomes zero.
• Friction reduces spring compression.

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Prepared for NEET Aspirants

Easy Notes • NCERT Based • Beginner Friendly

Conservation of Mechanical Energy Class 11 Notes, MCQs, Questions & Answers | CBSE & NEET

-  Dr Sanjay Kumar Pawar 

Conservation of Mechanical Energy Class 11 Physics Notes PDF | CBSE & NEET 

Conservation of Mechanical Energy explained with a falling ball showing the conversion of potential energy into kinetic energy.

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Link this page to related Class 11 Physics topics to improve SEO and user navigation:

1. Work, Energy and Power Class 11 Notes
2. Work-Energy Theorem Explained
3. Potential Energy Class 11 Notes
4. Kinetic Energy Formula and Examples
5. Conservative and Non-Conservative Forces
6. Gravitational Potential Energy Notes
7. Free Fall Motion Class 11
8. Laws of Motion Class 11
9. Newton's Laws of Motion Notes
10. Circular Motion Class 11 Notes
11. System of Particles and Rotational Motion
12. Gravitation Class 11 Notes
13. Mechanical Properties of Solids
14. Complete Class 11 Physics Notes Index 
15. Class 11 Physics MCQs with Answers
16. CBSE Class 11 Physics Important Questions
17. NEET Physics Chapter-wise Notes
18. NCERT Solutions for Class 11 Physics
19. Class 11 Physics Formula Sheet
20. Previous Year CBSE Class 11 Physics Questions

Chapter 5.8
Conservation of Mechanical Energy

Definition:
Mechanical Energy is the sum of Kinetic Energy (KE) and Potential Energy (PE).

Mechanical Energy = KE + PE

1. Work-Energy Theorem

Suppose a body moves through a small distance Δx under the action of force F. According to the Work-Energy Theorem,

ΔK = F(x) Δx

This means the work done by a force changes the kinetic energy of the body.

• Positive work increases kinetic energy.
• Negative work decreases kinetic energy.

2. Conservative Force

If the force is conservative, then potential energy can be defined.

−ΔU = F(x) Δx

The negative sign shows that whenever potential energy decreases, kinetic energy increases.

Example:
A falling stone loses potential energy and gains kinetic energy.

3. Combining the Equations

From the two equations:

ΔK = −ΔU

Therefore,

ΔK + ΔU = 0

or

Δ(K + U) = 0

4. Conservation of Mechanical Energy

Since Δ(K+U)=0, the total mechanical energy never changes.

K + U = Constant

This is called the Law of Conservation of Mechanical Energy.

5. Equation Between Two Positions

Ki + Ui = Kf + Uf

The total mechanical energy before motion equals the total mechanical energy after motion.

6. Conservative Force - Important Properties

• Potential energy can be defined.
• Work depends only on initial and final positions.
• Work does not depend on the path.
• Work done in a closed path is zero.
• Mechanical energy remains conserved.

7. Example - Falling Ball

A ball of mass m is dropped from height H. Initially the velocity is zero.

At Height H

PE = mgH

KE = 0

EH = mgH

At Height h

PE = mgh

KE = ½mv²h

Eh = mgh + ½mv²h

At Ground Level

PE = 0

KE = ½mv²f

E0 = ½mv²f

8. Conservation of Energy

EH = Eh = E0

Since only gravity acts on the body, mechanical energy remains constant.

mgH = mgh + ½mv²h = ½mv²f

9. Final Velocity

Using conservation of energy,

mgH = ½mv²f

After simplifying,

vf = √(2gH)

10. Velocity at Height h

mgH = mgh + ½mv²h

Therefore,

vh² = 2g(H − h)

11. Energy Conversion

Position	Potential Energy	Kinetic Energy
Top	Maximum	Zero
Middle	Decreasing	Increasing
Ground	Zero	Maximum

12. Important Points for NEET

• Mechanical Energy = KE + PE
• Gravity is a conservative force.
• Spring force is also conservative.
• Mechanical energy remains constant if only conservative forces act.
• Work done by a conservative force depends only on the initial and final positions.
• Work done in a closed path is zero.
• At the highest point, PE is maximum and KE is zero.
• At the ground, KE is maximum and PE is zero.
• Potential energy converts into kinetic energy during free fall.

13. Formula Sheet

Mechanical Energy = KE + PE

ΔK + ΔU = 0

K + U = Constant

Ki + Ui = Kf + Uf

PE = mgh

KE = ½mv²

vf = √(2gH)

vh² = 2g(H − h)

Work done in a Closed Path = 0

Prepared for NEET Physics Students • Chapter 5.8 - Conservation of Mechanical Energy

Conservation of Mechanical Energy

What is Mechanical Energy?

Mechanical Energy is the sum of Kinetic Energy (KE) and Potential Energy (PE).

