Name: Midterm II Solutions March 30, 1999 Correct Responses are given in bold type Useful constants: ρ water =1000 kg/m 3, G=6.67x10-11 Nm 2 /kg 2, N A =6.02x10 23, R=8.31 J/(mol K), k = 1.38x10-23 J/K, D isk = 1/2 MR 2, M e =5.98x10 24 kg, g=9.8 m/s 2. 1. A car with mass 1200 kg going 36 km/hr strikes a parked car with mass 1800 kg. After the collision the cars stick together and move off with the same speed. What is the speed of the two cars after the collision? A) 14 km/hr B) 36 km/hr C) 22 km/hr D) 24 km/hr E) 18 km/hr Conserve momentum: M 1 v 1 =(M 1 +M 2 )*V F ; V F =M 1 v 1 /(M 1 +M 2 ) V F =1200*36/(1200+1800) V F =14.4 km/hr 2. On the Rankine temperature scale, sometimes used in engineering applications, the freezing point of water is 491.67 R and the boiling point 671.67 R. What temperature on the Rankine scale corresponds to 0 K (absolute zero)? A) 491.67 R B) 478.67 R C) 0 R D) 32.67 R E) 273.15 R T = 671.67-491.67 =180; so degrees are same as Fahrenheit; Freezing: 491.67 R*5/9=273.15K. Thus 0 K=0 R 3. In a diesel engine cylinder the piston compresses air at 300 K to a volume onefifteenth (1/15) of its original and to a pressure 50 times the original pressure. What is the resulting temperature? (Use the ideal gas law) A) 90 K B) 450 K C) 1500 K D) 1000 K E) 135 K PV/T=nR; P 1 V 1 /T 1 =P 2 V 2 /T 2 ; or T 2 =T 1 *(P 2 /P 1 )(V 2 /V 1 )=300*(50/1)(1/15)=1000 K p. 1 of 1
4. Two identical containers held at the same temperature contain one mole each of nitrogen (N 2 ) and hydrogen (H 2 ) gases, respectively. Which of the following statements is most nearly true. A) The number of molecules of gas in each container (N 2 or H 2 ) is the same and equal to 2 x N A where N A is Avogadro s number. B) The average speed of the molecules in both containers is the same. C) The pressure in the hydrogen container is greater than that in the nitrogen container. D) The average kinetic energy of a molecule is equal to 1/2 k T, where k is Boltzman s constant and T is the temperature in K. E) The average kinetic energies of the molecules in both containers are the same. 5. The gravitational force acting on the moon to keep it in orbit around the earth is F=1.98x10 20 N. If the mass of the moon is 7.35x10 22 kg and the radius of its orbit (assumed to be circular) is 3.85x10 8 m, then what is the velocity of the moon in its orbit around the earth (in km/s)? A) 1040 km/s B) 61.5 km/s C) 32.2 km/s D) 10.2 km/s E) 1.02 km/s F C =MV 2 /R V = (F C xr/m) 1/2 = (1.98x10 20 x 3.85x10 8 /7.35x10 22 ) 1/2 = 1018 m/s=1.02 km/s 6. A satellite is put into a circular orbit around the sun at a radius R = 4 R e where R e is the radius of the earth s orbit about the sun. The period of the satellite around the sun is: A) 8 years B) 2.5 years C) 4 years D) 16 years E) 2 years Since T 2 R 3 ; T S 2 /T e 2 =R S 3 /R e 3 or T S = T e (R S /R e ) 3/2 = 1 (4) 3/2 = 8 years p. 2 of 2
7. The moment of inertia of a disk of radius R, rotated about an axis at its center and perpendicular to the disk is 1/2 MR 2. If the disk is rotated about an axis its edge perpendicular to its surface, then its moment of inertia is: A) MR 2 B) 3/2 MR 2 C) 2 MR 2 D) 2/3 MR 2 E) 3 MR 2 R From axis theorem: I = I CM + Md 2 = 1/2MR 2 +MR 2 =3/2MR 2 8. A girl sits on a stool spinning without friction holding a pair of weights in her outstretched arms. By bringing the weights in close to her body she increases her angular speed by a factor of two. In the process, her kinetic energy: (Hint: Angular momentum is