Physics 8.01X, Solutions to Problem Set 12

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01X Fall Term 2002 S OLUTIONS TO PROBLEM SET 12 1 Young & Friedman 11­24 Stress on a Mountaineer’s Rope A nylon rope used by mountaineers elongates 1.10 m under the weight of a 65.0­kg climber. If the rope is 45.0 m in length and 7.0 mm in diameter, what is Young’s modulus for this material? Y = = F⊥ L0 A ΔL � � (65.0 kg) 9.81 sm2 (45.0 m) π (3.5 × 10−3 m)2 (1.10 m) = 6.8 × 108 Pa 2 Young & Friedman 14­17 An object of average density ρ floats at the surface of a fluid of density ρfluid . (a) How must the two densities be related? If ρ < ρfluid and the object is initially entirely submerged, then the buyant force will exceed the gravitational force on the object. It will rise to the surface and float there. (b) In view of the answer to part (a), how can steel ships float in water? If a piece of steel is hollow, its overall density will be less than that of solid steel. Since the average density of an object determines buoyancy, rather than total weight or the density of some specific part of it, a piece of steel can be hollowed out until its average density is less than that of water and it will float. (c) In terms of ρ and ρfluid , what fraction of the object is submerged and what fraction is above the fluid? Check that your answers give the correct limiting behavior as ρ → ρfluid and as ρ → 0. When floating at a constant level, the total force on the object is zero. This means that the buoyant force and the gravitational force have the same magnitude. Fb = W ρfluidVsubmerged g = ρV g Vsubmerged ρ = ρfluid V where V is the object’s total volume. When the densities are the same, the object is not forced out of the fluid, Vsubmerged =1 ρ→ρfluid V lim and when the object is much less dense than the fluid, almost all of it will be floating, lim ρ→0 Vsubmerged =0 V (d) While aboard your yacht, your cousin Throckmorton cuts a rectangular piece (dimensions 5.0 × 4.0 × 3.0 cm3 ) out of a life preserver and throws it into the ocean. The piece has a mass of 42 g. As it floats in the ocean, what percentage of its volume is above the surface? g g The rectangular piece has a density of 42 60 cm3 , and water has by definition a density of 1 cm3 . Thus piece is submerged, and 30% is above the surface. 3 42 60 = 70% of the Young & Friedman 14­34 A soft drink (mostly water) flows in a pipe at a beverage plant with a mass flow rate that would fill 220 0.355­L cans per minute. At point 2 in the pipe, the gauge pressure is 152 kPa and the cross­section area is 8.00 cm2 . At point 1, 1.35 m above point 2, the cross­section area is 2.00 cm2 . (a) Find the mass flow rate. Since we are dealing with constant flow, the derivative ρ dV dt dV dt is the same as the discrete ΔV = ρ � Δt � kg (220) (0.355 L) = 1 L (60 s) kg = 1.30 s (b) Find the volume flow rate. dV dt (220) (0.355 L) (60 s) L = 1.30 s = ΔV Δt . (c) Find the flow speeds at points 1 and 2. dV dt = Av v = v1 = dV dt �A � 3 1.30 × 10−3 ms (2.00 × 10−4 m2 ) � v2 = 3 1.30 × 10−3 ms = 6.51 m s = 1.63 m s � (8.00 × 10−4 m2 ) (d) Find the gauge pressure at point 1. Since we have an incompressible beverage, Bernoulli’s equation can be used if we also ignore the small viscosity. 1 1 p1 + ρgy1 + ρv21 = p2 + ρgy2 + ρv22 2 2 � 1 � p1 = p2 + ρg (y2 − y1 ) + ρ v22 − v21 2 � �� � �2 � 1 1 dV 1 − = p2 + ρ gΔy + 2 dt A22 A21 � � � �� �� � � 3 2 15 1 m� 3 kg −3 m 8 1 = (152 kPa) + 10 3 − × 10 4 1.30 × 10 9.81 2 (−1.35 m) + s 64 m m 2 s = 152 kPa − 13.2 kPa − 19.9 kPa = 119 kPa � � Note that (y2 − y1 ) is negative because point 2 is lower than point 1, and v22 − v12 is negative because the pipe is wider at point 2 than at point 1. 4 Young & Friedman 14­56 The Great Molasses Flood On the afternoon of January 15, 1919, an unusually warm day in Boston, a 27.4­m high (90 ft), 27.4­m­diameter cylindrical metal tank used for storing molasses ruptured. Molasses flooded into the streets in a 9­m­deep stream, killing pedestrians and horses, and knocking down buildings. The molasses had a density of 1600 mkg3 . If the tank was full before the accident, what was the force the molasses exerted on its sides? (Hint: Consider the outward force on a circular ring of the tank wall of width dy at a depth y below the surface. Integrate to find the total outward force. Assume that before the tank ruptured, the pressure at the surface of the molasses was equal to the air pressure outside the tank.) At a depth y the molasses pressure was p (y) = p0 +ρgy, where p0 is atmospheric pressure. If the tank had a diameter d, the area of the tank wall at y (a ring of height dy) was dA = (πd) dy. Force being pressure times area, dF = (p0 + ρgy) (πd) dy � h dy (p0 + ρgy) (πd) � ρg � = πd p0 h + h2 2 �� � � � � 1 kg � m� N = π (27.4 m) 1.01 × 105 2 (27.4 m) + 1600 3 9.81 2 (27.4 m)2 m 2 s m F = 0 = 2.38 × 108 N + 5.07 × 108 N = 7.45 × 108 N This is the total force exerted by the molasses on the tank wall. Note that there was also an inward force of πd p0 h = 2 8 2.38× 108 N on the tank wall due to the air outside the tank. The net force on the wall was thus πd ρg 2 h = 5.07 × 10 N outward.