Mechanical Energy = KE + PE

Work-Energy Theorem

When a force acts on an object, its kinetic energy changes.

ΔKE = Work Done

For conservative forces, Potential Energy decreases when Kinetic Energy increases.

ΔKE + ΔPE = 0

Energy Conversion During Falling

Top
PE Maximum
KE Zero

Middle
PE ↓
KE ↑

Ground
PE Zero
KE Maximum

Visual Falling Ball

⬇

As the ball falls, Potential Energy continuously converts into Kinetic Energy.

Example

Position	Potential Energy	Kinetic Energy	Total Energy
Top	100 J	0 J	100 J
Middle	60 J	40 J	100 J
Ground	0 J	100 J	100 J

Important Formulae

PE = mgh

KE = ½mv²

KE + PE = Constant

vf = √(2gH)

vh² = 2g(H − h)

NEET Remember:

• Gravity is a conservative force.
• Total Mechanical Energy remains constant if only conservative forces act.
• At the highest point: PE is maximum and KE is zero.
• At the ground: KE is maximum and PE is zero.
• Potential Energy converts into Kinetic Energy during falling.

CBSE Class 11 Physics

Chapter 5.8 - Conservation of Mechanical Energy

Question Bank with Answers

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Part A - Multiple Choice Questions

1. Mechanical energy is the sum of

1. Potential energy and Heat energy
2. Kinetic energy and Potential energy
3. Heat energy and Electrical energy
4. Sound energy and Potential energy

Answer: B

2. Mechanical energy remains constant when

1. Friction acts
2. Air resistance acts
3. Only conservative forces act
4. External force acts

Answer: C

3. Which one is a conservative force?

1. Friction
2. Gravity
3. Air resistance
4. Viscous force

Answer: B

4. Work done by a conservative force depends on

1. Path followed
2. Distance travelled
3. Initial and final positions only
4. Speed

Answer: C

5. Work done in a closed path by gravity is

1. Positive
2. Negative
3. Zero
4. Infinite

Answer: C

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Part B - Very Short Answer Questions

1. Define mechanical energy.

Answer: Mechanical energy is the sum of kinetic energy and potential energy.

2. Write the formula of mechanical energy.

Answer: E = KE + PE

3. Name one conservative force.

Answer: Gravitational force.

4. Write the formula of kinetic energy.

Answer: KE = ½mv²

5. Write the formula of potential energy.

Answer: PE = mgh

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Part C - Short Answer Questions

1. What is conservation of mechanical energy?

Answer:
When only conservative forces act on a body, the total mechanical energy (kinetic energy + potential energy) remains constant throughout the motion.

2. What is a conservative force?

Answer:
A conservative force is a force whose work depends only on the initial and final positions and not on the path followed. Examples:

• Gravity
• Spring force

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Part D - Long Answer Questions

1. State the law of conservation of mechanical energy.

Answer:
If only conservative forces act on a body, its total mechanical energy remains constant. Mechanical Energy = Kinetic Energy + Potential Energy Initial Energy Ki + Ui Final Energy Kf + Uf Therefore, Ki + Ui = Kf + Uf Example: A freely falling body loses potential energy and gains kinetic energy. The total mechanical energy remains constant.

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Part E - Assertion and Reason

Assertion: Mechanical energy remains constant when only conservative forces act.

Reason: Gravity is a conservative force.

Answer: Both Assertion and Reason are true, and Reason is the correct explanation.

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Part F - Fill in the Blanks

1. Mechanical energy is the sum of ______ and ______.
2. Gravity is a ______ force.
3. Potential energy converts into ______ during free fall.
4. Work done in a closed path is ______.
5. Mechanical energy remains ______ when only conservative forces act.

Answers:

1. Kinetic energy, Potential energy
2. Conservative
3. Kinetic energy
4. Zero
5. Constant

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Part G - Match the Columns

Column A	Column B
Gravity	Conservative force
Friction	Non-conservative force
Potential Energy	mgh
Kinetic Energy	½mv²

Matching Answers 1 → Conservative force 2 → Non-conservative force 3 → mgh 4 → ½mv²

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Part H - Case Study

A ball of mass 2 kg is dropped from a height of 20 m. Ignore air resistance.

Q1. Which energy is maximum at the top?

Answer: Potential Energy

Q2. Which energy is maximum at the ground?

Answer: Kinetic Energy

Q3. Calculate total mechanical energy at the top. Take g = 10 m/s².

PE = mgh = 2 × 10 × 20 = 400 J Answer = 400 Joule

Q4. Is mechanical energy conserved?

Yes. Only gravity acts.

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Important Formulae

• E = KE + PE
• KE = ½mv²
• PE = mgh
• Ki + Ui = Kf + Uf
• Δ(KE + PE) = 0
• v = √(2gH)
• v² = 2g(H − h)
• Work in closed path = 0

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End of CBSE Class 11 Physics Question Bank