Physics 102 Homework solutions, problem set 2 Chapter 13: Exercises: 10: The blocks of metal both sink. This means that they displace their volume of water. Both blocks have the same volume so they displace the same amount of water. The lead is heavier, but that does not matter since it is not floating. 12: The blocks of metal still both sink. Lead is denser than aluminum so that a 10 N block of lead (about 1 kg or 2 pounds) takes up less volume than a 10 N block of aluminum. (This is more obvious if you think about styrofoam and metal. You need a much bigger piece of styrofoam to weight the same as a piece of metal.) Since the lead takes up less volume, it displaces less water than the aluminum. The difference is that in this problem the blocks have the same weight, not the same volume. 18: The stones hurt your feet less in deep water than in shallow water because more of your body is underwater (ie:you are displacing a larger volume of water) so that there is a larger buoyancy force. There are three vertical forces acting on your body: gravity, buoyancy, and the normal force from the stones. Since the net vertical force on you is zero, the normal force from the stones equals gravity minus the buoyancy force. The more buoyancy, the less normal force and the less pain. 28: Deeper. If you put more mass in a ship, it will need to displace more water in order to float, and thus it will ride deeper. (Since you are not changing the size of the ship.) 30: Fall. We did this in class with the `boat' in the fish tank. When we took the iron weight out of the boat, the boat displaced less water so the water level fell. When we put the weight back in the water, the water level rose, but not by as much. The reason for this is that when the iron is in the boat (ie: floating), it displaces its weight of water. When the iron has sunk, it displaces only its volume of water. Since iron is much denser than water, its weight of water is much larger than its volume. (For example, 1 m^3 of iron weighs about 10 tons. 10 tons will displace 10 m^3 of water if it is floating.) Thus, the iron displaces much less water when it sinks than when it floats and thus the water level will drop since less water is dispalced. 38: Ice cubes will float lower in a mixed drink than in water since adding alcohol will decrease the density of the liquid. Thus the ice cube will have to displace more volume in order to displace the same weight of fluid in order to have the same buoyancy force. If the ice cubes are at the bottom of the glass, then the drink is more than 50% alcohol (since the density of the water plus alcohol is less than the density of ice). 40: If you add something to a half-filled bucket, the water it displaces will stay in the bucket and the bucket will get heavier. The reading on the scale will increase. If you add a fish to a full bucket of water, the water it displaces will leave the bucket. Since a fish has the same density of water, the fish will have the same weight as the water it displaces. Therefore the reading of the scale will not change in this case. 42: Stay the same. There are at least two possible reasons (both are OK): a) the fish floats because it has the same density as the water. Changing gravity won't change this. b) The water displaced by the fish has the same weight as the fish. If gravity doubles, then the weight of the fish AND the weight of the displaced water will both double. However, the water displaced by the fish will still have the same weight as the fish. 44: If there is an amount of water floating motionless in the absence of gravity, it will tend to form a sphere. This is because surface tension tries to minimize the surface area of the water. The shape with the minimum surface area for any given volume is a sphere. Problem: 8: a) She weighs (including her belt) W = mg = 60 kg * 10 m/s^2 = 600 N. (Note that using 9.8 m/s^2 for g is more precise and perfectly acceptable.) b) Since she has the density of water (d = 1000 kg/m^3) and we know that m = d*V, we have V = m/d or V = 60 kg / (1000 kg/m^3) = 0.06 m^3 c) There are two ways to do this: 1) the buoyant force equals the weight of the displaced water or F_buoy = m(displaced water) * g = (d*V) * g = 1000 kg/m^3 * 0.06 m^3 * 10 m/s^2 = 600 N In this case the net force is zero because the bouyant force is equal to and opposite the force of gravity. The second way to do this is to realize that she is floating. This implies that she is not accelerating either upward or downward. Therefore, the net force on her is zero. Therefore, since gravity pulls down, it must be exactly counterbalanced by the buoyant force. Therefore F_buoy = 600 N and the net force is zero. --------------------------------------------- Estimation: There are 6*10^9 people in the world. First we need to figure out how much space one person needs if we are all really crowded together. If you draw an area on the floor that is 1 m by 1 m (or 3' by 3'), you can fit between 4 and 10 people in it. (Look at the linoleum floor tile. They are frequently 1 foot square. How many people can you fit in a space that is 3 by 3?) I will assume 6 people per m^2 because it makes the arithmetic easier and I refuse to worry about whether we are dealing with fat Americans, starving Zambians, adults, children, ... The idea here is to get an answer correct to within a factor of 10 and maybe even correct to within a factor of 2. With 6*10^9 people and 6 people/m^2, I have area needed = 6*10^9 people / (6 people/m^2) = 1 * 10^9 m^2 Now I need to convert my answer to km^2 from m^2. I know that there are 1000 m in 1 km. However, if I have a SQUARE km, then it is 1000 m from east to west and 1000 m from north to south, therefore it is 1 km^2 = (1 km)*(1 km) = (1000 m)*(1000 m) = 10^6 m^2 Therefore, I need area needed = 1*10^9 m^2 * (1 km^2)/(10^6 m^2) = 10^3 km^2 This is very close to the area of Virginia Beach! Again: your answer should be in scientific notation with at most one significant figure. If you write that the area is 1.02467*10^3 km^2, you are lying! You are claiming that your answer is MUCH more precise than it really is.