Work, Energy, Power, Gravitation and Pressure

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Work and power

In physics, work is done only when a force moves a body: W = F x s x cos θ, where θ is the angle between the force and the displacement. Work is a scalar measured in joules (1 J = 1 N x 1 m). It is zero when nothing moves (pushing a fixed wall) or when the force is perpendicular to the motion: a porter carrying a load on his head along level ground does no work against gravity, and the Earth's pull does no work on a satellite in a circular orbit. Work is negative when the force opposes motion, as friction does. Power is the rate of doing work, P = W/t, measured in watts (1 W = 1 J/s). 1 kilowatt = 1000 W and 1 horsepower = 746 W; the kilowatt hour is a unit of energy, not of power.

Energy and its conservation

Energy is the capacity to do work and is also measured in joules. Kinetic energy is 1/2 mv^2, so doubling the speed makes it four times; potential energy due to height is mgh, and a stretched spring or a bent bow stores elastic potential energy. The law of conservation of energy says energy can neither be created nor destroyed, only changed from one form to another; in a freely falling body potential energy turns into kinetic energy while their sum stays constant. A motor turns electrical energy into mechanical energy, a generator mechanical into electrical, a cell chemical into electrical, a solar cell light into electrical and photosynthesis light into chemical energy. Other units: 1 calorie = 4.186 J and 1 kWh = 3.6 x 10^6 J.

Gravitation and g

Newton's law of gravitation says any two bodies attract each other with a force F = G m1 m2 / r^2, proportional to the product of their masses and inversely proportional to the square of the distance between them. G, the universal gravitational constant, is 6.67 x 10^-11 N m^2/kg^2 everywhere; Henry Cavendish first measured it. The acceleration due to gravity, g = GM/R^2, is about 9.8 m/s^2 at the Earth's surface but it varies: it decreases with height and with depth, is zero at the Earth's centre, and is greater at the poles than at the equator because the equatorial radius is larger and the Earth's rotation reduces it at the equator.

Mass, weight and weightlessness

Mass is the quantity of matter in a body, measured in kilograms with a beam balance, and it is the same everywhere. Weight is the force of gravity on the body, W = mg, measured in newtons with a spring balance, so a body weighs more at the poles than at the equator and about one-sixth as much on the Moon. In a lift accelerating upward a person's apparent weight increases; in a freely falling lift it becomes zero. Astronauts in an orbiting spacecraft feel weightless because they and the craft are falling freely around the Earth together, not because gravity is absent.

Escape velocity and satellites

Escape velocity is the least speed at which a body must be projected to escape a planet's gravity; for the Earth it is 11.2 km/s, whatever the mass of the body. On the Moon it is only about 2.4 km/s, so gas molecules escape easily and the Moon has no atmosphere. A satellite orbiting close to the Earth needs about 7.9 km/s, and escape velocity is √2 times this orbital velocity. A geostationary satellite circles about 36,000 km above the equator, from west to east, with a period of 24 hours, so it appears fixed in the sky; such satellites serve communication, television and weather watch.

Pressure and Pascal's law

Pressure is force per unit area, P = F/A, measured in pascal. A sharp knife cuts well because a small area gives a large pressure, while wide bag straps and a camel's broad feet spread the force. Atmospheric pressure at sea level is about 1.013 x 10^5 Pa, equal to a 76 cm column of mercury; Torricelli invented the mercury barometer that measures it. It falls with height, so a fountain pen leaks in an aeroplane and water boils below 100 degrees C in the hills; a sudden fall in the barometer reading warns of a storm. Liquid pressure increases with depth (P = hρg), so dam walls are thicker at the bottom. Pascal's law says pressure applied to an enclosed fluid is transmitted equally in all directions; hydraulic lifts, presses and brakes work on it.

Density, buoyancy and Archimedes

Density is mass per unit volume; water is densest at 4 degrees C, at 1000 kg/m^3. Relative density has no unit and is measured with a hydrometer; a lactometer checks the purity of milk. A fluid pushes up on any body immersed in it, and Archimedes' principle says this upthrust equals the weight of the fluid displaced. A body floats if it is less dense than the fluid, and a floating body displaces its own weight of liquid. An iron ship floats because its hollow shape displaces a large weight of water, and it rises slightly when it sails from a river into the denser sea. Ice floats with about nine-tenths of its volume under water.

Key facts

Work W = Fs cos θ; no work is done when nothing moves or when the force is perpendicular to the motion (porter on level ground, satellite in a circular orbit).
KE = 1/2 mv^2, so doubling the speed makes KE four times; PE = mgh; energy only changes form (law of conservation of energy).
1 horsepower = 746 W; 1 kWh = 3.6 x 10^6 J; 1 calorie = 4.186 J; the kilowatt hour is a unit of energy, the watt a unit of power.
G = 6.67 x 10^-11 N m^2/kg^2 is universal (first measured by Cavendish); g is about 9.8 m/s^2, more at the poles, less at the equator, zero at the Earth's centre.
Mass (kg, beam balance) never changes; weight = mg (newton, spring balance) does: zero in a freely falling lift, about one-sixth on the Moon.
Escape velocity: Earth 11.2 km/s, Moon about 2.4 km/s (hence no lunar atmosphere); orbital velocity near the Earth about 7.9 km/s.
Geostationary satellite: about 36,000 km above the equator, period 24 hours, moves west to east, used for communication and TV.
Atmospheric pressure is about 1.013 x 10^5 Pa = 76 cm of mercury (barometer, Torricelli); a sudden fall in the barometer warns of a storm.
Pascal's law runs hydraulic lifts, presses and brakes; Archimedes: upthrust = weight of fluid displaced; a lactometer tests milk, a hydrometer measures the density of liquids.

Quick revision

W = Fs cos θ (joule); KE = 1/2 mv^2, PE = mgh; energy is conserved; P = W/t (watt); 1 hp = 746 W; kWh is energy; F = Gm1m2/r^2, G = 6.67 x 10^-11 (Cavendish); g = 9.8 m/s^2, more at poles, zero at centre; weight = mg changes, mass does not; escape velocity 11.2 km/s, orbital 7.9 km/s; geostationary 36,000 km, 24 h; pressure = F/A (pascal); Pascal's law: hydraulic brakes; Archimedes: upthrust = weight of fluid displaced.

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