Chapter 05 • Work and Energy

Work and Energy

Build a clear understanding of conservative and non-conservative forces, spring potential energy, Hooke's law, elastic energy, force-extension graphs and energy conversion through explanations, derivations, worked examples and interactive visuals.

Conservative Forces

Learn path independence, closed-path work and the connection between work and potential energy.

Non-Conservative Forces

Understand why friction depends on the path and converts mechanical energy into heat.

Hooke's Law

Understand force-extension proportionality, spring constant and restoring force.

Spring Potential Energy

Derive $U=\frac12kx^2$ using the average force during gradual stretching.

Force–Extension Graph

Use the area under the graph to calculate work done in stretching a spring.

Energy Conversion

Connect elastic potential energy and kinetic energy when energy losses are absent.

Learning Sequence

Classify forces
Decide whether work depends on path.

Understand spring behaviour
Connect force, extension and spring constant through Hooke's law.

Derive spring energy
Use the force-extension graph and average force to obtain elastic potential energy.

Apply energy conservation
Convert spring potential energy into kinetic energy when the source assumes no energy loss.

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5.1 Conservative and Non-Conservative Forces

Not every force changes mechanical energy in the same way. A useful first question is: Does the work done by the force depend on the path taken?

Conservative force: a force for which the work done does not depend on the path taken. Equivalently, the work done by the force around a closed path is zero.

The two defining ideas

Path independent

The work depends only on the initial and final positions, not on the particular route followed.

Closed path

For a conservative force, total work over a complete closed path is zero.

Work and Potential Energy

For a conservative force, the work done by the force is equal to the decrease in potential energy:

$$W=-\Delta U=-(U_f-U_i)=U_i-U_f$$

Equivalently:

$$\boxed{\Delta U=-W}$$

Remember the sign: if a conservative force does positive work, the associated potential energy decreases. If potential energy increases, the conservative force does negative work.

Examples from the chapter

Force	Classification	Why?
Gravitational force	Conservative	Its work is associated with a change in gravitational potential energy and is path independent.
Spring force	Conservative	Its work is associated with elastic potential energy.
Friction	Non-conservative	Work depends on the path and mechanical energy is converted into heat.
Drag	Non-conservative	It dissipates mechanical energy and depends on the motion/path.

Figure 5.1: Path independence of conservative force work: Work done between A and B is identical along all paths.

Think

A ball is thrown upward and later returns to the hand. The gravitational force is conservative, so the work depends on the initial and final positions rather than the exact path followed.

Quick Check — Conservative Forces

Define a conservative force.
Why is gravitational force called conservative?
What is the work done by a conservative force over a closed path?
Write the relation between conservative-force work and potential energy.

View Solutions in Exercise 5.1

Non-Conservative Forces and Energy Loss

Non-conservative force: a force for which the work done depends on the path taken.

Friction is the chapter's main example. A sliding book eventually stops because friction converts kinetic energy into heat.

Important: Saying that friction “destroys energy” is not the right idea. The energy is transferred from organised mechanical motion into thermal energy, so the original mechanical energy is not fully recoverable as mechanical motion.

Everyday connections

Sliding Book

Friction opposes motion and converts kinetic energy into heat.

Pendulum

Air resistance and friction gradually reduce its mechanical energy, so its swing decreases.

Lubrication

Lubrication reduces frictional losses in moving machine parts.

3D Interactive Simulation: Damped Pendulum (Non-Conservative Force & Heat Dissipation)

Three.js 3D Engine

Drag to orbit 3D pendulum

Mode: Non-Conservative (Friction/Air Drag)
Current Angle: 60.0° | Mechanical Energy: 100% (Energy dissipates as heat)

Quick Check — Non-Conservative Forces

Why is friction non-conservative?
What happens to mechanical energy when friction acts?
If there were no friction on Earth, how would motion be different?
Why do machines require lubrication?

View Solutions in Exercise 5.1

5.2 Potential Energy of a Spring

A spring stores energy when it is stretched or compressed, provided the spring remains within the behaviour described by Hooke's law.

Hooke's law: within the applicable elastic range, the magnitude of restoring force is proportional to the extension or compression.

$$F_{\text{restoring}}=-kx$$

Here $k$ is the spring constant and $x$ is the displacement from the natural position. The negative sign indicates that the restoring force acts opposite to the displacement.

Symbol	Meaning	SI unit
$F$	Force magnitude	N
$k$	Spring constant	N/m
$x$	Extension or compression	m

Stiffness: a large $k$ means the spring is stiff; a small $k$ means the spring is soft.

Deriving Elastic Potential Energy

When a spring is stretched gradually from zero extension to a maximum extension $x$, its force does not remain constant.

