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PHYSICS FOR ENGINEERS. B-EXAMS 2006-2007.

PHYSICS FOR ENGINEERS. B-EXAMS 2006-2007. B1. B2. B3. B4.  2. B.  1. C. A. 07B1Eng. HOME EXAM B1. PROBLEM 1. (a). (b). (c).

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PHYSICS FOR ENGINEERS. B-EXAMS 2006-2007.

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  1. PHYSICS FOR ENGINEERS. B-EXAMS 2006-2007. B1 B2 B3 B4

  2. 2 B 1 C A 07B1Eng HOME EXAM B1 PROBLEM 1 (a) (b) (c) You have a spring whose natural length is L0 = 10 cm -figure (a)-. When a mass M = 250 g is hung on the spring, its length increases by L = 40 cm –figure (b)-. Finally, the hanging mass oscillates after the spring is stretched A = 10 cm and then released –figure (c)-. Answer the following questions: B1 a) What is the constant of the spring? 0.5 p b) Find the period of the oscillation 0.5 p c) Find the position of the mass 6.98 s after the oscillations start. 1.0 p d) Find the period of the oscillation if you had hung the same mass M from two identical springs like this one disposed in a paralell way. 2.0 p PROBLEM 2 The mechanism depicted in the figure is composed by two rigid rods, each of lenght L. Rod AB girates around toggle A, whereas rod BC has a joint in point B and its end C slides on the floor. You know the lenght L, the angular velocity of rod AB, AB, and the angles 1 and 2. Find: a) The velocity of point B, vB, and the angle formed by vB with the horizontal. 2.0 p b) The angular velocity of rod BC, BC, and the velocity of point C, vC. 4.0 p

  3. 07B1_Eng HOME EXAM B1 PROBLEM 1 (a) (b) (c) You have a spring whose natural length is L0 = 10 cm -figure (a)-. When a mass M = 250 g is hung on the spring, its length increases by L = 40 cm –figure (b)-. Finally, the hanging mass oscillates after the spring is stretched A = 10 cm and then released –figure (c)-. Answer the following questions: B1 a) What is the constant of the spring? 0.5 p b) Find the period of the oscillation 0.5 p c) Find the position of the mass 6.98 s after the oscillations start. 1.0 p d) Find the period of the oscillation if you had hung the same mass M from two identical springs like this one disposed in a paralell way. 2.0 p a) From Hooke’s law: From Newton’s 2nd law: c) b) We choose t = 0 when y = A That implies  = 0

  4. where PROBLEM 1 (SOLUTION CONTINUED) From Newton’s 2nd law: B1 (The springs are identical) d) From Hooke’s law: Equation of the oscillation driven by two identical springs: Let us write the equation as The solution of this equation is The period is The set of two identical springs disposed in a parallel way (each constant = k) behaves as a single spring of constant 2 k.

  5. 2 1 PROBLEM 2 The mechanism depicted in the figure is composed by two rigid rods, each of lenght L. Rod AB girates around toggle A, whereas rod BC has a joint in point B and its end C slides on the floor. You know the lenght L, the angular velocity of rod AB, AB, and the angles 1 and 2. Find: B1 B a) The velocity of point B, vB, and the angle formed by vB with the horizontal. 2.0 p b) The angular velocity of rod BC, BC, and the velocity of point C, vC. 4.0 p B C A

  6. See that B 1 A PROBLEM 2 (solution continued) a) Find: The velocity of point B, vB, and the angle formed by vB with the horizontal. B1 From dot product definition: Rod AB

  7. 2 Vector B Let’s call 1 C A b) Find: The angular velocity of rod BC, BC, and the velocity of point C, vC. B1 We know: This vectorial equation can be splitted into two equations, one for each component: Although we don’t know their values, we can write for angular velocity and velocity of point C: BC and vC are the quantities to find. C end slides on the floor

