Class 9 › Science › Chapter 7

Work, Energy, and Simple Machines

Work, the work-energy theorem, kinetic and potential energy, conservation of mechanical energy, power, and simple machines (pulley, inclined plane, lever).

1. Work Done by a Constant Force

Work (Eq. 7.1, 7.2) $$W = F \times s$$ force applied $\times$ displacement in the direction of the force. SI unit: joule (J). $$1\ \text{J} = 1\ \text{N}\times 1\ \text{m} = 1\ \text{kg m}^2\,\text{s}^{-2}$$

Idea: lifting 3 bags separately to 1 m, or 3 bags together, or one bag to 3 m, all need 3 times the work of lifting one bag 1 m. More force over the same distance, or the same force over more distance, means proportionally more work.
Lifting slowly: the applied upward force equals the weight $mg$.
Always specify which force does the work and on which object.
Work has no direction: only a number with a positive or negative sign.
Graph: work = area under the force-displacement graph (eg: 10 N $\times$ 1 m $= 10$ J); also valid for varying forces.

When is work zero?	Example
Force is zero ($F = 0$)	No force acting
Displacement is zero ($s = 0$)	Pushing a rigid wall: you feel tired (muscles use internal energy) but do no work on the wall; holding a barbell steady
Force perpendicular to displacement (beyond)	Carrying a box horizontally: the upward supporting force does no work

Positive work

Displacement along the force

Pushing a wheelchair; lifting a dumbbell up; striker pushing a coin.

Negative work

Displacement opposite to the force

Goalkeeper stopping a ball; lowering a dumbbell; friction on a sliding object.

Examples 7.1 and 7.2

7.1: the girl does positive work lifting the dumbbell (force and displacement both up) and negative work lowering it (force up, displacement down).

7.2: goalkeeper's hand moves back 15 cm while applying 200 N: $W = 200\times(-0.15) = -30$ J.

By Newton's third law the ball does positive work on the goalkeeper, while she does negative work on the ball.

2. The Work-Energy Theorem

Energy: the capacity to do work. SI unit: joule (J) (same as work). Positive work on an object increases its energy; it can then do work on another object (energy transfer).

Work-energy theorem (Eq. 7.3) $$\text{work done on an object} = \text{change in its energy}$$ Valid for systems of objects and for non-constant forces as well.

Examples: a ball thrown by a fielder gains energy from the work done in throwing it; a flowerpot raised to a height gains energy from the work done in lifting it.
Carrom (Example 7.3): striker does positive work on the white coin (its energy rises) and the white coin does negative work on the striker (its energy falls); same between white and black coin.
Other ways to transfer energy (beyond): heat (hotter to colder), radiation (Sun to Earth), electric circuits, sound waves, nuclear reactions.
James Prescott Joule: the SI unit of work and energy is named after him; he studied how mechanical and thermal energy are related and interconvertible.

3. Forms of Energy

Form	Meaning
Mechanical	Energy due to motion or position of objects
Thermal	Energy that makes things warm or hot
Light	Energy that allows us to see
Sound	Energy of vibrations of air or other molecules
Electrical	Energy related to position or motion of charges
Nuclear	Energy stored in the nuclei of atoms
Chemical	Energy stored in fuels and food (chemical bonds between atoms)

Energy changes form: bulb (electrical → light), water heater (electrical → thermal), food (chemical → mechanical in muscles), bell (mechanical → sound). Other chain examples: solar panel (light → electrical), photosynthesis (light → chemical), burning matchstick (chemical → heat and light), microphone (sound → electrical).
Mechanical energy $=$ kinetic energy $+$ potential energy.

4. Kinetic Energy

Kinetic energy (Eq. 7.6) $$K = \dfrac{1}{2}mv^2$$ Energy due to motion; unit joule; no direction. A body at rest has $K=0$.

Derivation

From $v^2 = u^2 + 2as$: $s = \dfrac{v^2-u^2}{2a}$.

$W = F\,s = ma\times\dfrac{v^2-u^2}{2a} = \tfrac12 m(v^2 - u^2)$ = change in energy.

If $u = 0$ this is the final kinetic energy: $K = \tfrac12 mv^2$.

