Calculate work as a transfer of energy, track energy transformations in a system, and account for energy dissipated as thermal energy

Learning goal: Calculate work from force and displacement when they are along the same line; use work as the change in a system's energy; represent the energy transformations of a phenomenon with bar or pie graphs; combine work, kinetic energy and potential energy with conservation of energy to solve problems; and explain why the energy dissipated as thermal energy is still conserved but less available for doing further work.

Work is a transfer of energy. In middle school, work was introduced as the way energy moves into or out of a system when an outside force moves an object over a distance. Now it gets a number. When a constant force F (newtons) acts on an object that moves a displacement Δx (meters) along the same line as the force, the work done is W = F Δx, in newton-meters, which are joules. The condition matters: this equation is for a force in the same direction as the motion or exactly opposite to it. A force at right angles to the motion, such as the upward pull of a rope on a bucket carried horizontally, does no work by this equation, and a force with no displacement does none either. Pushing hard against a wall that does not move transfers no energy to the wall, however tired you get. A force opposite to the motion, such as friction on a sliding box, does negative work: it takes energy out of the box's motion.

Work equals the change in energy. Because work is energy transferred, when work is the only way energy enters or leaves a system, the work done on the system equals the change in its energy: W = ΔE. This gives a second way to calculate work with no force in sight. Lifting a 5.0 kg box 2.0 m at steady speed needs an upward force equal to its weight, 5.0 × 9.8 = 49 N, so W = 49 × 2.0 = 98 J; the gravitational potential energy of the box (strictly, of the box and Earth together, the system the lifting force does work on) rose by 5.0 × 9.8 × 2.0 = 98 J. Both routes agree, and they must, because the same energy was transferred. If an engine does 500 J of work on a stationary cart on a level road with no friction, the cart's kinetic energy becomes 500 J.

Transformations and where the energy goes. A phenomenon can be followed as a chain of transformations: at the top of a roller-coaster hill the car's energy is almost all gravitational potential; on the way down it becomes kinetic; climbing the next hill it becomes potential again. A bar graph with one bar per form of energy at each stage, or a pie graph showing the shares, makes the accounting visible, and the total height or the whole pie must be the same at every stage as long as no energy enters or leaves the system being tracked. Such a system is often called closed; the stricter word is isolated. In any real transfer, some energy is transformed into thermal energy by friction and air resistance, and it warms the track, the wheels and the air. Thermal energy is the random motion of many trillions of particles. It is still energy and is still counted, so the total stays constant; but it is spread among so many particles in so many directions, at close to the temperature of the surroundings, that almost none of it can be gathered up and organized to move the car again. (Thermal energy can still do work where there is a temperature difference to use, as in an engine; once it has spread out at the temperature of the surroundings, that chance has gone.) That is why, even though energy is conserved, less of it is available for useful work after every transfer, and why no roller coaster can climb a second hill as high as the first without a motor.

Estimating dissipated energy. The energy transferred out of a system as thermal energy can be estimated in two ways. From energy differences: compare the energy the system should have (from conservation with no loss) with the energy it actually has (from a measurement), and the gap is the energy dissipated. From work done by friction: the friction force times the distance over which it acts is the work done against friction, all of which becomes thermal energy. A sled pushed 6.0 m against a 15 N friction force transfers 15 × 6.0 = 90 J of the pusher's work into thermal energy in the runners and snow. A bouncing ball gives the first method: it rises less after each bounce, and the drop in height gives the energy dissipated in that bounce.

A fully worked model: a roller coaster with losses. A hypothetical 500 kg car is at rest at the top of a 20 m hill, with the bottom of the hill as reference. At the top: E_g = 500 × 9.8 × 20 = 98,000 J, Eₖ = 0. With no losses the speed at the bottom would satisfy ½ × 500 × v² = 98,000, so v² = 392 and v ≈ 19.8 m/s. A speed sensor at the bottom instead reads 18 m/s, so the actual kinetic energy is ½ × 500 × 18² = ½ × 500 × 324 = 81,000 J. The gap, 98,000 − 81,000 = 17,000 J, is the energy transferred out of the car as thermal energy through work done by friction and air resistance, about 17 % of the start. The bar chart shows the three amounts. A redesign that lowers friction (smoother wheels, a shorter track) would shrink the third bar and raise the second. The car can then climb a second hill no higher than 81,000 ÷ (500 × 9.8) ≈ 16.5 m, and less if losses continue on the climb.