conserved) A) Increases by a factor of two B) Decreases by a factor of two C) Stays the same D) Increases by a factor of four E) Decreases by a factor of 4 KE = 1/2 I ω 2 = L 2 /(2 I); If L is conserved, I 1 ω 1 = I 2 ω 2 and I 2 /I 1 =ω 1 /ω 2 = 1/2 and KE 2 /KE 1 =I 1 /I 2 =2 9. A 1-kg mass hangs from a string wound around a 1-kg circular disk, which is free to spin without friction around its axis. If the radius of the disk is 1 m what is the acceleration of the hanging mass? A) 4.9 m/s 2 B) 3.3 m/s 2 C) 1.0 m/s 2 D) 0.5 m/s 2 E) 6.5 m/s 2 y x If T is tension in string: Force on mass M: Mg-T = Ma; Torque on disc = Tx R = I α = 1/2 MR 2 a/r; T = Ma/2 and Mg-Ma/2=Ma gives: a = 2/3 g = 6.5 m/s 2 p. 3 of 3
10. Guests at a party are offered a punch which consists of a mixture of ethyl alcohol (density = 806 kg/m 3 ) and water (density = 1000 kg/m 3 ). It is noted that the ice cubes (density = 917 kg/m 3 ) are neutrally buoyant (they have the same density as the punch). What is the percentage by volume of alcohol in the punch? A) 75% B) 43% C) 25% D) 57% E) 33% ρ P = 917 = (ρ W x V W + ρ A xv A )/(V W +V A ) %=100x V A /(V A +V W )=100x(ρ W -ρ P )/( (ρ W -ρ A ) =43% 11. An incompressible fluid is flowing smoothly through the arrangement of pipes shown in a vertical view in the drawing. At which of the points indicated is the pressure the lowest? (Use Bernoulli s Equation) A) A up B) B A C B C) C D) D D E E) E 12. A hypodermic syringe of radius R and length L is used to inject a patient with a fluid with viscosity 1.5x10-3 Pa s. A flow rate of 3.3 x 10-7 m 3 /s is achieved with a pressure difference of 1200 Pa. If the syringe is replaced by one with twice the length and twice the radius, what is the pressure difference needed to maintain the same flow rate? A) 9600 Pa B) 1200 Pa C) 2400 Pa D) 4800 Pa E) 150 Pa Bernoulli: P +1/2 ρv 2 + ρ g h = const. Pick place where h, v are high: D Poiseuille s Law: Flow rate P x R 4 /L; P 1 x R 1 4 /L 1 = P 2 R 2 4 /L 2 ; Thus P 2 = P 1 x (L 2 /L 1 )(R 1 /R 2 ) 4 = 1200 (2/1)(1/2) 4 = 150 Pa p. 4 of 4
13. A 1 m long aluminum rod with Young s Modulus 6.9 x 10 10 N/m 2 and cross sectional area 0.5 cm 2 is used to lift a 20-ton tank. How much does the rod stretch when the tank is picked up? A) 56 cm B) 0.57 cm C) 0.058 cm D) 5.7 cm E) 2.9 cm F/A = Y L/L or L = L/Y x F/A = 1 m/ 6.9 x 10 10 N/m 2 x 20,000kg x 9.81m/s 2 /0.5x10-4 m 2 = 0.057 m = 5.7 cm 14. In a 1-dimensional collision a ball of mass M/2 with velocity +2V collides elastically with a ball of mass M and velocity V. After the collision, which of the statements below is most nearly correct? A) The center of mass of the two balls moves in the +x direction. B) The final momentum is less than the initial C) The final kinetic energy is less than the initial D) The center of mass remains stationary P CM = M/2 x 2V M x V = 0 E) Both balls are at rest M/2 M 2V V 15. A fly clings to the rim of a bicycle wheel with radius R that is turning with uniform angular velocity ω. Choose the most nearly correct statement below. A) The fly experiences no acceleration since its speed is constant. B) The fly experiences an acceleration in a direction toward the center of the wheel with magnitude ω 2 R. C) The fly experiences an acceleration directed toward the center of the wheel with magnitude ω 2 /R. D) The fly is accelerated in a direction tangential to the rim of the wheel with magnitude ω/r. E) The angular momentum of the fly is directed toward the center of the wheel. R ω p. 5 of 5