Positive work raises velocity and $K$; negative work lowers them; zero work leaves $K$ unchanged (constant velocity means constant $K$, whatever the position).
$K \propto v^2$: doubling the velocity makes kinetic energy 4 times (Example 7.4). Equal $K$ for masses $m$ and $4m$ means the velocity ratio is $2:1$.
Example 7.5: ball 0.2 kg at 154.8 km h⁻¹ $= 43$ m s⁻¹: $K = \tfrac12\times0.2\times43^2 \approx 184.9$ J.

Example 7.6: aircraft carrier arrester wire

Jet 15000 kg stopped in 100 m by a backward force of 367500 N (negative work: $W = -367500\times100$).

$-\tfrac12(15000)v^2 = -367500\times100 \Rightarrow v^2 = 4900$, so $v = 70$ m s⁻¹ $= 252$ km h⁻¹.

5. Potential Energy

Potential energy: energy stored by an object because of its deformation or in a system of objects because of their relative positions.

Deformation: stretched rubber band or slingshot (gulel), bent bow, compressed or stretched spring. Work done to deform it is stored; on release it becomes kinetic energy. Internal forces undo the deformation.
Relative positions: separated unlike magnetic poles, separated electric charges, and the Earth-ball system (gravitational). Work against the interaction is stored.
Work done against friction is not stored as potential energy.

Gravitational potential energy (Eq. 7.8) $$U = mgh$$ Work to raise an object slowly to height $h$ is $mg\times h$, which appears as its potential energy (taken as zero on the ground).

Activity 7.1: a ball dropped from a greater height makes a deeper depression in sand: more height, more potential energy.
Valid only near the Earth's surface (where $g$ is nearly constant).
$U$ does not change for horizontal motion at constant height; it increases as the object is raised. It depends on height, not on the path taken (lift or staircase give the same gain).
Example 7.7: 200 g ball at 10 m, $g=10$: $U = 0.2\times10\times10 = 20$ J.

6. Conservation of Mechanical Energy

Mechanical energy $$E = K + U = \text{constant}$$ If no external forces (like friction or air resistance) act, the mechanical energy is conserved.

Falling object: potential energy (blue) turns into kinetic energy (red); the sum stays mgh.

Free fall from height $h$ (derivation)

At A ($t=0$): $U = mgh$, $K=0$, $E = mgh$.

After time $t$ at B: $v = gt$, fallen distance $= \tfrac12 gt^2$, so $U = mgh - \tfrac12 mg^2t^2$ and $K = \tfrac12 mg^2t^2$.

Sum: $E = mgh$. The decrease in $U$ equals the increase in $K$.

Pendulum (Activity 7.2): at extreme points (P, R) only $U$ (K = 0); at the lowest point (Q) only $K$ (U = 0). The bob almost reaches its starting height; in real life it stops due to friction at the support and air resistance.
Ball roller coaster: successive peaks are lower as some energy is lost to friction.
Total energy of an isolated system (no external forces) stays constant in all forms.

Example 7.8: speed at the bottom of a slide of height $h$

$\tfrac12 mv^2 = mgh \Rightarrow \boxed{v = \sqrt{2gh}}$ (ignoring friction).

The speed depends only on $h$: not on the child's mass or the shape of the slide.

Example 7.9: truck on an escape ramp (30° incline)

Truck 10000 kg at 72 km h⁻¹ $= 20$ m s⁻¹: $K = \tfrac12\times10000\times400 = 2\,000\,000$ J. Sand force 50000 N. Height gained $= d/2$ (30° incline).

Work by sand $=$ change in total energy: $-50000\,d = 50000\,d - 2\,000\,000$, so $d = 20$ m (minimum ramp length).

Gharat / panchakki (Himalayan watermill): water's potential energy → kinetic energy → turns the wheel and the grinding stone. Dams convert potential energy of stored water into kinetic energy to generate electricity.

7. Power

Power (Eq. 7.11) $$P = \dfrac{W}{t}$$ Rate at which work is done. SI unit: watt (W), $1\ \text{W} = 1\ \text{J s}^{-1}$. 1 horsepower (hp) $= 746$ W.

Same work in less time needs more power (running up stairs vs walking). More work in the same time also needs more power.
The watt is named after James Watt, who invented an efficient steam engine; horsepower compared engines with real horses.

Examples 7.10 and 7.11

7.10: 75 kg lifted 2 m in 5 s: $W = mgh = 75\times10\times2 = 1500$ J; $P = 1500/5 = 300$ W.

