These questions deal with fluids, thermal physics, ideal gases, and thermodynamics.
The numerical answers to these questions are given in square brackets, like [ 17.8 m/s ].
1. A solid ball with a mass m = 0.53 kg floats in a tank of water. The ball is made of material with a density of 400 kg/m3. The density of water is 1000 kg/m3.
(a) What fraction of the volume of the ball is below the waterline? [ 0.4 ]
(b) Now the ball is tied to the bottom of the tank by a thin string so that 90% of the ball is submerged. What is the tension in the string? [ 6.49 N ]
2. A slab of wood with mass m = 1.7 kg floats 78% submerged.The density of water is 1000 kg/m3.
(a) What is the density of the wood? [ 780 kg/m3 ]
(b) An iron weight placed upon the slab makes the slab float 100% submerged. What is the mass of the iron weight? [ 0.48 kg ]
3. A container of height h = 1.3 m is completely full of water and is resting on the floor. The density of water is 1000 kg/m3.
(a) What is the gauge pressure at the bottom of the container? [ 1.27 x 104 Pa ]
(b) A horizontal jet of water sprays from a hole 1/3 of the way down the side and hits the floor a distance x from the container. Calculate x. [ 1.23 m ]
4. A container is made with two chambers, each with volume of 0.004 m3. The container is insulated so that no energy can be exchanged between the contents of the container and the outside world. There is an opening in the wall between the two sides. Initially this opening is plugged. One side is at vacuum; the other side is filled with 2.4 moles of a certain monatomic ideal gas at 127 °C.
(a) What is the pressure of the gas? [ 2.0 x 106 Pa]
(b) Find the total translational kinetic energy of the gas. [ 1.20 x 104 J ]
The opening between the two chambers is unplugged and the gas expands to fill both sides.
(c) After the gas has reached equilibrium, what is the final temperature of the gas?
(d) What is the final pressure of the gas? [ T = 400 K; P = 1.0 x 106 Pa]
5. A gas container consists of an open-topped vertical cylinder with a piston that can move up and down without friction and which makes a good seal to the cylinder walls. The area of the piston is 0.01 m2 and its mass is 18 kg. The piston and cylinder are well insulated so that no energy is exchanged between the gas and the rest of the world. The cylinder contains 0.13 moles of argon gas at a temperature of 20 °C. The pressure outside the cylinder is one atmosphere.
(a) What is the pressure in the argon gas? [118,900 Pa]
(b) A heater is turned on for a while and the temperature of the argon is increased to 50 °C. How far does the piston rise as a result of this? [0.027 m]
6. One mole of argon gas is contained in a 1.0 x 10-3 m3 container at a pressure of 24.0 atmospheres.
(a) What is its temperature? [293 K]
(b) Heat is applied to the container and the internal energy of the gas increases by 18,000 J. What is the final pressure of the gas? [1.44 x 107 Pa, or 142 atm.]
Note - this question was set up on the assumption that the ideal gas law would be applied to solve it, but at very high pressures the ideal gas law breaks down. In other words, this isn't a particularly well designed question! It would be hard to justify question 4 on these grounds also, seeing as the pressure starts off at about 20 times atmospheric pressure.
7. A cube of aluminum, m = 0.37 kg, T = 62 °C, is dropped in a jar containing m = 0.50 kg of water at T = 14 °C. What is the final temperature of the aluminum - water mixture? The specific heat capacity of aluminum is 900 J / (kg °C) and the specific heat capacity of water is 4186 J / (kg °C). [ 20.6 °C ]
8. A heat engine operates between a cold temperature of 280 K and a hot temperature of 320 K. In one cycle, QH = 45,300 J was removed from the higher temperature reservoir, and work in the amount W = 3,000 J was done.
(a) How much heat QL was dumped into the low temperature reservoir? [42,300 J]
(b) What is the maximum amount of work W (according to the Second Law of Thermodynamics) that the engine could have done in that cycle? [5700 J]
9. A beaker of water at 100 °C sits on an aluminum cylinder of cross-sectional area A = 0.072 m2 and height of 0.100 m. The cylinder in turn sits on a hot-plate maintained at a temperature of 175 °C. The thermal conductivity of aluminum is k = 240 J / (s m °C).
(a) How much heat is transferred from the hot plate to the water in 51 seconds?
(b) What is the mass of water boiled away in 51 seconds? The latent heat of vaporization is Lv = 22.6 x 105 J /k g. [Q = 661,000 J; m = 0.292 kg]