CH 10: PRESSURE, GRAVITY AND MOMENTS Exercise 10.1: Page 104 1. Convert each of the following to kg: (i) 200 g (ii) 4 g (iii) 2 x 10 5 g (iv) 24 mg 2. Convert each of the following to m 3 : (i) 1 cm 3 (ii)120 cm 3 (iii)4 litres (iv) 2 x 10 6 cm 3 3. Convert each of the following to m 2 : (i) 1 cm 2 (ii) 220 cm 2 (iii) 4 mm 2 (iv) 3x 10 4 cm 2 4. Find the density of a piece of wood of mass 4 kg and volume 0.012 m 3. 5. A piece of metal has a volume of 1.61 m 3 and a mass of 1.8 x 10 4 kg. Find its density. 6. The density of silver is 1.05 x 10 4 kg m -3. What is the volume of 4 kg of silver? What is the mass of 3 cm 3 of silver? 7. What is the mass of 1cm 3 of Mercury? The density of Mercury is 1.36 x 10 4 kg m -3. 8. A sheet of steel has a layer of tin electroplated onto one side of it. The thickness of the layer is 2 x 10-6 m and 20 grams of tin is used in all. What is the area of one side of the sheet of steel? (Density of tin = 7.3 x 10 3 kg m -3 ) Exercise 10.2, Page 105 1. A force of 100 N acts uniformly over a surface of area 5 m 2. Calculate the pressure at any point on the surface. 2. A force of 60 N acts uniformly over an area of 25 cm 2. What is the pressure at any point on that area? 3. Calculate the pressure on the table caused by the block shown in Fig 10.2 if: (i) Side A (ii) Side B and (iii)side C is on the table. The mass of the block is 4 kg and acceleration due to gravity is 9.8 m s -2. 4. The pressure on a metal sheet is 400 Pa. If the area of the sheet is 0.06 m 2, calculate the force on the sheet. 5. The pressure on the cover of a book due to the Earth s atmosphere is 1 x 105 Pa. If the area of the cover of the book is 621 cm 2, calculate the force on the cover of the book due to the atmosphere. 6. The pressure at any point on one side of a disc of radius 10 cm is 500 Pa. What is the total force on this side of the disc? 7. A tank of oil, in the shape of a cube of side 2.2 m, sits on a horizontal surface. The pressure at the bottom of the tank due to the oil is 556 Pa. What is the weight of the oil? What is the density of the oil? 8. A cylindrical metal block of mass 32 kg stands with its base on a horizontal table. The radius of the base of the cylinder is 4 cm. Calculate the value of the pressure on the table.
Exercise 10.3, Page 100 (g = 9.8 m s -2 ) 1. A tank contains 100 kg of water. If the area of the base of the tank is 0.5 m2, calculate the pressure of the water at the bottom of the tank. 2. Find the pressure, due to the water, at a depth of 33 m in water. (Density of water = 1000 kg m-3) 3. Find the pressure at a depth of 20 cm in a graduated cylinder full of (i) water (ii) Mercury. Assume the pressure at the top of each liquid is zero. (Density of water = 10 3 kg m -3, Density of Mercury = 13.6 x 10 3 kg m -3 ) 4. HL: A rectangular block has dimensions 10 cm x 20 cm x 30 cm. It is placed with its upper surface at a depth of 1 m in water as in Fig 10.7. Find: (i) The pressure at the upper surface of the block to the water (ii) The pressure at the lower surface of the block due to the water (iii) The force on the upper surface due to the water (iv) The force on the lower surface due to the water If the weight of the block is 70 N determine whether the block will float or sink. (Density of water = 1000 kg m -3 )
Exercise 10.4, Page 111 1. A fixed mass of Oxygen has a volume of 3 m 3 at a pressure of 1 x 10 5 Pa. What is its volume if the pressure is increased to 3 x 10 5 Pa, the temperature remaining the same? 2. A certain mass of methane has a volume of 40 cm 3 at a pressure of 1 x 10 5 Pa. At what pressure would its volume be: (i) 106 cm 3 (ii) 80 cm 3 (iii) 1 cm 3 Assume the temperature remains the same. 3. Fig 10.14 (A) shows some air trapped in a bicycle pump. No air escapes and the temperature remains constant when it is compressed (Fig 10.14 (B)). Describe what changes, if any, take place in: (i) The mass of the air (ii) The volume of the air (iii) The pressure of the air (iv) The density of the air 4. The volume of a certain mass of gas is 700 cm 3 at a pressure of 1 x 10 5 Pa. Find its volume if the pressure changes to: (i) 2 x 10 5 Pa (ii) 7 x 10 5 Pa (iii) 5 x 10 4 Pa With the temperature remaining constant throughout. 5. The value of pressure x volume (i.e. pv) of a fixed mass of gas is 20 Pa m 3 at a temperature of 20 o C. If the pressure is trebled and the temperature remains the same, find the new value of pv for the mass of gas. Exercise 10.5, Page 115 Take: G = 6.7 x 10-11 N m 2 kg -2, g = 9.8 m s -2, Radius of the Earth = 6.4 x 10 6 m, Mass of Earth = 6 x 10 24 kg, Mass of Moon = 7 x 10 22 kg, Radius of Moon = 1.7 x 10 6 m 1. Find the gravitational attraction between two steel spheres each of mass 1 kg, when the distance between their centres is 1 m. 2. Find the gravitational attraction between a woman of mas 76 kg and the Earth when she is standing on the surface of the Earth. 3. Find the gravitational force of attraction between a sphere of mass 90 kg and another sphere of mass 1000 kg when the distance between their centres is 2 m. 4. Find the gravitational attraction between a man of mass 76 kg and the Moon when he is standing on the surface of the Moon. 5. Find the gravitational attraction between the Earth and the Moon. (Distance from the centre of Earth to centre of Moon = 3.8 X 10 8 m) 6. Find the gravitational attraction between the Earth and the Sun given that the distance between their centres is 1.5 x 10 11 m. Compare with the answer to Q5. (Mass of Sun = 1.9 x 10 30 kg)
