Work, Energy, and Simple Machines

Every time you lift a schoolbag, push a stalled scooter, pull a bucket up from a well, or watch a fielder's throw crash into the stumps, you are watching work and energy trade places right in front of you. This chapter gives the everyday words work, energy and power the precise, testable meanings science needs, and then uses them to explain something genuinely useful: instead of tracking every push and pull with Newton's laws step by step, you can often just keep score of one running total called energy, and let it hand you the answer directly. We start by pinning down work done by a constant force as W = F times s, force multiplied by displacement in its own direction, noticing the counter-intuitive cases where a huge force does zero work, or where work comes out negative. The work-energy theorem then links that work to a change in energy, opening the door to the two forms of mechanical energy: kinetic energy, the energy of motion, and potential energy, the energy stored in height, stretching or squeezing. Together they explain why a falling flowerpot can dent a step, why a stretched slingshot fires a stone, and why a pendulum keeps swinging back to nearly the same height it started from. Along the way you will meet power, the rate of doing work, why sprinting up a staircase feels so different from strolling up it even though both use exactly the same energy. The chapter closes with simple machines: the pulley, the inclined plane and the lever, three everyday devices that never create energy and never reduce total work, but trade force for distance so a task feels easier to a human body. By the end, you should be able to calculate work, kinetic energy, potential energy and power from given numbers, apply conservation of mechanical energy to falling and swinging objects, and compute the mechanical advantage of a ramp, a pulley or a lever.

Key takeaways
  • Work done by a constant force is W = F times s, force multiplied by displacement in the direction of the force; its SI unit is the joule, 1 J = 1 N times 1 m.
  • Work is zero if the force is zero, if there is no displacement, or if the force is perpendicular to the displacement, as when a girl carries a box horizontally while pushing straight up.
  • Work is positive when force and displacement point the same way, negative when opposite, such as a goalkeeper's hand slowing a ball.
  • The work-energy theorem states work done on an object equals the change in its energy, W = change in E, and it holds even when the force is not constant.
  • Kinetic energy is the energy of motion, K = one-half m v squared; gravitational potential energy is U = mgh.
  • Mechanical energy, kinetic plus potential, is conserved for an object acted on only by gravity, letting you find a falling or swinging object's speed directly.
  • Power is the rate of doing work, P = W over t, measured in watts, 1 W = 1 J per second; 1 horsepower equals 746 W.
  • Simple machines such as the pulley, inclined plane and lever change the direction or size of the force needed by trading force for distance, but never reduce total work.
  • Mechanical advantage equals load over effort; for an inclined plane it is length over height, and for a lever it is effort arm over load arm.
  • Key idea: work done by a force equals force times displacement along the force, and this work becomes a change in an object's kinetic and or potential energy, so mechanical energy is conserved under gravity alone while simple machines only trade force for distance without ever reducing the total work required.
The core insight

Work done by a force, W = F times s along the direction of the force, becomes a change in an object's energy through the work-energy theorem, so under gravity alone kinetic energy of one-half m v squared and gravitational potential energy of mgh trade into each other while their sum, the mechanical energy, stays constant, power measures how fast that work is done as P = W over t, and simple machines such as the pulley, inclined plane and lever change only the size or direction of the needed force, never the total work.

What this chapter covers

This chapter has one running thread: work done on an object always shows up as a change in its energy, and everything else follows from that single idea. Work through it in this order.

  • Work done by a constant force — W = F times s, its unit the joule, and when work is zero, positive or negative.
  • The work-energy theorem — work done equals change in energy, the statement that connects everything else in the chapter.
  • Kinetic and potential energy — the energy of motion and the energy of position, and their formulas K = one-half m v squared and U = mgh.
  • Conservation of mechanical energy — how kinetic and potential energy trade places as an object falls or swings, while their sum stays fixed.
  • Power — the rate at which work is done, and why doing the same work faster needs more power.
  • Simple machines — the pulley, the inclined plane and the lever, and how mechanical advantage tells you how much a machine trades force for distance.

The core ideas

Work done by a constant force

Picture lifting a 5 kg bag of wheat straight up by 1 m: you apply an upward force equal to its weight, mg, and it moves 1 m in that same direction — you have done some work. Lift three such bags to the same height and you do three times the work; lift one bag three times as high instead and you again do three times the work. A larger force over the same distance, or the same force over a larger distance, both do proportionally more work, exactly as captured below.

