Work, Energy & Simple Machines
Understand how forces do work, how energy is stored and transferred, and how simple machines make tasks easier — all explained clearly for school students.
In earlier chapters, you learnt how forces change the motion of objects using Newton's laws. But when forces change with time or act in complicated ways, applying those laws directly can get tricky. The ideas of work, energy and power give us a simpler and more powerful way to understand such situations. Energy — the capacity to do work — lies at the heart of almost every activity in daily life.
In everyday language, "work" means any physical or mental effort. In science, the word has a precise meaning. Let's build that meaning step by step.
Imagine lifting a 5 kg wheat bag to a height of 1 m. You apply an upward force equal to its weight mg, and the bag moves 1 m in the direction of that force. You have done work.
From this experience, two clear patterns emerge:
- The larger the force applied over the same distance, the more work is done.
- The larger the distance over which the same force acts, the more work is done.
Work done on an object by a constant force = Force applied × Displacement in the direction of the force
Where W = work done (in joules), F = force (in newtons), and s = displacement in the direction of force (in metres).
The SI unit of work is the joule (J).
1 J = 1 N × 1 m = 1 kg m² s⁻²
One joule of work is done when a constant force of 1 newton displaces an object by 1 metre in the direction of the force.
Even when the force is not constant, work done can be found by calculating the area under the force-displacement graph between the initial and final positions.
Work done by a force is zero in three situations:
The sign of work done tells us the relationship between force direction and displacement direction.
While describing work done, always specify the force doing the work and the object on which work is done. Force and displacement have magnitude and direction, but work only has a sign (positive or negative).
When a force does work on an object, the object gains energy. Think about a cricket ball thrown by a fielder — it hits the wickets and makes them fall. The moving ball has acquired the capacity to do work. This capacity is called energy.
Work done on an object = Change in its energy (Equation 7.3)
This holds for a system of objects and even for non-constant forces. The SI unit of energy is the same as work — the joule (J).
The unit "joule" is named after scientist James Prescott Joule, who studied how mechanical energy and thermal energy are related and can convert into each other — a foundational insight in physics.
Energy is the capacity to do work. It exists in many forms and can change from one form to another. Here are the main forms:
Energy conversions happen all around us: electrical → light (bulb), chemical → mechanical (muscles), mechanical → sound (bell ringing).
Mechanical energy is the energy an object has because of its motion or its position. It has two components: kinetic energy and potential energy.
The energy an object possesses due to its motion is called kinetic energy. All moving objects (bicycle, rolling ball, cricket ball) have kinetic energy.
Starting from rest (u = 0), applying Newton's second law (F = ma) and the kinematic equation v² = u² + 2as, then using the work-energy theorem:
For the general case (initial velocity u, final velocity v), the change in kinetic energy is:
• KE has no direction — it is always positive or zero.
• Positive work → velocity increases → KE increases.
• Negative work → velocity decreases → KE decreases.
• If velocity doubles, KE becomes 4 times (because KE ∝ v²).
The energy stored by an object as a result of its deformation (stretched rubber band, compressed spring) or by a system of objects due to their relative positions (ball raised above ground) is called potential energy.
Examples of potential energy:
- A stretched slingshot — stores elastic potential energy, releases it as kinetic energy when let go.
- A compressed spring — stores energy in its deformed shape.
- A ball held at a height — stores gravitational potential energy due to its position above the ground.
- Separated unlike magnetic poles — store energy due to relative positions.
Gravitational Potential Energy is the most common form you will use. When you raise an object of mass m to a height h above the ground, you do work against gravity. By the work-energy theorem, this work gets stored as potential energy:
The expression U = mgh is valid only near the Earth's surface where g is approximately constant. Further from Earth, g decreases and the formula changes (you will learn this in higher grades).
Mechanical Energy = Kinetic Energy + Potential Energy
When only gravitational force acts (no friction or external forces), the total mechanical energy of an object remains constant. This is the Law of Conservation of Mechanical Energy.
As an object falls: PE decreases → KE increases, but (KE + PE) stays the same.
In real life, the pendulum loses energy to friction at the pivot and air resistance. This is why it eventually stops. If there were no friction or air resistance, it would swing forever (conservation of mechanical energy would hold perfectly).
Running up a staircase in 1 minute feels very different from walking up the same staircase in 5 minutes, even though the same work is done. The difference is described by power.
Power is defined as the rate at which work is done. The more work done per second, the greater the power.
The SI unit of power is the watt (W), named after James Watt.
1 W = 1 J s⁻¹ (one joule per second).
Another common unit: horsepower (hp) where 1 hp = 746 W.
The unit watt is named in honour of James Watt, who invented an efficient steam engine that could generate rotational motion and move wheels — a major milestone of the Industrial Revolution.
