Gravitation and Pressure: Laws, Formulas and Facts

The correct answer is B, Pascal. Pressure is thrust per unit area, so its unit is one newton per square metre, and this unit is named the pascal after Blaise Pascal. Atmospheric pressure at sea level is about one hundred and one thousand pascals, and the commercial unit bar equals one hundred thousand pascals.

Option A, the newton, is the SI unit of force itself, not of force spread over an area, and it is the numerator in the definition of pressure. Option C, the joule, is the unit of work and energy and equals a newton metre, which is a force multiplied by a distance rather than divided by an area. Option D, the watt, is the unit of power and equals one joule per second. Learn the four together because papers frequently ask which unit goes with which quantity, and pascal, newton, joule and watt appear in the same option set.

The correct answer is B, the poles. The earth is flattened at the poles, so the distance from the centre is least there, and since g varies inversely as the square of that distance, g is largest at the poles. The rotation of the earth also reduces the effective value of g, and that reduction is greatest at the equator and zero at the poles, which adds to the difference.

Option A, the equator, is where g has its smallest surface value, for exactly the opposite reasons. Option C, the centre of the earth, is where g becomes zero, since the mass of the earth then surrounds the body on all sides and its pulls cancel out; a body there has weight zero although its mass is unchanged. Option D, the top of Mount Everest, is far above sea level, and g decreases with height, so it is smaller there than at sea level. The order worth remembering is poles greatest, equator smaller, height smaller still, centre zero.

The correct answer is C, 11.2 kilometres per second. Escape velocity is the minimum speed a body must be given so that it leaves the gravitational pull of the planet and never returns. It depends only on the mass and radius of the planet and not on the mass of the body projected or on the direction of projection, and for the earth it works out at about 11.2 kilometres per second.

Option A, about 7.9 kilometres per second, is the orbital velocity for a satellite close to the earth's surface, and it is smaller than the escape velocity by a factor of the square root of two. Option B, 9.8, is the numerical value of the acceleration due to gravity in metres per second squared and has nothing to do with escape velocity except as a familiar number. Option D is far too large for the earth. The moon's escape velocity is only about 2.4 kilometres per second, which is why it cannot retain an atmosphere.

The correct answer is B, Pascal's law. Pascal's law states that pressure applied to an enclosed fluid is transmitted undiminished in every direction to all parts of the fluid and to the walls of the container. In a hydraulic brake a small force on the narrow master piston creates a pressure that reaches the wider pistons at the wheels, where the same pressure over a larger area gives a much larger force, and all wheels are braked equally at the same moment.

Option A, Archimedes' principle, deals with the upward buoyant force on a body immersed in a fluid and explains ships, submarines and the lactometer. Option C, Bernoulli's principle, relates the speed of a flowing fluid to its pressure and explains the lift of an aerofoil and the working of an atomiser. Option D, Newton's law of gravitation, concerns the attraction between masses. The hydraulic lift and the hydraulic press are the other two standard applications of Pascal's law.

The correct answer is B, Archimedes' principle. It states that a body immersed wholly or partly in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced, so the apparent loss in weight of the body equals the weight of the displaced fluid. A body floats when the weight of fluid displaced equals its own weight, which is the law of flotation, and the same idea explains the ship, the submarine, the hydrometer and the lactometer.

Option A, Pascal's law, deals with the transmission of pressure through an enclosed fluid and explains hydraulic machines. Option C, Hooke's law, is about elasticity and states that within the elastic limit stress is proportional to strain. Option D, Boyle's law, is a gas law stating that at constant temperature the pressure of a fixed mass of gas is inversely proportional to its volume. The two fluid laws, Pascal and Archimedes, are the ones most often swapped in objective papers.

The correct answer is C, one-sixth. Weight equals mass times the acceleration due to gravity, and the acceleration due to gravity on the moon is about one-sixth of its value on the earth because the moon has a much smaller mass and a smaller radius. A body of sixty kilograms therefore still has a mass of sixty kilograms on the moon but weighs only about a sixth of what it weighs on the earth, which is why astronauts can take long bounding steps there.

Options A and B, one-half and one-fourth, are invented fractions with no basis in the moon's mass and radius. Option D, the same, would be true of the mass of the body but never of its weight, and this is exactly the confusion the question is designed to catch. Remember the one-line rule: mass does not change with place, weight does, and on the moon the change is a factor of about one-sixth.

The correct answer is A, atmospheric pressure. A mercury barometer balances the pressure of the atmosphere against a column of mercury, which stands about 76 centimetres high at sea level. Mercury is chosen because it is very dense, does not wet glass and has a low vapour pressure; a water barometer would need a tube more than ten metres tall. A sudden fall in the barometer reading warns of a storm.

Option B, humidity, is measured by a hygrometer, the wet and dry bulb type being the common one. Option C, wind speed, is measured by an anemometer, usually with rotating cups. Option D, ocean depth, is found with an echo sounder or fathometer, which times a sound pulse reflected from the sea bed. Note also that the aneroid barometer, which uses no liquid, becomes an altimeter when its scale is marked in metres of height.

The correct answer is B, Henry Cavendish. In 1798 Cavendish used a delicate torsion balance to measure the extremely small attraction between lead spheres in a laboratory and from it obtained a value of G, about 6.674 times ten to the power minus eleven newton metre squared per kilogram squared. Since G appears in the formula for g, the same experiment also gave the mass and mean density of the earth, which is why it is popularly called weighing the earth.

Option A, Isaac Newton, stated the law of universal gravitation in the seventeenth century but never measured G, which is why the constant remained unknown in his lifetime. Option C, Galileo Galilei, showed that bodies of different masses fall with the same acceleration and studied projectile motion, but did not work on G. Option D, Johannes Kepler, gave the three laws of planetary motion that Newton later explained using gravitation. Keep the pair Cavendish and G firmly apart from Newton and the law itself.

The correct answer is C, the shape and area of the base of the vessel. The pressure at a point inside a liquid equals the depth multiplied by the density of the liquid and by the acceleration due to gravity. Nothing in that expression refers to the shape of the container, so a narrow tube and a wide tank filled to the same level show the same pressure at the same depth, a result demonstrated by the classic hydrostatic paradox.

Option A is wrong as an answer because pressure does rise with depth, which is why a dam is built thicker at the bottom and why a diver feels increasing pressure on descending. Option B is wrong because a denser liquid such as mercury gives a far greater pressure than water at the same depth. Option D is wrong because the term g appears directly in the formula, so the same column of liquid would exert less pressure on the moon. The question asks for the factor that does not matter, so read the wording carefully.

The correct answer is B, the spacecraft and the astronaut are both in free fall around the earth. In orbit the gravitational pull of the earth is the very force that keeps the craft curving around the planet, and the craft, the astronaut and everything inside accelerate towards the earth at the same rate. Since the floor no longer pushes up on the astronaut, nothing presses on anything else and the sensation of weight disappears, although true weight is not zero.

Option A is a common misconception: gravity at the height of a low orbit is only slightly less than at the surface, certainly not absent. Option C is wrong because mass is the quantity of matter in a body and does not change with position; it is the apparent weight that vanishes. Option D is wrong because leaving the atmosphere has no bearing on gravitational force, which acts through empty space. The phrase to remember is that weightlessness is a state of free fall, not a state without gravity.