Work, Energy, Power and Simple Machines

A student sits for six hours reading a book and says, 'I have done so much work today.' In everyday language that is true, but in Physics he has done no work at all. Physics has a stricter meaning. Work is done only when a force acts on a body and the body is displaced in the direction of the force (or has a component of displacement along it). Two conditions must be satisfied together: (1) a force must act, and (2) the body must move through some displacement. If either is missing, the work done is zero.

Work = Force × Displacement in the direction of the force: W = F s cos θ, where θ is the angle between the force and the displacement. If the force and displacement are along the same line, W = F × s. Work is a scalar quantity, even though force and displacement are vectors, because it is the dot product of the two. Its sign tells the story: the sign shows whether the force helped the motion, opposed it, or did nothing.

Unit of work. The SI unit of work is the joule (J). 1 joule is the work done when a force of 1 newton moves a body through 1 metre in the direction of the force: 1 J = 1 N × 1 m. The unit is named after James Prescott Joule. Its dimension is [M L² T⁻²], which is the same as that of energy, since work and energy are measured in the same units. A bigger unit is the kilojoule: 1 kJ = 1000 J.

Work against gravity. To lift a body of mass m slowly through a height h, you must apply a force equal to its weight mg upward, so the work you do is W = mgh. The work done by gravity on the same body is −mgh, because gravity points down while the displacement is up. If the body comes down by h, gravity does +mgh. The important point is that this work depends only on the height difference, not on the path taken: climbing a hill by a straight steep path or by a long winding road, the work against gravity is the same. (The winding road simply spreads the same work over a longer distance, which is why it feels easier.)

A hammer lying on the table cannot drive a nail; the same hammer swung fast can. A stretched bow does nothing until released, and then it can send an arrow far. In each case the body has a capacity to do work, and that capacity is called energy. Energy is a scalar, and its SI unit is the same as that of work, the joule (J). Energy exists in many forms: mechanical, heat, light, sound, electrical, chemical, nuclear. In this topic we study mechanical energy, which comes in just two types: kinetic and potential.

Why KE = ½mv²? When a constant force F moves a body from rest through a distance s, the work done is F × s. With a = F/m and the equation v² = u² + 2as (u = 0), we get s = v²/2a, so the work is F × v²/2a = ma × v²/2a = ½mv². That work is stored as the energy of motion. So kinetic energy = the work needed to bring the body from rest to speed v, and it is exactly equal to the work it can do before coming to rest.

Gravitational potential energy. To raise a body of mass m through a height h, the work done against gravity is mgh, and this work is stored as gravitational PE = mgh. The value of h is measured from a chosen reference level (usually the ground), so what matters in practice is the change in PE between two heights. If an object is lifted, PE rises; if it falls, PE falls.

Elastic potential energy. A spring (or rubber band) that is stretched or compressed by a distance x stores PE = ½ k x², where k is the spring constant. Because of the x², halving the stretch cuts the stored energy to one-fourth, and doubling the stretch makes it four times. For a deformed solid body, the elastic PE per unit volume is ½ × stress × strain.

Drop a ball from your hand. As it falls, it moves faster and faster, so its kinetic energy grows. Where does this new energy come from? From the potential energy it had at the height, which keeps shrinking. When the ball hits the ground, the energy has not vanished: some of it has become sound and heat. This is the great idea of this topic, the Law of Conservation of Energy: energy can neither be created nor destroyed; it can only be changed from one form to another. The total energy of an isolated system always remains constant.

For a falling body (no air resistance), mechanical energy KE + PE stays constant. At every instant, whatever PE is lost is gained as KE. This is why a body dropped from height h reaches the ground with speed v = √(2gh), which does not depend on its mass. A heavy and a light stone dropped together land together, with the same speed.

Work–energy theorem. The net work done on a body equals the change in its kinetic energy: W(net) = ΔKE = ½mv² − ½mu². Positive net work increases the speed; negative net work (friction, brakes) decreases it. This is just another face of conservation of energy, and it is the quickest way to solve many numericals.

Conservative and non-conservative forces. A force is conservative if the work it does depends only on the starting and ending positions, not on the path, so the work over a complete round trip is zero. Gravity, the spring force and the electrostatic force are conservative, and only for these can we define a potential energy. Friction and air drag are non-conservative: the longer the path, the more work they take. Under conservative forces alone, the total mechanical energy is conserved. Under friction, mechanical energy falls, but the lost part appears as heat, so the total energy is still conserved.

Two boys carry the same bag of cement up to the same terrace. One takes 10 seconds and the other takes 30 seconds. The work done is the same, because the weight and the height are the same, yet we say the first boy is more powerful. What differs is the speed of doing work. This is power: the rate of doing work, or the rate at which energy is transferred or transformed.

Power P = Work / Time = W / t. Since W = F s, for a body moving at constant speed v we also get P = F v. Power is a scalar. Its SI unit is the watt (W): 1 watt = 1 joule per second (1 W = 1 J/s). The unit honours James Watt, the improver of the steam engine. Dimension of power: [M L² T⁻³]. For a fixed amount of work, power is inversely proportional to time: double the power and the same work takes half the time.

Commercial unit of electrical energy. Energy used = power × time. If power is in kilowatts and time in hours, the energy comes out in kilowatt-hours (kWh), which is the 'unit' charged by electricity boards. For example, a 60 W lamp used for 5 hours every day for 30 days uses 60 W × 5 h × 30 = 9000 Wh = 9 kWh, that is, 9 units. With the electrical formulas, power can be found from P = VI = I²R = V²/R. For a bulb at 200 V drawing 0.1 A, P = 200 × 0.1 = 20 W.

Nobody can lift a car by hand, but a small jack lifts it with a few pumps of the handle. A tiny nail cannot be pulled out with fingers, but a claw hammer does it easily. These tools are machines. A machine is a device that makes work easier or more convenient by changing the size or the direction of the force, or by changing the speed. The simplest ones are called simple machines: the lever, pulley, inclined plane, wheel and axle, wedge and screw.

A machine never gives you extra work. If it lets you use a small effort, you must move that effort through a longer distance, so what you gain in force you lose in distance (work = force × distance stays about the same). In fact, because of friction, the work you put in is always more than the work you get out. A machine is therefore a way of trading force for distance, never of getting energy for free. This is the conservation of energy again.

The lever. A lever is a rigid bar that can turn about a fixed point called the fulcrum (F). The effort and the load act at two other points. The turning effect of a force about the fulcrum is its moment (torque) = force × perpendicular distance from the fulcrum, measured in newton-metre. A lever balances when the clockwise moment equals the anticlockwise moment, which is the principle of moments:

Effort × effort arm = Load × load arm, so MA = Load / Effort = Effort arm / Load arm.

So a long effort arm and a short load arm give a big MA. That is why pushing a door near its handle (far from the hinge) is easy, but near the hinge is hard.