W=F×sW = F \times s

where: W is work done in joules (J); F is the constant force in newtons (N); and s is displacement in the direction of that force, in metres (m). Since 1 N = 1 kg m/s squared, 1 J = 1 N m = 1 kg m squared per second squared. Force and displacement have direction, but work does not; it is a plain signed number. Three special cases matter more than the formula. Work is zero whenever the force is zero, or whenever the object does not move, s = 0, however large the force — pushing hard against a rigid wall does zero work, even though your muscles tire from repeatedly contracting. It is also zero whenever the force is exactly perpendicular to the displacement, as when a girl carries a box while walking: her upward force balances its weight, but the box moves horizontally, so no work is done. Work is positive when force and displacement point the same way, as when you push a wheelchair forward, and negative when they point opposite ways, as when a goalkeeper's hand pushes backward against a ball still moving toward the goal, decelerating it.

Figure
Three labelled panels showing positive work when force and displacement point the same way, negative work when a goalkeepers hand opposes the motion of the ball, and zero work when a force pushes on a rigid wall that does not move
The sign of work done depends on the angle between force and displacement Positive work chair force F displacement s force and motion same way Negative work ball displacement s, ball moving in force F, goalkeeper force opposes motion Zero work force F wall does not move, s = 0

The work-energy theorem and forms of energy

When work is done on an object, does it gain something it can later use to do work of its own? A fielder throws a cricket ball toward the wicket; the moving ball knocks the wicket down. A flowerpot raised to a height can damage something below it if it falls. In each case the object has gained the capacity to do work, called energy. Work done on an object always appears as a change in its energy, a relationship called the work-energy theorem.

W=ΔEW = \Delta E

where: W is the work done on an object or system, and Delta E is the resulting change in its energy, measured in the same unit, the joule. This holds even when the force is not constant, and even for a whole system, which is why it can replace long chains of Newton's-law reasoning. Energy shows up in several everyday forms: mechanical (motion or position), thermal, light, sound, electrical, nuclear, and chemical (stored in food and fuel), and these convert into one another — electrical energy becomes light in a bulb, chemical energy in food powers your muscles, and a ringing bell turns mechanical energy into sound. This chapter focuses on mechanical energy, since it connects most directly to forces and motion already studied.

Kinetic energy

The energy an object possesses because of its motion is called kinetic energy. Every moving object, from a rolling ball to a speeding car, has kinetic energy, and an object at rest has zero. The formula, derived in full below, is one of the most-used results in this chapter.

K=12mv2K = \frac{1}{2} m v^2

where: K is kinetic energy in joules, m is mass in kilograms, and v is speed in metres per second. Because speed is squared, kinetic energy grows much faster than speed does: doubling speed makes kinetic energy four times as large, not merely twice. It increases when positive work is done on the object, decreases when negative work is done, and stays constant when no net work is done.

Potential energy

Stretch a slingshot, bend an archer's bow, or compress a spring, and the object has not moved by itself yet, but it has gained the capacity to do work once released. This stored capacity, from deformation or from the relative positions of interacting objects, is called potential energy. A ball lifted above the ground and the Earth together form a system storing energy this way, since raising the ball against gravity needs a force, and the system can do work (deform sand, break glass) when released. Since the Earth barely moves toward the ball, this is usually just called the gravitational potential energy of the ball.

U=mghU = mgh

where: U is potential energy in joules, m is mass in kilograms, g is 10 m/s squared unless stated otherwise, and h is the height above a chosen reference level, usually the ground, in metres. Potential energy is always relative to a reference level you choose to call zero; raising that reference changes only the numbers, not the physics. Raising a ball higher stores more potential energy, which is why it makes a deeper depression when dropped from a greater height.

Conservation of mechanical energy

The sum of an object's kinetic and potential energy is its mechanical energy. Consider a ball released from height h at rest: at that instant it has potential energy mgh and zero kinetic energy, so its mechanical energy is mgh. As it falls, its height and potential energy decrease while its speed and kinetic energy increase. The remarkable fact, proved step by step in the second derivation below, is that these two changes exactly cancel: mechanical energy stays equal to mgh throughout the fall, provided no force such as air resistance does work on it. This is conservation of mechanical energy, and it works just as well for a swinging pendulum, whose bob trades potential for kinetic energy and back on every swing, returning almost to its starting height.

