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1 PHYSICS FOR ENGINEERS. B-EXAMS 2006-2007. B1B2B3B4

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2 B1 (a)(b)(c) You have a spring whose natural length is L 0 = 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: What is the constant of the spring? Find the period of the oscillation Find the position of the mass 6.98 s after the oscillations start. 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. a) b) c) d) 0.5 p 1.0 p 2.0 p PROBLEM 1 07B1EngHOME EXAM B1 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: b) The angular velocity of rod BC, BC, and the velocity of point C, v C. 4.0 p a) The velocity of point B, v B, and the angle formed by v B with the horizontal. 2.0 p PROBLEM 2 A C 11 22 B

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3 B1 (a)(b)(c) You have a spring whose natural length is L 0 = 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: What is the constant of the spring? Find the period of the oscillation Find the position of the mass 6.98 s after the oscillations start. 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. a) b) c) d) From Hooke’s law: From Newton’s 2 nd law: a) b) c) 0.5 p 1.0 p 2.0 p We choose t = 0 when y = A That implies = 0 PROBLEM 1 07B1_EngHOME EXAM B1

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4 B1 From Newton’s 2 nd law: d) From Hooke’s law: Equation of the oscillation driven by two identical springs: The solution of this equation is Let us write the equation as where 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. (The springs are identical) PROBLEM 1 (SOLUTION CONTINUED)

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5 B1 A B C 11 22 B 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: b) The angular velocity of rod BC, BC, and the velocity of point C, v C. 4.0 p a) The velocity of point B, v B, and the angle formed by v B with the horizontal. 2.0 p PROBLEM 2

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6 B1 A 11 B Rod AB See that From dot product definition: a) Find: The velocity of point B, v B, and the angle formed by v B with the horizontal. PROBLEM 2 (solution continued)

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7 B1 C 22 b) Find: The angular velocity of rod BC, BC, and the velocity of point C, v C. We know: A 11 B Vector Although we don’t know their values, we can write for angular velocity and velocity of point C: C end slides on the floor BC and v C are the quantities to find. Let’s call This vectorial equation can be splitted into two equations, one for each component:

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8 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). PROBLEM 1 HOME EXAM B2 (2007) MODEL A 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 I zz 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 PROBLEM 2 R = 14,7 cm, = 50 kg·m -2, 0 = 0,60 rad/s, = 0.15

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9 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 PROBLEM 1 Solution HOME EXAM B2 (2007) MODEL A

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10 B2 Z axis normal to the plane, not shown Moment of inertia of a disc ( surface density, a radious) About to a normal axis passing through its simmetry center (I zz ) 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: Z axis 3R3R 2R2R R 2R2R 2R2R PROBLEM 2 Solution (continued)

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

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12 B2 Translation & Rotation equations t > 0, but before rolling without slipping Rolling condition When rolling without slipping begins, we have PROBLEM 2 Resolution R = 14,7 cm, = 50 kg·m -2, 0 = 0,60 rad/s, = 0.15 I ZZ = 1.65 kg·m 2 = 6.67 rad·s -2 t f = 0.060 s v CM (t f ) = 8.82·10 -2 m/s (t f ) = 0.20 rad/s

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13 B3 07B3_EngHOME EXAM B3 PROBLEM 1 (4 p) A hollow sphere (inner radius R 1 = 10 cm, outer radius R 2 = 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 m 2. 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: 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). a) How long does the surface of the water take to reach the spillway (in case it reaches).

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14 B3 A hollow sphere (inner radius R 1, outer radius R 2 ) 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. 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 M 0 g. V L is the volume of the sphere’s submerged portion Setting E equal to the weight, we find V L (a) Volume and mass of the hollow sphere: Submerged fraction PROBLEM 1 SOLUTION

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15 B3 (b) When the interior of the sphere is filled with material of density m, the sphere just floats completely submerged. 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 PROBLEM 1 SOLUTION Numerical solutions

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16 B3 A cylindrical-shaped tank is used in a farm as water reservoir. The area of the base of the tank is 10 m 2. 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 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). Spillway Continuity equation: 0.500.60 PROBLEM 2 SOLUTION

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17 B3 Spillway 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 h (m) =1 PROBLEM 2 SOLUTION

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

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19 B4 07B4HOME EXAM B3Only spanish version available Un gas ideal de coeficiente adiabático = 1.4 ejecuta un ciclo de potencia formado por las siguientes etapas: 1→2. El gas se expande politrópicamente (índice de politropía k 1 = 1.35) desde las condiciones V 1 = 1 litro, P 1 = 7.87 bar, hasta que su volumen se duplica. 2→3. El gas se enfría a volumen constante, hasta que su temperatura es T 3 = 280 K. 3→1. El gas se comprime politrópicamente hasta restituir las condiciones iniciales (sea k 2 el índice de politropía de este proceso, que deberá determinarse). 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). 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). B) Determinar el índice k 2 y representar gráficamente el ciclo en un diagrama de Clapeyron (P-V) (2 p).

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

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