At start

Extension $=0$, so the force magnitude is $0$.

At maximum extension

Extension $=x$, so the force magnitude is $kx$.

Because the force increases linearly, the average force is:

$$F_{\text{average}}=\frac{0+kx}{2}=\frac{kx}{2}$$

Work done is average force multiplied by extension:

$$W=F_{\text{average}}\times x$$

$$W=\frac{kx}{2}\times x=\boxed{\frac12kx^2}$$

The work done in stretching the spring is stored as elastic potential energy:

$$\boxed{U=\frac12kx^2}$$

Memory connection: the factor $\frac12$ appears because the force increases from zero to $kx$, so the average force is half the final force.

Worked Example

A spring has $k=30\,N/m$. A force of $100\,N$ is applied. Find the extension using Hooke's law.

$$F=kx$$

$$x=\frac{F}{k}=\frac{100}{30}=\boxed{3.33\,m}\text{ (approximately)}$$

3D Interactive Simulation: Hooke's Law & Elastic Potential Energy ($U = \frac{1}{2}kx^2$)

Three.js 3D Engine

Drag to orbit 3D spring

Spring Constant: k = 50 N/m | Extension: x = 2.00 m
Restoring Force: F = −kx = −100.0 N | Stored Elastic PE: U = ½kx² = 100.00 J

Displacement (x):

Spring Constant (k):

Quick Check — Spring Potential Energy

State Hooke's law.
What does the negative sign in $F=-kx$ indicate?
What is the SI unit of the spring constant?
Derive $U=\frac12kx^2$ using average force.

View Solutions in Exercise 5.1

Force–Extension Graph and Work Done

For a Hooke's-law spring, the force-extension graph is a straight line passing through the origin.

Graph rule: Work done is represented by the area under the force-extension graph.

For a straight-line Hooke's-law graph

The region from $0$ to $x$ is a triangle:

$$W=\frac12(\text{base})(\text{height})=\frac12x(kx)=\frac12kx^2$$

Work from 2 cm to 6 cm

To find the work done in stretching the spring from $2\,cm$ to $6\,cm$, calculate the area under the force–extension graph between these two extensions.

$$W_{2\to6}=\text{area under the graph between }x=0.02\,m\text{ and }x=0.06\,m$$

Using the graph: Work done between two extensions is found from the area between the force–extension curve and the x-axis over that interval.

Quick Check — Force–Extension Graph

What does the slope of a Hooke's-law $F-x$ graph represent?
What does the area under an $F-x$ graph represent?
Why is the area a triangle for a spring obeying Hooke's law from zero extension?

View Solutions in Exercise 5.1

Energy Conversion: Spring Potential Energy to Kinetic Energy

When a stretched spring is released and the source assumes no loss of energy, the stored elastic potential energy can be converted into kinetic energy.

$$U_{\text{spring}}=\frac12kx^2$$

$$K=\frac12mv^2$$

With no energy loss:

$$\frac12kx^2=\frac12mv^2$$

This relation allows us to calculate the maximum speed under the assumptions stated in the source.

Worked Example

For $k=100\,N/m$, $x=0.06\,m$, and $m=0.5\,kg$:

$$U=\frac12(100)(0.06)^2=0.18\,J$$

At maximum speed, all this potential energy becomes kinetic energy:

$$0.18=\frac12(0.5)v^2$$

$$v^2=0.72\quad\Rightarrow\quad \boxed{v\approx0.85\,m/s}$$

3D Interactive Simulation: Energy Conversion ($\frac{1}{2}kx^2 \to \frac{1}{2}mv^2$)

Three.js 3D Engine

Drag to orbit 3D launch track

Spring PE Stored: U = ½kx² = 312.50 J | Mass: m = 0.5 kg
Max Velocity (100% Conversion): v = √(kx²/m) = 35.36 m/s (Kinetic Energy K = ½mv² = 312.50 J)

Quick Check — Energy Conversion

What happens to spring potential energy when a stretched spring is released?
Write the equation used when all spring potential energy becomes kinetic energy.
Why is the “no loss of energy” assumption important?
If elastic potential energy is $2\,J$, what is the kinetic energy at the point where all of it has converted to kinetic energy?

View Solutions in Exercise 5.1

Work, Force and Energy — Quick Review

Conservative Force

Work is path independent and can be associated with potential energy.

Non-Conservative Force

Work depends on path and can transfer mechanical energy into thermal energy.

Hooke's Law

Spring force magnitude is proportional to extension within the stated range.

Spring Energy

Stretching a spring stores $U=\frac12kx^2$.

Graph

Area under an $F-x$ graph gives work done.

Energy Conversion

With no loss, spring potential energy can become kinetic energy.

