Work, power and conservation of energy

How fast will a load be moving at the bottom of a slope? We could calculate acceleration along each section and the descent time. Without friction, a much shorter argument gives the answer: follow the energy transformation as the load loses height.

A moving cart can compress a spring; a raised load can drive a mechanism as it falls. Energy connects these possibilities. Kinetic energy describes motion; potential energy describes interactions and relative positions. Their sum is mechanical energy. Raising a load or compressing a spring changes the system into a state that can later produce motion. Potential energy is not a special substance inside the load.

Pull a cart first along its motion, then at an angle. With the same force magnitude, only part of the angled pull acts along the motion. For constant force, work is A = Fs cos α, where s is displacement magnitude and α the angle between force and displacement. Work describes energy transfer by a force. Doubling force or displacement at the same angle doubles work; with no displacement work is zero, even if your muscles tire.

A force along the motion does positive work; one opposing it does negative work; a perpendicular force does zero work. Why does the sign matter? The work of the net force equals the kinetic-energy change. Braking reduces kinetic energy, so net work is negative. In uniform circular motion, the inward force turns velocity without changing its magnitude. It is perpendicular to motion at each instant and does no work.

A 1 N force acting along a 1 m displacement does 1 J of work: 1 Дж = 1 Н·м. Dimensions help check a calculation but cannot replace the formula's conditions. If force increases while a spring stretches, its initial value does not describe the whole motion. Add the work over sections or use a justified average force. For constant force, A = Fs cos α uses displacement, rather than any winding path length.

Two motors lift the same load through the same height, one twice as fast. Work is equal, but the rate of energy transfer differs. Average power is N = A/t, measured in watts: 1 Вт = 1 Дж/с. For the same work, halving time doubles power; for the same time, doubling work also doubles power. The formula gives an average over the chosen interval; power can vary within it.

Why does doubling speed increase braking distance so much? The work of a constant accelerating force, together with v² − v₀² = 2as, gives Eₖ = mv²/2 for translational motion. Mass in kilograms and speed in m/s give joules. Doubling mass doubles Eₖ; doubling speed quadruples it. At v = 0 it is zero. Squaring removes the velocity sign: equally fast motion in opposite directions has equal Eₖ.

Slowly raising a load through height h requires work against gravity. For nearly constant g, the potential-energy increase is mgh; with a chosen zero we write Eₚ = mgh. Here h is the vertical coordinate in metres relative to that level and may be negative. Doubling height above the same zero doubles Eₚ. Moving the zero changes energy values but not the difference between two positions or the predicted motion.

Stretching an ideal spring increases force from zero to k|Δl|. Its average over this interval is therefore half the final force, and the work gives Eₚ = k(Δl)²/2 relative to the undeformed state. Compression stores energy too: the square removes the deformation sign. Doubling |Δl| quadruples energy; at Δl = 0 it is zero. This requires Hooke's law and cannot describe arbitrarily large stretching.

As a load descends, gravity does positive work: kinetic energy increases by the amount potential energy decreases. If potential energy includes gravity and elasticity and all other forces together do zero work, Eₖ + Eₚ = const. This is conservation of mechanical energy in the chosen model. It compares states without requiring the travel time. First check friction, moving supports and possible external work.

Imagine feeling a bicycle brake after stopping: it has warmed up. Has the energy of motion vanished? It became internal energy of brakes, wheels and surroundings. Such friction usually reduces mechanical energy, but total energy of the system including its surroundings is conserved. We cannot simply equate initial and final Eₖ + Eₚ without accounting for energy transfer. 'Loss' means leaving the chosen mechanical form.

A load starts sliding from rest at a height of 1.8 m. There is no friction and g = 10 м/с². Find its speed at the zero level.

Initial speed is zero, so energy relative to the bottom level is mgh. There is no friction; the normal force from the stationary smooth surface is perpendicular to motion and does no work.

At the bottom Eₚ = 0, and the lost potential energy has become kinetic: mgh = mv²/2. Mass cancels, giving v = √(2gh).

v = √(2 · 10 · 1,8) = √36 = 6 м/с.

The result depends on the height drop, rather than travel time or the shape of the smooth slope, provided the load stays on it. Friction converting some energy into heat would reduce the final speed.

Speed at the zero level is 6 m/s: in this model the 1.8 m height drop converts all the corresponding potential energy into energy of motion.

Work accounts for force direction and energy change; even a large force may do zero work.

An energy calculation compares states and often avoids finding the travel time.

Friction does not destroy energy, but a chosen system can have less mechanical energy.

Periodic energy exchange will explain pendulum motion.