Gravitation and Pressure: Laws, Formulas and Facts

Gravitation explains why an apple falls and why the moon does not. Pressure explains why a sharp knife cuts, why a hydraulic jack lifts a car and why a ship of steel floats. Both chapters give the same kind of exam question: a formula, a unit, a value or a device that works on a named principle. This note keeps them together and states each fact plainly.

Newton's law of gravitation

Every body in the universe attracts every other body with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. In symbols the force is G times m one times m two divided by r squared. The constant G is the universal gravitational constant, the same everywhere in the universe, with a value of about 6.674 times ten to the power minus eleven newton metre squared per kilogram squared. Henry Cavendish measured it in 1798 with a torsion balance, an experiment often described as weighing the earth. Because the force falls as the square of the distance, doubling the separation makes the force one-fourth of what it was. Gravitation is the weakest of the four fundamental forces, yet it governs the motion of planets because it is always attractive and never cancels out.

The value of g and how it changes

The acceleration produced in a freely falling body by the earth's attraction is g, equal to GM divided by R squared, about 9.8 metres per second squared at the surface. It does not depend on the mass of the falling body, which is why a stone and a feather fall together in vacuum. The value is not the same everywhere.

Mass is the quantity of matter in a body, is measured in kilograms and never changes with place. Weight is the force with which the earth pulls the body, equals mass times g, is measured in newtons and changes from place to place. A body of sixty kilograms has the same mass on the moon but weighs one-sixth as much there. At the centre of the earth the weight of a body is zero although its mass is unchanged.

Escape velocity and satellites

Escape velocity is the least speed with which a body must be thrown so that it never returns, and for the earth it is about 11.2 kilometres per second. It does not depend on the mass or the direction of projection, only on the mass and radius of the planet. The moon's escape velocity is only about 2.4 kilometres per second, too small to hold gas molecules, which is why the moon has no atmosphere. An astronaut in an orbiting spacecraft feels weightless not because gravity is absent but because the craft and everything in it are in continuous free fall around the earth, so nothing presses on anything else.

Thrust, pressure and liquids

The force acting normally on a surface is called thrust, and the thrust per unit area is pressure. Pressure equals force divided by area, and its SI unit is the pascal, which is one newton per square metre. Because area is in the denominator, the same force gives a large pressure on a small area: a knife is sharpened, a nail is pointed and a needle is thin for this reason, while a camel's broad foot, a wide tyre and the broad base of a dam reduce pressure by spreading the load.

The pressure inside a liquid at a depth h is h times the density times g. It therefore increases with depth and with the density of the liquid, is the same at all points at one depth, and does not depend on the shape of the vessel or on the area of its base. A liquid also presses sideways on the walls, which is why a dam is built thicker at the bottom.

Pascal's law and Archimedes' principle

Pascal's law states that pressure applied to an enclosed fluid is transmitted undiminished in all directions to every part of the fluid and to the walls of the container. A small force on a narrow piston therefore produces a large force on a wide piston, and this is the working principle of the hydraulic lift, the hydraulic press and hydraulic brakes.

Archimedes' principle states that a body immersed wholly or partly in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. The apparent loss of weight of a body in water equals the weight of water displaced. A body floats when the weight of the fluid it displaces equals its own weight, which is the law of flotation; this is why a steel ship floats while a steel nail sinks, since the hollow hull displaces a large volume of water. Relative density is the ratio of the density of a substance to that of water, and the instruments based on flotation include the hydrometer for the relative density of liquids and the lactometer for the purity of milk.

Atmospheric pressure

The weight of the column of air above us exerts atmospheric pressure, about one hundred and one thousand pascals at sea level, which supports a column of mercury 76 centimetres high. Torricelli demonstrated this with the first barometer, and mercury is used in it because it is very dense, does not wet glass and has a low vapour pressure; a water barometer would need a tube over ten metres tall. Atmospheric pressure falls with height, which is why a fountain pen may leak in an aircraft, why nose bleeding can occur at high altitude, and why water boils below one hundred degrees Celsius in the hills so that cooking takes longer. A pressure cooker works the other way, raising the pressure above the liquid so that the boiling point rises and food cooks faster. The aneroid barometer is a barometer without liquid, and when graduated to read height it becomes the altimeter used in aircraft.