Exam strategy: identify the force, write the correct energy/work relation, keep extension in metres when using $k$ in N/m, and check the final unit.

OVERVIEW EXERCISE 5.1

Exercise 5.1 — Work and Energy

1. Define a conservative force and give one example.

Solution: A conservative force is one for which work does not depend on path. Gravitational force is an example.

2. Why is gravitational force called conservative?

Solution: Its work depends on the initial and final positions rather than the path, and its work around a closed path is zero.

3. Why is friction called a non-conservative force?

Solution: The work done by friction depends on the path taken, and it converts mechanical energy into heat.

4. What happens to energy when a non-conservative force acts on an object?

Solution: Mechanical energy can be transferred into other forms such as thermal energy.

5. If there were no friction on Earth, how would motion be different?

Solution: Moving objects would not lose mechanical energy through friction in the same way, so motion would persist much more readily unless another force changed it.

6. State Hooke's law and explain the negative sign in $F=-kx$.

Solution: Within the stated elastic range, restoring force is proportional to displacement. The negative sign shows that the restoring force acts opposite to the displacement.

7. A spring has $k=30\,N/m$. A force of $100\,N$ is applied. Calculate the extension.

Solution: $x=F/k=100/30=\boxed{3.33\,m}$ approximately.

8. Derive the elastic potential energy stored in a spring.

Solution: Final force $=kx$, initial force $=0$. Average force $=kx/2$. Therefore $W=(kx/2)x=\boxed{\frac12kx^2}$. This work is stored as elastic potential energy.

9. A $5\,kg$ object is placed on top of a $30\,m$ high building. Calculate its gravitational potential energy taking the base as zero.

Source-data note: The supplied question requires the value of $g$, which is not explicitly specified in that question. Using the common value $g=9.8\,m/s^2$, $U=mgh=5(9.8)(30)=\boxed{1470\,J}$.

10. What is the increment in its potential energy?

Solution: Since the base is taken as the zero level, the increase in potential energy is \[ \Delta U=mgh=5 imes9.8 imes30=oxed{1470\,J}. \]

11. A $10\,kg$ weight is hung from a $5\,m$ wire, causing it to stretch by $1\,mm$. Calculate the energy stored.

Source-data note: The pasted source does not provide the spring constant $k$ or enough information to determine it from the stated data. Since $U=\frac12kx^2$, a numerical answer cannot be obtained without $k$ (or equivalent force-extension information).

12. Calculate the work done by an external force to lift a uniform $2\,m$ long rod from horizontal to vertical.

Solution: The centre of mass of a uniform rod is at its midpoint. During the rotation, it rises through $1\,m$. Hence, \[ W=mgh=mg(1)=oxed{mg\,J}. \] If the mass of the rod is $M$, then $W=Mg$.

13. For a spring stretched gradually from zero extension to $x$, show why the average force is $kx/2$.

Solution: Hooke's law gives initial force $0$ and final force $kx$. Because force varies linearly, average force $=(0+kx)/2=kx/2$.

14. A spring has $k=100\,N/m$ and is stretched to $0.06\,m$. Calculate the elastic potential energy.

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

15. A $0.5\,kg$ body is attached to the spring in Question 14 and all stored spring energy converts to kinetic energy. Find its maximum speed.

Solution: $0.18=\frac12(0.5)v^2$, so $v^2=0.72$ and $\boxed{v\approx0.85\,m/s}$.

CONCEPTS WORKSHEETS

Worksheets

Practice sheets on conservative forces, Hooke's law, spring energy, force-extension graphs and energy conversion will be added here.

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EXERCISE 5.1 REVISION

Quick Revision

Concept	Remember
Conservative Force	Work is path independent; closed-path work is zero.
Potential Energy Relation	$W=-\Delta U$.
Non-Conservative Force	Work depends on path; friction is an example.
Hooke's Law	$F=-kx$; negative sign indicates restoring direction.
Spring Constant	$k$ has SI unit N/m; larger $k$ means a stiffer spring.
Average Spring Force	From 0 to $x$: $F_{\text{avg}}=kx/2$.
Spring Work / Energy	$U=\frac12kx^2$.
Force–Extension Graph	Area under the graph gives work done.
Kinetic Energy	$K=\frac12mv^2$.
No-Loss Conversion	$\frac12kx^2=\frac12mv^2$.

WORKSHEETS TESTS

Test Yourself

A full test covering conservative and non-conservative forces, Hooke's law, spring potential energy, graphs and energy conversion will be added here.

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REVISION NEXT CHAPTER

Symbol	Meaning	SI unit
\(F\)	Force magnitude	N
\(k\)	Spring constant	N/m
\(x\)	Extension or compression	m