  8. HOME EXAM B2 (2007) PROBLEM 1 MODEL A A rigid thin rod of lenght L = 1.80 m, mass M = 6 kg is articulated on a toggle (point O in the figure). The rod is kept tilted by an steel towline attached to the wall. The angles between the towline, the rod and the wall are 1 = 60º and 2 = 50º respectively. A counterweight m = 4 kg hangs from the other end of the rod. B2 a) Draw the free body diagram for the rod (2 p). b) Find the tension on the towline and the rectangular components of the reaction in the point O (2 p). PROBLEM 2 The picture shows a disc of radious 3R, with four circular holes, each of radious R, lying on the indicated positions. The surface density of the disc is  (kg·m-2). The disc moves on the floor. Answer the following questions for the numerical values given below: a) Calculate the Izz moment of inertia of the disc, where Z is the perpendicular axis which passes through its simmetry center (not shown in the figure). (2 p) b) The initial angular velocity of the disc when it takes contact with the floor is 0 (clockwise direction), meanwhile the initial linear velocity of its center of mass is zero. The dynamics friction coefficient is . Make a FBD taking into account the external forces acting on the disc, and find the time it takes the disc to roll without slipping, the linear velocity of its center of mass and the angular velocity of the solid at the moment in which rolling without slipping begins. (4 p) Find the numerical results for the above questions, being R = 14,7 cm,  = 50 kg·m-2, 0 = 0,60 rad/s,  = 0.15

  9. HOME EXAM B2 (2007) PROBLEM 1 Solution MODEL A B2 A rigid thin rod of lenght L = 1.80 m, mass M = 6 kg is articulated on a toggle (point O in the figure). The rod is kept tilted by an steel towline attached to the wall. The angles between the towline, the rod and the wall are 1 = 60º and 2 = 50º respectively. A counterweight m = 4 kg hangs from the other end of the rod. a) Draw the free body diagram for the rod (2 p). b) Find the tension on the towline and the rectangular components of the reaction in the point O (2 p). FBD

  10. Z axis 3R R 2R 2R 2R Z axis normal to the plane, not shown PROBLEM 2 Solution (continued) B2 Moment of inertia of a disc ( surface density, a radious) About to a normal axis passing through its simmetry center (Izz) In our problem, we have two different types of disc: 1. A solid disc of radious a = 3R and surface density  = . All of them are simmetrically disposed around, being 2R the distance between each center and the center of the body. Momentum of inertia of every hole about to the Z axis passing through its simmetry center We have now to calculate the momentum of inertia of every hole about to the Z axis passing through the center of the body. We apply Steiner theorem:

  11. Y X C Z Initial situation t > 0, but before rolling without slipping PROBLEM 2 Solution (continued) Initial picture of the problem B2 This means that initially point C moves towards the left, therefore the friction force goes to the right. CM 2nd Newton’s law: Rotation of the disk: The moment of inertia ICM is the same IZZ previously found, because our disc is a flat figure, so the angular acceleration can be written as: Angular velocity decreases at the same time that velocity of the center of mass increases Where the mass is given by and the moment of inertia is Anticlockwise

  12. R = 14,7 cm,  = 50 kg·m-2, 0 = 0,60 rad/s,  = 0.15 IZZ = 1.65kg·m2 = 6.67 rad·s-2 tf= 0.060 s vCM(tf) = 8.82·10-2 m/s (tf) = 0.20 rad/s t > 0, but before rolling without slipping PROBLEM 2 Resolution Translation & Rotation equations B2 Rolling condition When rolling without slipping begins, we have

  13. 07B3_Eng HOME EXAM B3 B3 PROBLEM 1 (4 p) A hollow sphere (inner radius R1 = 10 cm, outer radius R2 = 20 cm) is made of material of density 0 = 0.80 g·cm-3 and is floating in a liquid of density L = 1.60 g·cm-3. The interior is now filled with material of density m so that the sphere just floats completely submerged. (a) Find the volume fraction of the floating hollow sphere under the liquid surface level before filling its interior. (b) Find density m. PROBLEM 2 (6 p) A cylindrical-shaped tank is used in a farm as water reservoir. The area of the base of the tank is 10 m2. A faucet on the top lets go in 0.60 kg of water per second, meanwhile the outcoming flux across the sink lying on the bottom pours 0,50y kg/s outside, where y means the height of the liquid above the flat bottom of the tank. The tank has also a spillway 1 m above the bottom. Assuming that at the beginning of the day the tank contains 100 liters, then we open at the same time the incoming faucet and the sink, find: a) How long does the surface of the water take to reach the spillway (in case it reaches). b) In case it would not have any spillway, find the maximum height the surface of the water could reach. c) Plot the height of the surface of the water versus time, and mark in your plot the values you obtained as results for the paragraphs a) and b).

