(a) Suppose you find that you can only suck wa- ter up to a height of 1 . 0 m. What is the minimum pressure inside the straw? √ (b) Superman, strong as he is, can suck out all the air from a (strong-walled) straw. What is the tallest straw through which Superman can drink water out of a lake? √ 10-d2 The first transatlantic telegraph cable was built in 1858, lying at a depth of up to 3.2 km. What is the pressure at this depth, in at- mospheres? 10-d3 One way to measure the density of an unknown liquid is by using it as a barometer. Suppose you have a column of length L of the unknown liquid (inside a vacuum tube), which provides the same pressure as atmospheric pres- sure P 0 . (a) What is the density of the unknown liquid? √ (b) Mercury barometers have L = 760 mm at standard atmospheric pressure P 0 = 1 . 013 × 10 5 Pa. Given these data, what is the density of mercury to three significant figures? √ 10-d4 A U-shaped tube, with both ends ex- posed to the atmosphere, has two immiscible liq- uids in it: water, and some unknown liquid. The unknown liquid sits on top of the water on the right side of the tube in a column of height H . Also, the water extends a height h above the unknown-water interface. If the density of water is ρ w , what is the density of the unknown liquid? √ 10-d5 Typically the atmosphere gets colder with increasing altitude. However, sometimes there is an inversion layer , in which this trend is reversed, e.g., because a less dense mass of warm air moves into a certain area, and rises above the
122 CHAPTER 10. FLUIDS Problem 10-d4. denser colder air that was already present. Sup- pose that this causes the pressure P as a func- tion of height y to be given by a function of the form P = P o e - ky (1 + by ), where constant tem- perature would give b = 0 and an inversion layer would give b > 0. (a) Infer the units of the con- stants P o , k , and b . (b) Find the density of the air as a function of y , of the constants, and of the acceleration of gravity g . (c) Check that the units of your answer to part b make sense. . Solution, p. 156 10-d6 Estimate the pressure at the center of the Earth, assuming it is of constant density throughout. The gravitational field g is not con- stant with repsect to depth. It equals Gmr/b 3 for r , the distance from the center, less than b , the earth’s radius. Here m is the mass of the earth, and G is Newton’s universal gravitational constant, which has units of N · m 2 / kg 2 . (a) State your result in terms of G , m , and b . √ (b) Show that your answer from part a has the right units for pressure. (c) Evaluate the result numerically. √ (d) Given that the earth’s atmosphere is on the order of one thousandth the earth’s radius, and that the density of the earth is several thousand times greater than the density of the lower atmo- sphere, check that your result is of a reasonable order of magnitude.