7.11: car 1000 kg from rest to $20$ m s⁻¹ in 10 s: $W = \tfrac12(1000)(20)^2 = 200\,000$ J; $P = 200000/10 = 20\,000$ W.

8. Simple Machines

Simple machines make a task easier by changing the magnitude or direction of the force needed. They do not reduce the total work (no machine creates energy). Effort = force applied; load = force to be overcome.

Mechanical advantage (Eq. 7.12) $$\text{MA} = \dfrac{\text{load}}{\text{effort}}$$

Fixed pulley

Changes direction

Grooved wheel guiding a rope. Effort = load, so MA = 1; only the direction changes (pulling down is easier). Movable pulleys or pulley systems (cranes, elevators) have MA > 1.

Inclined plane

Longer path, smaller force

$F'L = mgh$, so $\text{MA} = \dfrac{mg}{F'} = \dfrac{L}{h}$. A longer, gentler ramp needs less effort over a larger distance; work is the same.

Lever

Rigid bar about a fulcrum

$F_1d_1 = F_2d_2$; $\text{MA} = \dfrac{\text{effort arm}}{\text{load arm}}$. A longer effort arm gives a larger MA.

Inclined plane (Activity 7.3)

Pulling a cart up a plank needs a smaller force than lifting it vertically; the less steep the plank, the smaller the force, but the longer the distance.

Example 7.12: step height $h = 30$ cm, ramp width 40 cm, so length $L = 50$ cm (right triangle); $\text{MA} = 50/30 \approx 1.67$.

Real-life: winding hill roads with gentle slopes; inclined ladder easier than vertical.

A lever: small effort on the long arm balances a large load on the short arm.

Lever parts: fulcrum (fixed pivot), load and effort; distances from the fulcrum are the load arm and effort arm. The smaller force moves through a larger distance; total work is unchanged.
Beam balance (Activity 7.5): beam balances when $n_1L_1 = n_2L_2$, i.e. effort $\times$ effort arm $=$ load $\times$ load arm. Eg: 1 coin at 20 cm balances 2 coins at 10 cm.
Everyday: a spoon opens a can lid (long effort arm); push a hard object close to the scissors' fulcrum (smaller load arm, larger cutting force).

Example 7.13: seesaw balancing

15 kg child at A (2 m): $15\times2 = 30\times L \Rightarrow L = 1$ m, so the 30 kg child sits at D (1 m on the other side).

Lever class	Arrangement	Examples
Class I	Fulcrum in between	Scissors, crowbar, pliers, balance scale, seesaw
Class II	Load in between	Lemon squeezer, wheelbarrow, bottle opener
Class III	Effort in between	Tweezers, tongs, broom, oar

Many everyday machines combine two or more simple machines. Perpetual motion machines cannot work: real machines lose energy to friction and eventually stop.

9. Quick Revision

Quantity	Formula	SI unit
Work	$W = Fs$ (in direction of force)	joule (J)
Work-energy theorem	$W = \Delta E$ (change in energy)	J
Kinetic energy	$\tfrac12 mv^2$	J
Potential energy (gravitational)	$mgh$	J
Mechanical energy	$K + U$ (constant when conserved)	J
Power	$P = W/t$	watt (W)
Mechanical advantage	load / effort; ramp $L/h$; lever effort arm / load arm	no unit

Tip

Speed of fall

$v = \sqrt{2gh}$ for free fall or a smooth slide; independent of mass.

Tip

Units

Convert km h⁻¹ to m s⁻¹ and grams to kg before using $K=\tfrac12mv^2$ or $U=mgh$.

Remember

Machines

They trade force for distance. Work in = useful work out (ignoring friction).

Common mistakes:
• Saying work is done whenever a force is applied (displacement is needed).
• Forgetting that opposite force and displacement give negative work.
• Saying kinetic energy doubles when velocity doubles (it becomes 4 times).
• Confusing work (joule) and power (watt).
• Using $U=mgh$ far from the Earth's surface.
• Saying a machine reduces the work done (it reduces the force required).
• Taking mechanical advantage of a fixed pulley as more than 1.
• Using the arm lengths the wrong way round in $\text{effort}\times\text{effort arm} = \text{load}\times\text{load arm}$.