  14. PROBLEM 1 SOLUTION A hollow sphere (inner radius R1, outer radius R2) is made of material of density 0 and is floating in a liquid of density L. The interior is now filled with material of density m so that the sphere just floats completely submerged. (a) Find the volume fraction of the floating hollow sphere under the liquid surface level before filling its interior. (b) Find density m. B3 (a) Volume and mass of the hollow sphere: From Archimedes’ Principle, the sphere is buoyed up by a force E equal to the weight of the displaced fluid. As the sphere floats, E should be equal to its weight M0g. VL is the volume of the sphere’s submerged portion Setting E equal to the weight, we find VL Submerged fraction

  15. Numerical solutions PROBLEM 1 SOLUTION (b) When the interior of the sphere is filled with material of density m, the sphere just floats completely submerged. B3 Now the sphere is buoyed up by a force E’ equal to the weight of the displaced fluid, which coincides with the weight of an amount of fluid whose volume is the same that the sphere’s volume. The mass of the material filling the hollow is Setting equal both expressions for M’ we have

  16. Spillway 0.60 0.50 PROBLEM 2 SOLUTION A cylindrical-shaped tank is used in a farm as water reservoir. The area of the base of the tank is 10 m2. A faucet on the top lets go in 0.60 kg of water per second, meanwhile the outcoming flux across the sink lying on the bottom pours 0,50y kg/s outside, where y means the height of the liquid above the flat bottom of the tank. The tank has also a spillway 1 m above the bottom. Assuming that at the beginning of the day the tank contains 100 liters, then we open at the same time the incoming faucet and the sink, find: B3 a) How long it takes the surface of the water to reach the spillway (in case it reaches). b) In case it would not have any spillway, find the maximum height the surface of the water could reach. c) Plot the height of the surface of the water versus time, and mark in your plot the values you obtained as results for the paragraphs a) and b). Continuity equation:

  17. Spillway PROBLEM 2 SOLUTION B3 a) Find how long takes the surface of the water to reach the spillway (in case it reaches). b) In case it would not have any spillway, find the maximum height the surface of the watercould reach. Numerical values

  18. Maximum height in case there weren’t any spillway y (m) y = h = 1 m The level of the water reaches the spillway t (s) PROBLEM 2 SOLUTION c) Plotting: level of water versus time B3 1.20 m t = 35668 s

  19. 07B4 HOME EXAM B3 Only spanish version available B4 Un gas ideal de coeficiente adiabático  = 1.4 ejecuta un ciclo de potencia formado por las siguientes etapas: El gas se expande politrópicamente (índice de politropía k1 = 1.35) desde las condiciones V1 = 1 litro, P1 = 7.87 bar, hasta que su volumen se duplica. 1→2. 2→3. El gas se enfría a volumen constante, hasta que su temperatura es T3 = 280 K. El gas se comprime politrópicamente hasta restituir las condiciones iniciales (sea k2 el índice de politropía de este proceso, que deberá determinarse). 3→1. Se supone que todas las etapas son reversibles. La masa de gas es n = 0.20 moles, y la constante universal de los gases es R = 8,314 J/(K·mol). Se pide: A) Calcular las coordenadas de presión y temperatura en todos los puntos notables del ciclo (2 p). B) Determinar el índice k2 y representar gráficamente el ciclo en un diagrama de Clapeyron (P-V) (2 p). C) Calcular el trabajo asociado con cada una de las etapas del ciclo, discutiendo su signo (2 p). D) Calcular el calor asociado con cada una de las etapas del ciclo, discutiendo su signo (2 p). E) Determinar el rendimiento del ciclo (1 p). F) Calcular la variación de entropía de cada una de las etapas del ciclo (2 p).

  20. EXAMEN DE CASA B4 (2007) 1 2 3 B4 - + + 1. 2. - + - Rendimiento Incremento de entropía (isocórica) Incremento de entropía (politrópicas) donde

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