In everyday life, we often need to do work against gravity or friction — like lifting or moving heavy objects. Although the total work required cannot be reduced, it can be made easier by changing the magnitude or direction of the applied force. Devices that do this are called simple machines.
• Effort — the force we apply to the machine.
• Load — the force that needs to be overcome.
• Mechanical Advantage (MA) — ratio of load to effort.
A machine does not create energy. It only helps us use energy more effectively. Conservation of mechanical energy always holds: work input = useful work output (ignoring friction).
A pulley is a wheel with a groove that guides a rope. It allows you to change the direction of the force you apply.
• A fixed pulley only changes the direction of the effort. Effort = Load, so MA = 1.
• A movable pulley system has MA > 1 — it can lift heavy loads with much smaller effort. Used in elevators and cranes.
An inclined plane is a sloped surface that makes it easier to raise a heavy object to a height. Instead of lifting it straight up (which requires a large force), you push it along the slope (smaller force, larger distance).
The total work done remains the same whether you lift directly or use an inclined plane. If force decreases, the displacement increases proportionally. Trade-off: less force but more distance.
This is why hill roads are built in gentle winding slopes (large L, small angle) rather than straight up — it reduces the force needed to drive or walk uphill.
A lever is a rigid bar that rotates about a fixed point. By placing the fulcrum cleverly, a small effort can overcome a large load.
There are three classes of levers based on the relative positions of effort, fulcrum, and load:
| Class | What is in the Middle? | Examples | MA |
|---|---|---|---|
| Class I | Fulcrum is between Load and Effort | Scissors, crowbar, pliers, seesaw, balance scale | Can be >1, =1, or <1 |
| Class II | Load is between Fulcrum and Effort | Bottle opener, wheelbarrow, lemon squeezer | Always > 1 |
| Class III | Effort is between Fulcrum and Load | Tweezers, broom, hammer, oar | Always < 1 (but increases speed/range) |
📝 Exercises — Questions & Solutions
(ii) Lifting a bucket vertically upward results in positive work done on the bucket.
(iii) The SI unit for both work and energy is joule (J).
(iv) A motionless stretched rubber band has kinetic energy.
(v) Energy can change from one form to another.
(i) False — Work requires both force AND displacement. If there is no displacement (s = 0), work done = 0 even if you apply force.
(ii) True — The applied force is upward and displacement is also upward (same direction), so work done is positive.
(iii) True — Both work and energy have the same SI unit: joule (J).
(iv) False — A motionless object has zero velocity, so zero kinetic energy. A stretched rubber band has potential (elastic) energy, not kinetic energy.
(v) True — Energy can change from one form to another (e.g., chemical → mechanical, electrical → light).
(ii) 1 joule of work is done when a force of ______ newton displaces an object by 1 metre in the direction of the force.
(iii) The expression for kinetic energy of a body of mass m and velocity v is ______.
(iv) The potential energy of an object of mass m at a small height h from the Earth's surface is ______.
(v) Power is defined as the ______ at which work is done.
(i) Work done = Force × Displacement
(ii) 1 newton
(iii) ½mv²
(iv) mgh
(v) rate
(ii) The acceleration of the ball is zero.
(iii) Its kinetic energy is zero.
(iv) Its potential energy is maximum.
(iii) and (iv) are correct.
(i) False — Gravity (mg downward) still acts on the ball at the highest point.
(ii) False — The acceleration due to gravity (g = 10 m/s²) still acts; it never becomes zero.
(iii) True — At the highest point, velocity = 0, so KE = ½mv² = 0.
(iv) True — The ball is at its greatest height, so gravitational PE = mgh is maximum.
- (i) Truck uphill: Chemical (fuel) → Mechanical (kinetic + potential)
- (ii) Watch spring unwinding: Elastic potential energy → Mechanical (kinetic)
- (iii) Photosynthesis: Light energy → Chemical energy (stored in food/glucose)
- (iv) Water from dam: Gravitational potential energy → Kinetic energy → Electrical energy (in hydroelectric plants)
- (v) Burning matchstick: Chemical energy → Light + Thermal (heat) energy
- (vi) Fire cracker explosion: Chemical energy → Sound + Light + Thermal + Kinetic energy
- (vii) Speaking into microphone: Sound energy → Electrical energy
- (viii) Glowing bulb: Electrical energy → Light + Thermal energy
- (ix) Solar panel: Light energy → Electrical energy
(ii) Find the gain in PE when climbing stairs.
(iii) What can you conclude about potential energy and path taken?