K+U=constantK + U = \text{constant}

where: K is kinetic energy, U is potential energy, and their sum stays fixed at every point of the motion, provided only gravity does work. In real pendulums and falling objects, friction and air resistance slowly remove energy from this total as heat and sound, which is why a real pendulum eventually stops swinging.

Figure
A pendulum swinging between two extreme points labelled P and R where it has only potential energy and zero kinetic energy, and its lowest point labelled Q where it has only kinetic energy and zero potential energy, showing that kinetic plus potential energy is the same constant throughout the swing
Mechanical energy stays constant as a pendulum swings between P and R reference line, height of release P all potential energy zero kinetic energy Q all kinetic energy zero potential energy R all potential energy again zero kinetic energy Kinetic energy plus potential energy is the same constant value at P, Q and R

Power

Carrying your bag up a flight of stairs by running takes one minute; walking up slowly takes five minutes; yet the work done against gravity is exactly the same in both cases. What differs is how quickly that work is done, a quantity called power.

P=WtP = \frac{W}{t}

where: P is average power in watts (W), W is work done in joules, and t is time in seconds. One watt is one joule per second, 1 W = 1 J/s. Doing more work in the same time, or the same work in less time, both need more power. Engines are often rated in horsepower instead, where 1 horsepower equals 746 W.

Simple machines: pulley, inclined plane and lever

Many everyday tasks require working against gravity or friction. Total work needed cannot be reduced, but it can feel easier by changing the size or direction of the force you apply. Devices that do this are simple machines. The force you apply is the effort; the force you overcome, usually a weight, is the load.

mechanical advantage=loadeffort\text{mechanical advantage} = \frac{\text{load}}{\text{effort}}

where: mechanical advantage is a plain ratio with no unit. A value greater than 1 means less force than the load itself, at the cost of a larger hand-distance, since work put in must equal useful work delivered, ignoring friction. A pulley is a wheel with a grooved rim guiding a rope. A single fixed pulley does not reduce the force needed, only its direction, so pulling downward replaces lifting up, giving mechanical advantage exactly 1. A movable pulley, or a system of pulleys, can exceed 1, letting a small effort lift a much heavier load — why cranes and elevators use them. An inclined plane, such as a ramp, raises a load to height h using a smaller force F prime over the longer length L instead of the full weight mg over height h. Since F prime times L must equal the potential energy gained, mgh, the mechanical advantage works out to L divided by h: a longer, shallower ramp needs a smaller force. A lever is a rigid bar that rotates about a fixed point called the fulcrum, with the load arm and effort arm being the distances of the load and the effort from the fulcrum. Because the work done at one end transfers to the other, the following balance holds for a lever in equilibrium.

F1×d1=F2×d2F_1 \times d_1 = F_2 \times d_2

where: F1 is the smaller force applied at the end that moves a larger distance d1, and F2 is the larger force delivered at the end that moves the smaller distance d2. By making the effort arm much longer than the load arm, a lever lets a small applied force lift a much heavier load, and its mechanical advantage equals the effort arm divided by the load arm.

Figure
A lever diagram showing a fulcrum with a long effort arm on the left carrying a small applied force and a short load arm on the right carrying a larger load force so that force times distance stays equal on both sides
A lever trades a small effort over a large distance for a larger load over a small distance fulcrum effort F1 small force load F2 large force effort arm, d1 (large) load arm, d2 (small) F1 times d1 equals F2 times d2, so a long effort arm needs only a small F1
Derivation ADeriving kinetic energy from the work-energy theorem

The goal is to find a formula for the kinetic energy of a moving object purely from the work-energy theorem and known kinematic equations, without introducing any new law of physics. Start an object from rest, let a constant force act over some distance, and use the work-energy theorem to identify the work done as the change in energy, then express that work using only mass and speed.

Step 1. An object of mass m starts with speed u and, under a constant force F and acceleration a, reaches speed v after displacement s. The kinematic equation v squared equals u squared plus 2 a s rearranges to s equal to (v squared minus u squared) over 2a.

Step 2. Work done, W equals F times s; using Newton's second law F equals m a, this becomes W equal to m a times s.

Step 3. Substitute s from Step 1: the acceleration a cancels top and bottom, leaving W equal to one-half m times (v squared minus u squared), depending only on mass and the two speeds.