Exercise 10.6, Page 117 Take: G = 6.7 x 10-11 N m 2 kg -2, g = 9.8 m s -2, Radius of the Earth = 6.4 x 10 6 m, Mass of Earth = 6 x 10 24 kg) 1. Find the value of acceleration due to gravity on Jupiter, given that the radius of Jupiter is 7 x 10 7 m and the mass of Jupiter is 1.9 x 10 27 kg. Hence find the weight of a 90 kg man on Jupiter. 2. Find the value of acceleration due to gravity on the Moon, given that the radius of the Moon is 1.7 x 10 6 m and the ass of the Moon is 7 x 10 22 kg. Hence find the weight of a 60 kg woman on the Moon. 3. Calculate the value of acceleration due to gravity on the surface of the Sun, given that mass of the Sun = 1.9 x 10 30 kg and the radius of the Sun = 7 x 10 8 m). 4. HL: Prove the formula g = GM/R 2 5. HL: Prove that acceleration due to gravity g d at a distance d from the centre of a planet of mass M and radius R is given by g d = GM/d 2. Hence derive a formula for the acceleration due to gravity at height h above the surface of a planet of radius R in terms of G, R, M and h. 6. HL: Find acceleration due to gravity 100 km above the Earth s surface. 7. HL: At what height above the surface of the Earth is acceleration due to gravity equal to: (i) Half its value and (ii) One tenth its value on the surface of the Earth? Take g = 9.8 m s -2 8. HL: Find the mass of the Earth given that: G = 6.7 x 10-11 N m 2 kg -2, g = 9.8 m s -2 and the radius of the Earth = 6.4 x 10 6 m 9. HL: Given that the value of g at the surface is 9.8 m s -2, calculate a value for acceleration due to gravity at a height above the Earth equal to twice the radius of the Earth. 10. HL: Is the value of acceleration due to gravity a constant on the surface of the Earth? Explain your answer. 11. HL: The mass of Mercury is 0.04 times the mass of the Earth. The radius of Mercury is 0.37 times the radius of the Earth. If acceleration due to gravity on the surface of the Earth is 9.8 m s-2, find the value of acceleration due to gravity on the surface of Mercury. 12. HL: Describe how the weight of an object changes as it moves from the surface of the Earth to the surface of the Moon. At what point between them is the resultant gravitational force on an object zero? (Distance between Earth and Moon = 3.8 x 10 8 m and Mass of Earth = 81 x mass of Moon.)
Exercise 10.7, Page 119 1. What is the moment of the force shown in Fig 10.31 about: (i) The point A (ii) The point B (iii) The point C 2. Find the sum of the moments of forces shown in Fig 10.32 about the point: (i) a (ii) b (iii) c 3. In each part of Fig 10.33 the metre stick is in equilibrium. Find the value of X in each case. 4. Find the weight of the uniform metre stick shown in Fig. 10.34. It is in equilibrium. 5. If the uniform metre stick in Fig 10.35 is in equilibrium, find the weight of the metre stick. Verify that the sum of the moments of the forces about each of the points A, B, C and is zero. 6. A uniform wooden beam of length 1 m is resting on supports as shown in Fig 10.36. Find the force that each support exerts on the beam.
7. HL: A uniform mast 10 m long and of mass 100 kg is being carried horizontally by a man and a boy. If the man is 3 m from one end of the mast and the boy is at the far end, find the upward force exerted by the boy and the man on the mast. Where should the man support the mast if he is to carry three times the weight of the boy, the boy remaining at one end? 8. HL: A man and a boy hold opposite ends of a pole 8 m long which is horizontal. A mass of 200 kg is tied to the pole. If the man is to support four times as much weight as the boy, where along the pole must the weight be tied? Neglect the weight of the pole. Exercise 10.8, Page 123 1. In Fig 10.43 the man exerts a force of 300 N on the crowbar. What is the force exerted on the stone by the crowbar? 2. Two forces of magnitude 40 N acting in opposite directions act on a steering wheel. If the perpendicular distance between the forces is 0.5 m, find the torque on the wheel. 3. One of the nuts holding a wheel on a car is to be tightened to a torque of 85 N m. A woman uses a wrench of length 40 cm to tighten the nut (Fig 10.44). If she exerts two equal but opposite forces on the wrench find the size of one of the forces. 4. The torque of the couple acting on an electric motor is 600 N m. If the perpendicular distance between the forces is 10 cm, find the magnitude of one of the forces.