Chapter 5 — Work and Energy

1. Conservative forces

A conservative force has work that does not depend on the path taken.

For a conservative force:$$W=-\Delta U$$

or$$\Delta U=-W$$

Work done around a closed path is zero.

Examples:

• gravitational force
• spring force

2. Non-conservative forces

For a non-conservative force, work depends on the path.

The major example in the chapter is friction.

When a book slides across a table, friction gradually converts its kinetic energy into thermal energy. This energy transfer makes the original mechanical energy unavailable for complete recovery.

3. Hooke’s law

For a spring operating within its elastic limit:$$F=kx$$

For the restoring force:$$F=-kx$$

where:

• $k$ = spring constant
• $x$ = extension/compression.

A large $k$ means a stiffer spring; a smaller $k$ corresponds to a softer spring.

The SI unit of $k$ is:$$N\,m^{-1}$$

4. Force-extension graph

For a spring obeying Hooke’s law, force is proportional to extension, so the force-extension graph is a straight line through the origin.

If the force increases from zero to $kx$, the average force during gradual stretching is:$$F_{\text{average}}=\frac{kx}{2}$$

Therefore:$$W=F_{\text{average}}x$$$$W=\frac12kx^2$$

The work done is stored as elastic potential energy:$$\boxed{U=\frac12kx^2}$$

5. Energy transformation

Energy can change form.

For a stretched spring, elastic potential energy can become kinetic energy.

If there are no energy losses:$$PE=KE$$

For the example in the chapter, a spring with $k=100\,N/m$, stretched by $0.06$ m, stores:$$U=\frac12(100)(0.06)^2=0.18J$$

If attached to a $0.5$ kg body, this energy can become kinetic energy, giving a maximum speed of approximately $0.85\,m/s$.

Key idea

Conservative forces store and recover mechanical energy; non-conservative forces transfer mechanical energy into other forms such as heat.

Chapter 5 — Work and Energy

Questions

A. Multiple Choice Questions

1. A conservative force is one for which work depends on:
a) The path followed
b) Initial and final positions rather than the path
c) Time only
d) Speed only

2. The work done by a conservative force around a closed path is:
a) Zero
b) Maximum
c) Infinite
d) Always negative

3. Which is a non-conservative force?
a) Gravity
b) Ideal spring force
c) Friction
d) Electrostatic force

4. Within the elastic limit, Hooke’s law gives the magnitude relation:
a) $F=k/x$
b) $F=kx$
c) $F=x/k$
d) $F=k+x$

5. The SI unit of spring constant is:
a) N
b) J
c) N/m
d) m/N

6. Elastic potential energy stored in a stretched spring is:

a) $kx$
b) $kx^2$
c) $\frac12kx^2$
d) $2kx^2$

B. Fill in the Blanks

7. Gravitational force is a ______ force.

8. Friction is a ______-conservative force.

9. A spring with a large value of $k$ is relatively ______.

10. Hooke’s law applies within the ______ limit of a spring.

11. The energy stored in a stretched spring is called ______ potential energy.

C. True or False

12. Friction is a conservative force.

13. Work done by gravity between two fixed points is independent of the path.

14. A larger spring constant represents a softer spring.

15. The negative sign in $F=-kx$ indicates the restoring direction of spring force.

D. Assertion–Reason

16. Assertion: Friction is non-conservative.
Reason: Work done against friction depends on the path travelled.

17. Assertion: A stiff spring requires more force for the same extension.
Reason: A stiff spring has a larger spring constant.

E. Short Answer

18. What is a conservative force?

19. Why is friction called non-conservative?

20. What does the spring constant tell us about a spring?

21. Explain how work done in stretching a spring becomes stored energy.

F. Numerical Questions

22. A spring has $k=100N/m$ and is stretched by $0.06m$. Calculate its elastic potential energy.

23. A spring with $k=30N/m$ is stretched by a force of $100N$. Find the extension.

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Answers

A. MCQ Answers

1. b) Initial and final positions rather than the path
2. a) Zero
3. c) Friction
4. b) F=kxF=kx
5. c) N/m
6. c) 12kx2\frac12kx^2

B. Fill in the Blanks

7. conservative
8. non
9. stiff
10. elastic
11. elastic

C. True/False

12. False
13. True
14. False
15. True

D. Assertion–Reason

16. Both are true, and the reason correctly explains the assertion.
17. Both are true, and the reason correctly explains the assertion.

E. Answers

18. A conservative force is one whose work between two positions is independent of the path taken.
19. Friction is non-conservative because the work done against it depends on the path and mechanical energy is dissipated.
20. The spring constant measures the stiffness of the spring. A larger $k$ means a stiffer spring.
21. During stretching, the applied work is stored in the spring as elastic potential energy:

$$U=\frac12kx^2$$

F. Numerical Answers

$$U=\frac12kx^2$$$$=\frac12(100)(0.06)^2$$$$\boxed{0.18J}$$

$$F=kx$$$$x=\frac{F}{k}$$$$=\frac{100}{30}$$$$\boxed{3.33m}$$