Work and Energy

3.1

Work

PMDC: Describe the concept of work in terms of the product of force F and displacement d in the direction of the force.

Work is done when a force moves something along its own direction. Push a box $2\mathrm{m}$ with $10\mathrm{N}$ along the floor: $W=20$ J. Hold a heavy bag still: displacement is zero, work is zero. Carry the bag at constant height: the lift is vertical, the walk is horizontal, that lift does no work while you walk.

If F and s are at an angle $\theta$, only the part of F along s counts.

• $W=Fs\cos \theta (F\cdot s)$
• $\theta =0^{\circ }\Rightarrow W=Fs(\max )$
• $\theta =90^{\circ }\Rightarrow W=0$
• $\theta$ > $90^{\circ }$ ⇒ W negative (force opposes the motion)

3.2

Energy

PMDC: Describe energy.

Energy is what a system has if it can do work. It comes in forms — kinetic, potential, heat, chemical — and the total of an isolated system stays put. Work is one way energy moves from one store to another.

• SI unit of energy: joule (same as work)

3.3

Kinetic energy

PMDC: Explain kinetic energy.

Kinetic energy is energy of motion. Double the speed, four times the KE — it depends on v². A truck and a bike at the same speed: the truck has more KE because of mass.

• $K=\frac{1}{2}m{v}^2$
• K → $4\mathrm{K}$ if v → 2v

3.4

Potential energy vs gravitational potential energy

PMDC: Explain the difference between potential energy and gravitational potential energy.

Potential energy is stored energy of position or arrangement. A stretched spring, a charged pair, a mass on a shelf — all have potential energy.

Gravitational potential energy is the special case for a mass in a gravity field. Near the ground we write it as m g h above a chosen floor. A spring’s ½ k x² is potential energy too, just not gravitational.

• $U_g$ = m g h (near Earth, uniform g)
• $U_{spring}=\frac{1}{2}k{x}^2$

3.5

Absolute potential energy and the reference level

PMDC: Describe that gravitational potential energy is measured from a reference level and can be positive or negative, to denote the orientation from the reference level.

You pick the zero. Floor, table, ground — any fixed level. Above it, U is positive. Below it, U is negative. Only changes in U matter for work.

The ‘absolute’ gravitational PE from infinity is $U=-G$ M m / r, which is negative because we set zero at infinity. MDCAT mostly wants the m g h picture plus this sign idea.

• $\Delta U$ = m g $\Delta h$ (near Earth)
• $U=-$ G M m / r (zero at infinity)

3.6

Power

PMDC: Express power as the scalar product of force and velocity.

Power is work per second — how fast you spend energy. If a force F moves a body at velocity v, the instant power is F · v. A small force at high speed can still be large power.

• $P=W/t$
• $P=Fv\cos \theta (F\cdot v)$
• $1\mathrm{W}=1\mathrm{J}{s}^{-1}$

3.7

Work–energy in a resistive medium

PMDC: Explain that work done against friction is dissipated as heat in the environment.

Friction opposes sliding. The work you do against friction does not sit in a store you can get back. It becomes thermal energy in the surfaces and the air. That is why a sliding block warms and why machines never give back all the work you put in.

Work–energy with friction: the drop in mechanical energy equals the work against friction.

• $W_{friction}=-f_k$ s (usually)
• $K_i+U_i+W_{other}=K_f+U_f$
• energy lost to friction → heat

3.8

Energy losses and efficiency

PMDC: State the implications of energy losses in practical devices.

Every real machine leaks energy — heat, sound, extra vibration. Efficiency tells you the useful fraction. A 40 % efficient motor turns $40\mathrm{J}$ of each $100\mathrm{J}$ of electrical energy into useful mechanical work. The rest is loss.

Implication: you must supply more energy than the useful job needs. Thicker cables, lubrication, and lower current (Unit 11) are ways to cut the leak. You cannot reach 100 % in a practical device.

• η = (useful output energy / input energy) × 100 %
• η = (useful power / input power) × 100 %
• η < 1 for any real device