Step 4. By the work-energy theorem, work done equals change in energy. Starting from rest, u equals zero, so the change in energy is simply the final kinetic energy: K equal to one-half m v squared.

Step 5. Kinetic energy depends on mass directly but on speed through its square, so doubling speed makes kinetic energy four times as large, not twice: one-half m (2v) squared equals 4 times one-half m v squared — why braking distances grow so sharply with speed.

Go deeper · Derivation B · 1 minConservation of mechanical energy during free fall
Derivation BConservation of mechanical energy during free fall

The goal is to show, using only kinematics and the definitions of kinetic and potential energy already introduced, that as an object falls freely from height h, its kinetic plus potential energy stays exactly equal to mgh at every instant, even though each part keeps changing. This is the algebraic proof behind conservation of mechanical energy.

Step 1. At the start, point A, time t = 0, the object of mass m is at height h and has just been released, so its velocity is zero. Potential energy is mgh, kinetic energy is zero, so mechanical energy at A is mgh.

Step 2. Falling freely for time t from rest, its velocity is v = gt, and the distance fallen is one-half g t squared, so its height at this instant, point B, is h prime = h minus one-half g t squared.

Step 3. Potential energy at B: m g times (h minus one-half g t squared), which expands to mgh minus one-half m g squared t squared.

Step 4. Kinetic energy at B: one-half m v squared = one-half m (g t) squared = one-half m g squared t squared.

Step 5. Adding Steps 3 and 4, the minus and plus one-half m g squared t squared terms cancel exactly, leaving mgh. Since B and t were arbitrary, mechanical energy equals mgh at every instant of the fall: whatever potential energy is lost is gained back exactly as kinetic energy.

Worked examples

Board level - stopping an escape ramp truck (NCERT Example 7.9)

A truck of mass 10000 kg is moving at 72 km/h when its brakes fail. The driver steers it onto an escape ramp inclined at 30 degrees, filled with sand that exerts a constant resistive force of 50000 N opposite to the truck's motion, and the truck comes to rest on the ramp. Using g = 10 m/s squared and the fact that for this 30-degree incline the truck rises 1 m vertically for every 2 m it travels along the ramp, find the minimum length of ramp needed to stop the truck.

Step 1. Convert the speed: 72 km/h = 20 m/s. Initial kinetic energy = one-half times 10000 times 20 squared = 2,000,000 J; initial potential energy is zero, so total initial mechanical energy is 2,000,000 J.

Step 2. Let the truck travel distance d along the ramp. By the hint, the height risen is d over 2, so final potential energy = 10000 times 10 times d over 2 = 50000 d joules; final kinetic energy is zero.

Step 3. The sand's negative work, minus 50000 d, equals the change in mechanical energy, (50000 d) minus 2,000,000. So minus 50000 d = 50000 d minus 2,000,000, giving 2,000,000 = 100000 d, so d = 20.

Answer: the minimum length of the ramp is d = 20 m.

Board level - power needed to lift a weight (NCERT Example 7.10 extended)

A weightlifter lifts a 75 kg mass through a height of 2 m in 5 seconds. Take g = 10 m/s squared. Find (a) the work done and (b) the power she develops, and (c) the power she would need if she lifted the same mass to the same height in half the time.

Step 1. Work done against gravity: W = mgh = 75 times 10 times 2 = 1500 J.

Step 2. Power for the 5 s lift: P = W over t = 1500 over 5 = 300 W.

Step 3. The same 1500 J done in 2.5 s instead needs 1500 over 2.5 = 600 W: double the power for half the time, since the work is unchanged.

Answer: work done = 1500 J; power for the 5 s lift = 300 W; power for the 2.5 s lift = 600 W.

Higher-order thinking (HOTS / olympiad-style) - balancing a seesaw with two loads on one side (extends NCERT Example 7.13)

A 4 m seesaw pivots at its centre. A 20 kg child sits at the far left end, 2 m from the fulcrum. On the right, a 10 kg child sits at the far right end, also 2 m out, and a 30 kg child must sit somewhere on that same right side so the seesaw is exactly balanced. Find the 30 kg child's distance from the fulcrum.

Step 1. A lever balances when load times distance on one side equals the total on the other; since every child feels the same g, mass times distance compares directly, as NCERT's Table 7.1 does with coins. The left side gives 20 times 2 = 40 kg m.