(i) PE gain in elevator:
U = mgh = 50 kg × 10 m/s² × 72.5 m = 36,250 J = 36.25 kJ
(ii) PE gain on stairs:
U = mgh = 50 × 10 × 72.5 = 36,250 J = 36.25 kJ (same result)
(iii) Conclusion: Gravitational potential energy depends only on the height gained, NOT on the path taken to reach that height. Whether you go straight up or take winding stairs, the PE gained is exactly the same.
Let height of one floor = H. So 10th floor is at height 10H, and 20th floor is at 20H.
Energy for 10th floor: E₁ = mg × 10H = 10mgH
Energy for 20th floor: E₂ = mg × 20H = 20mgH
∴ Energy required for 20th floor = 2 times the energy for 10th floor.
Power for 10th floor: P₁ = 10mgH / t
Power for 20th floor: P₂ = 20mgH / 2t = 10mgH / t
∴ Power required is the same (P₁ = P₂) — doubling the height but also doubling the time means the rate of doing work stays equal.
Factors determining energy: The energy required = mgh (weight of flag × height of flagpole). So it depends on the mass of the flag and the height of the flagpole only.
Does speed change work done? No. Work done = mgh regardless of whether the flag is raised slowly or quickly. Speed affects the time taken, not the total work.
If speed doubles: The same work (mgh) is done in half the time. So Power = W/t doubles. Power becomes 2 times.
Day 1 total mass: m₁ = 60 + 100 = 160 kg
Day 2 total mass: m₂ = 60 + 40 + 100 = 200 kg
KE = ½mv², so:
KE on Day 1: K₁ = ½ × 160 × v²= 80v²
KE on Day 2: K₂ = ½ × 200 × v² = 100v²
Since all energy comes from fuel:
Ratio of fuel used = K₂ / K₁ = 100v² / 80v² = 5/4
Day 2 uses 5/4 times the fuel of Day 1 (25% more fuel with the extra passenger).
Let mass of child = m, so mass of adult = 2m.
For balance: effort × effort arm = load × load arm
Let child sit at distance d₁ from fulcrum, adult at d₂.
mg × d₁ = 2mg × d₂
∴ d₁ = 2d₂
The child must sit twice as far from the fulcrum as the adult.
If d₂ = 1 m, then d₁ = 2 m. The child sits 2 m from the fulcrum; the adult sits 1 m from the fulcrum.
(ii) If the ball reaches only 19.4 m (not the theoretical maximum), how much work was done by air resistance? (g = 10 m/s²)
(i) Sign of work done by gravity:
— Upward motion: Ball moves up, gravity acts down (opposite directions) → Work by gravity = Negative
— Downward motion: Ball moves down, gravity acts down (same direction) → Work by gravity = Positive
(ii) Work done by air resistance:
Initial KE = ½mv² = ½ × 2 × (20)² = 400 J
Theoretical max height (no air resistance): Using ½mv² = mgh → h = v²/2g = 400/20 = 20 m
But actual height = 19.4 m
PE at 19.4 m = mgh = 2 × 10 × 19.4 = 388 J
Change in energy = Final PE − Initial KE = 388 − 400 = −12 J
By work-energy theorem: Work done by air resistance = −12 J (negative because air resistance acts opposite to displacement)
Speed at 0 m:
KE = 180 J = ½mv²
180 = ½ × 10 × v²
v² = 36
v = 6 m/s
Work done by force (from graph — area under F-x graph):
From 0 to 2 m: Area = 50 N × 2 m = 100 J
From 2 to 3 m: Triangle of height 50 N, base 1 m → Area = ½ × 50 × 1 = 25 J
From 3 to 4 m: Triangle below axis (negative force), height ~50 N, base 1 m → Area = −½ × 50 × 1 = −25 J
Total work by force = 100 + 25 − 25 = 100 J
KE at 4 m = KE at 0 m + Work done = 180 + 100 = 280 J
280 = ½ × 10 × v²
v² = 56
v = √56 ≈ 7.48 m/s
Negative acceleration: Yes, in the region 3 to 4 m where the applied force is negative (opposite to motion). A negative force causes deceleration (negative acceleration), so the block slows down in this region.
Using conservation of energy: ½mv² = mgh (KE converts to PE)
So h = v²/(2g). For the same initial velocity v, height h ∝ 1/g.
On Earth: h_Earth = 8 m, g_Earth = g
On Moon: g_Moon = g/6
h_Moon / h_Earth = g_Earth / g_Moon = g / (g/6) = 6
h_Moon = 6 × 8 = 48 m
The ball will travel 48 m high on the Moon — 6 times higher than on Earth.
(ii) KE at A?
(iii) Work done by brakes between B and C?
(iv) What does KE transform into?
(i) A to B: The car moves at constant speed of 35 m/s for 1 second (the driver's reaction time before brakes are applied). No change in KE during this phase.
