Chapter 5 — Work, Energy and Power
Work, kinetic and potential energy, power and collisions — a way of solving mechanics problems by bookkeeping instead of by tracking forces moment by moment.
A child at the top of a water slide, a cricket ball leaving the bowler's hand, a lift climbing twenty floors, two railway wagons clanging together in a shunting yard. Newton's laws (Chapter 4) can describe every one of them — but only if you know every force at every instant, and the forces in a slide that twists, a bat that flexes or a coupling buffer that crushes are anything but simple. Is there a way to predict the outcome without following the motion step by step?
There is, and it is one of the most powerful ideas in all of science. When a force pushes a body through a distance it does work, and that work changes the body's kinetic energy. Some forces — gravity, springs — store the work done against them as potential energy and give it back later; while only they act, the total mechanical energy never changes. That single bookkeeping rule gives the speed of the child at the bottom of any frictionless slide, whatever its shape. Add the rate at which work is done — power — and the conservation of momentum from Chapter 4, and you can rate motors and engines and predict what happens when bodies collide, from snooker balls to neutrons in a reactor.
- compute and use the scalar product $\mathbf{A}\cdot\mathbf{B}=AB\cos\theta=A_xB_x+A_yB_y+A_zB_z$, including the angle between two vectors;
- find the work done by a constant force, and by a variable force as the area under an $F$–$x$ graph, with the correct sign;
- state, prove and apply the work–energy theorem $W_\text{net}=\Delta K$;
- distinguish conservative and non-conservative forces, use $U=mgh$, $U=\tfrac12kx^2$ and $F=-\dfrac{dU}{dx}$, and solve problems with conservation of mechanical energy (including the vertical circle and energy lost to friction);
- calculate average and instantaneous power, $P=\mathbf{F}\cdot\mathbf{v}$, efficiency and energy in kWh;
- solve elastic and perfectly inelastic collisions in one dimension and simple two-dimensional collisions;
- tackle Board, JEE Main, JEE Advanced and NEET-style problems on work, energy, power and collisions (with potential-energy curves and the coefficient of restitution as exam extensions).
Sections in this chapter
Work, Kinetic Energy and the Work–Energy Theorem
The scalar product, work by constant and variable forces, kinetic energy, and the theorem linking them.
Potential Energy and Conservation of Mechanical Energy
Conservative forces, gravitational and spring potential energy, energy conservation and the energy of a block on a spring.
Power and Collisions
Average and instantaneous power, efficiency, and elastic and inelastic collisions in one and two dimensions.