Step 2. The right side has two children, so their turning effects must sum to 40 kg m. The 10 kg child gives 10 times 2 = 20 kg m, leaving 40 minus 20 = 20 kg m for the 30 kg child alone.

Step 3. Let L be the 30 kg child's distance from the fulcrum: 30 L = 20, so L = 20 over 30 = 0.67 m.

Answer: the 30 kg child must sit about 0.67 m (roughly 67 cm) from the fulcrum, between the fulcrum and the 10 kg child. Timing note: budget about three minutes. The trap is treating the seesaw as balancing only one mass against one mass, as in the textbook's original two-child version; once a side carries two loads, net their turning effects first, as in Step 2. Check the answer is less than half the seesaw's length — 0.67 m comfortably fits within 2 m.

Common traps

  • Believing work is done whenever a force is applied, even without displacement — pushing a locked door with all your strength does zero work on it.
  • Forgetting the perpendicular case: an upward force balancing a carried box's weight while it moves horizontally does zero work.
  • Dropping the sign of work: it is negative whenever the force opposes the displacement, as with friction or a goalkeeper's save.
  • Assuming doubling speed doubles kinetic energy; since K is proportional to v squared, doubling speed actually quadruples it.
  • Treating potential energy as absolute rather than relative to a chosen reference height, usually the ground.
  • Mixing units before substituting into a formula, especially leaving a speed in km/h instead of m/s.
  • Believing a simple machine reduces total work needed; it only changes the size or direction of the force, and real machines need slightly more input work, because of friction.
  • Assuming mechanical advantage greater than 1 means "getting energy for free," instead of the smaller force moving a proportionally larger distance.

You should be able to…

  • Calculate work using W = F times s, and identify when it is positive, negative or zero.
  • State and apply the work-energy theorem, W equals change in E.
  • Derive and apply K = one-half m v squared, including how it scales with the square of speed.
  • Derive and apply U = mgh, choosing a sensible reference height.
  • Apply conservation of mechanical energy to falling and swinging objects to find speed or height directly.
  • Calculate power using P = W over t, and explain why faster work needs more power.
  • Explain why a pulley, an inclined plane or a lever eases a task without changing total work.
  • Calculate mechanical advantage for a pulley, an inclined plane (L over h) and a lever (effort arm over load arm).
  • Solve a lever or seesaw balance problem with more than one load on the same side.

Ask the tutor: if you would like to work through another lever, pulley or free-fall energy problem with different numbers, just ask.

If I push against a wall as hard as I can and get tired, why does science say I did zero work on the wall?

Because the scientific definition of work needs two things together: a force and a displacement in the direction of that force. The wall does not move at all, so however large a force you apply, the work done on the wall is zero. You still feel tired because your muscles do internal work, repeatedly contracting and using up stored chemical energy, even though none of that shows up as work on the wall itself. Feeling tired is a biological signal, not a physics measurement.

Does the shape of a water slide change how fast a child is moving at the bottom?

No, as long as friction is small enough to ignore. Conservation of mechanical energy says the potential energy at the top, mgh, converts entirely into kinetic energy at the bottom, so v equals the square root of 2 g h. The mass m cancels out completely, and the shape of the slide never enters the calculation, only the height h. So a heavier and a lighter child released from the same height reach the bottom with the same speed, whether the slide is straight, curved or bumpy.

Why does kinetic energy become four times as large when speed doubles, instead of just twice as large?

Because kinetic energy depends on the square of the speed, not the speed itself: K equals one-half m v squared. Doubling v means squaring a number that has doubled, and (2v) squared works out to 4 v squared, four times the original. This is why braking distances grow so sharply with speed, and why a modest jump from 40 km/h to 60 km/h means a much bigger jump in kinetic energy than the speed increase alone suggests.

How can a much lighter child lift a heavier stapler or rock using a simple lever like a scale balanced on a pencil?

Because a lever trades a smaller applied force for a larger distance moved, and the two ends of a rigid lever must do equal amounts of work on each other. If the push end is much farther from the fulcrum than the load end, it moves through a much larger distance for the same work, so a much smaller force can still lift a much larger load. This is F1 times d1 equals F2 times d2: a long effort arm d1 lets a small F1 balance a large F2, exactly what you feel when a long-handled lever shifts a heavy rock your bare hands could never budge.

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Where this leads