Investigation: measure the energy dissipated by a bouncing ball. Materials: a tennis ball or rubber ball of measured mass (a balance), a meter stick against a wall, a phone camera at a known frame rate or a partner watching at eye level, goggles. Steps: drop the ball from rest at a measured height of 1.00 m and record the height of the top of the first bounce; repeat five times and average; then use the average first-bounce height as a new release height and record the second bounce, and so on for three bounces. Record: release height and bounce height for each trial, and for each bounce E_g before (m g h_release) and after (m g h_bounce). For a 0.060 kg ball released from 1.00 m that bounces to 0.64 m, E_g before is 0.060 × 9.8 × 1.00 = 0.588 J, after is 0.060 × 9.8 × 0.64 ≈ 0.376 J, and the energy dissipated in the bounce is about 0.212 J, roughly 36 % of the energy before it. A result that supports the claim: each bounce returns a roughly constant fraction of the energy before it, and the energy dissipated per bounce falls as the bounces get smaller, consistent with energy leaving as thermal energy (and a little sound) in the ball and floor. Safety: goggles; drop, do not throw; keep the drop zone clear. What the model cannot show: it does not measure the thermal energy directly or show where in the ball it appears; a video at 30 frames per second may miss the exact top of a bounce by a few millimeters, which sets the precision of the estimate.

Worked example 1 — work from force and displacement. A student pushes a crate with a steady 40 N force along a floor for 3.0 m in the direction of the push. W = F Δx = 40 N × 3.0 m = 120 J. This is the energy transferred to the crate system; if the crate moved at constant speed the whole 120 J went into thermal energy through friction, and if there were no friction it all became kinetic energy. If the student pushed the same crate against a wall and it did not move, Δx = 0 and W = 0 J.

Worked example 2 — work from a change in energy. A crane lifts a 5.0 kg block 2.0 m at steady speed. The force needed equals the weight, 5.0 × 9.8 = 49 N, so W = 49 × 2.0 = 98 J. Check by energy, taking the block and Earth together as the system the crane does work on: ΔE_g = 5.0 × 9.8 × 2.0 = 98 J and ΔEₖ = 0 because the speed did not change, so W = ΔE = 98 J. Both methods give the same answer because work is the energy transferred.

Worked example 3 — energy dissipated from a speed measurement. The 500 kg coaster car from the lesson starts from rest at 20 m, so it has 98,000 J, and reaches the bottom at a measured 18 m/s, so Eₖ = ½ × 500 × 324 = 81,000 J. Energy transferred out as thermal energy = 98,000 − 81,000 = 17,000 J. As a bar graph: at the top one bar of 98 kJ labeled E_g; at the bottom two bars, 81 kJ labeled Eₖ and 17 kJ labeled thermal, with the same total height. The maximum height of the next hill without a motor is 81,000 ÷ (500 × 9.8) ≈ 16.5 m.

Worked example 4 — dissipation from work against friction. A sled, starting from rest, is pushed 6.0 m across level snow with a 15 N friction force opposing it. Work done against friction = 15 N × 6.0 m = 90 J, and all of it is transformed into thermal energy in the runners and snow. If the pusher applied 25 N over the same 6.0 m, the pusher did 25 × 6.0 = 150 J of work, 90 J of which became thermal energy and 60 J of which is the sled's gain in kinetic energy; because the sled started from rest, its kinetic energy at the end is ½ m v² = 60 J.

Representation task: energy bar graphs for a ride. For the 500 kg coaster car, draw three sets of energy bars, each with bars for E_g, Eₖ and thermal energy: at the top of the 20 m hill (98 kJ, 0, 0), at the bottom (0, 81 kJ, 17 kJ) and at the top of a second hill 12 m high, using conservation to find the kinetic energy there if a further 5 kJ is dissipated on the climb. Label every bar with its value, check that each set totals 98 kJ, and write one sentence explaining why the thermal bar never shrinks.

Independent performance: build, predict and improve a model coaster. Using foam pipe insulation or track, a marble of measured mass, a meter stick and a photogate or video, build a track with a starting hill, two loops and a second hill. Success criteria: the potential energy at the start is calculated from measured mass and height, once with the lowest point of the track as the reference level and once with the floor, with a sentence explaining why the predicted speeds do not depend on that choice; the speed at the bottom of each hill, and at the top of the second hill, is predicted from conservation of energy with no losses; the speed at the bottom of each hill is then measured, compared with its prediction and used to calculate the kinetic energy there; the energy missing from the marble's motion at the bottom of the first hill is reported in joules and as a percentage, with a note that part of it is the marble's spin rather than energy dissipated (for a solid ball rolling without slipping, spin holds 2/7 of its kinetic energy, so ½mv² alone understates the kinetic energy); the highest point the marble reaches after the first hill is compared with the prediction from conservation; the track diagram is labeled with at least three places where energy changes form; and a redesign is proposed, tested and reported that reduces the percentage dissipated, with the two percentages compared.