(ii) KE at A:
KE = ½mv² = ½ × 1000 × (35)² = ½ × 1000 × 1225 = 612,500 J = 6.125 × 10⁵ J
(iii) Work done by brakes (B to C):
At B: speed = 35 m/s → KE = 612,500 J
At C: speed = 0 → KE = 0 J
Change in KE = 0 − 612,500 = −612,500 J
By work-energy theorem, work done by brakes = −6.125 × 10⁵ J (negative because braking force opposes motion)
(iv) KE transforms into: The kinetic energy is converted mainly into thermal energy (heat) due to friction between the brake pads and wheels. A small amount may also go into sound energy.
Since the track is frictionless, Mechanical Energy (ME) = KE + PE = constant = 30 + 0 = 30 J
At P: PE = 20 J
KE = ME − PE = 30 − 20 = 10 J
½mv² = 10 → ½ × 0.5 × v² = 10 → v² = 40 → v = √40 ≈ 6.32 m/s
At Q: PE = 30 J
KE = 30 − 30 = 0 J
v = 0 m/s (momentarily at rest — same height as starting point)
At R: PE = 40 J > ME = 30 J
KE = 30 − 40 = −10 J — this is impossible! A negative KE means the ball cannot reach point R. The ball will turn back before reaching R. The maximum height it can reach corresponds to PE = 30 J (same as point O).
(i) Velocity just before hitting sand:
Using conservation of energy (neglecting air resistance):
mgh = ½mv²
v² = 2gh = 2 × 10 × 10 = 200
v = √200 ≈ 14.14 m/s
(ii) Depth of depression:
KE of coconut just before impact = ½mv² = ½ × 1.5 × 200 = 150 J
This KE is used to do work against the sand's resistive force over depth d.
Work done by sand = Force × depth = 3000 × d
By work-energy theorem:
3000 × d = 150
d = 150 / 3000 = 0.05 m = 5 cm
The coconut makes a depression of 5 cm in the sand.
- What is the velocity at the bottom of the blue slide? Using conservation of energy: v = √(2gh), where h is the height of the slide. This depends only on the height, not the shape.
- Will two children of different masses reach the bottom with the same velocity? Yes! Since v = √(2gh), mass cancels out. Both reach the bottom with the same speed.
- Which slide gives the largest velocity? The slide with the greatest vertical height h — because v = √(2gh) is largest when h is largest.
- Is work done on the barbell? No. The barbell is stationary (s = 0), so W = F × 0 = 0. The weightlifter does no work on the barbell in the scientific sense, though muscles use internal energy and she feels tired.
- Work done by friction — positive, negative or zero? Friction acts opposite to the direction of motion, so work done by friction is negative. It removes energy from the coin stack.
- Your muscles convert chemical energy (food) into mechanical energy.
- This appears as: Kinetic energy (forward motion), a small amount of thermal energy (heat due to friction in chain, tyres), and sound energy (tyre-road noise).
- On an uphill, some also becomes gravitational potential energy.
- KE_A = ½mv_A² = KE_B = ½(4m)v_B²
- v_A² = 4v_B² → v_A/v_B = 2/1
- Ratio of velocities v_A : v_B = 2 : 1 (the lighter object moves faster)
- Constant velocity horizontally: Height h does not change, so U = mgh stays constant. PE does not change.
- Raised vertically: h increases, so U = mgh increases. PE increases.
- A winding road has a longer length L for the same vertical height h, giving a higher mechanical advantage MA = L/h.
- A greater L means a smaller angle of inclination, which means a smaller force (F' = mgh/L) is needed to move the vehicle uphill.
- This reduces engine effort and makes driving safer and easier.
- An inclined ladder acts like an inclined plane. The effective component of your weight opposing your motion along the incline is less than your full weight.
- You do the same total work (mgh) but with a smaller force over a larger distance, which feels much easier on your muscles.
- A spoon acts as a Class I lever. The rim of the can acts as the fulcrum.
- By pushing down at the far end of the spoon (long effort arm), you generate a large upward force on the lid (short load arm), making it easy to pop open.
- MA = effort arm / load arm > 1, so the effort needed is much less than the load being lifted.
- Scissors are a Class I lever. When you move the object closer to the fulcrum (closer to the pivot/screw), the load arm becomes shorter.
- MA = effort arm / load arm. Smaller load arm → larger MA → greater cutting force applied to the object with the same hand effort.
- All real machines have friction — between moving parts, air resistance, etc.
- Friction converts mechanical energy into thermal energy (heat), which disperses into the environment and cannot be fully recovered.
- So the total mechanical energy of the machine decreases over time. Without an external energy source to compensate, the machine loses energy and eventually stops.
- This is why a perpetual motion machine is impossible — it would violate the law of conservation of energy.