1. 1 PHYSICS FOR ENGINEERS. A-EXAMS 2006-2007. A1A2A3A4

2. 2 A1 A student raises one end of a board slowly, where a block lies. The block starts to move when the angle is 30º. The static friction coefficient between the block and the board surface should be: 30º 1. a) b) c) d) e)None of above An object falls freely from rest on a planet without atmosphere where the gravity acceleration is 10 m s -2. Its velocity when it crashes against the ground is 5 m s -1, therefore its initial height was: 2. a) b) c) d) e)None of above 3. The angle formed by the vectorsandis: a) b) c) d) e)None of above “An object at rest stays at rest unless acted on by an external force”. This is: 4. a) b) c) d) e) None of above Newton’s third law Law of inertia Law of motion along a curved path The work-energy theorem A punctual mass follows a circular trajectory with constant speed. As for its acceleration, it is true that a) b) c) d) e) None of above The punctual mass is not accelerated Its tangential acceleration is positive The modulus of its normal acceleration is constant There is no normal acceleration 5. A particle whose mass is (1.00±0.01)  10 -2 kg is moving along a straight path at (1.00±0.10) m s -1. Its momentum is 6. a) b) c) d) e)None of above AGR FOR

3. 3 A1 A spring initially at rest, attached to a mass m, is stretched to distance x. As for the mass m, it is true that: 7. a) b) c) d) e) None of above The potential energy of m only varies in case the mass hangs vertically on the spring The potential energy of m varies if the mass hang vertically on the spring, but even if the spring and the mass lie on a horizontal surface. The potential energy of m never changes, only its kinetic energy undergoes some variation. Neither the potential energy of m nor its kinetic energy undergo any variation. Two blocks of masses m 1 and m 2 lie on a horizontal table. On m 1 we apply a horizontal force F 0 as shown in the picture. The kinetic friction coefficients for both masses are  1 y  2, respectively. Using the numerical values given below, answer the following questions: m1m1 m2m2 F0F0 Assuming the force F 0 is big enough to move the set of two blocks, find its acceleration. b) c) Find the force exerted by the first block (m 1 ) on the second one (m 2 ) and the force exerted by the second one on the first one.  1 = 0.075  2 = 0.040 m 1 = 8 kgF 0 = 2,50 kpm 2 = 6 kg Separately draw the free body diagram for the set of two blocks, for m 1 and for m 2. a) PROBLEM GRADING: PROBLEM: 6 POINTS QUESTIONS: 4 POINTS EACH CORRECT ANSWER: +0.500 EACH WRONG ANSWER: -0.125 A harmonic oscillator obeys the equationwhere every quantity is given in S.I. units. The frequency and the period are: 8. 4.5 Hz and 0.22 sa)0.5 Hz and 2.0 s b) 6.3 Hz and 0.16 sc)5 Hz and 9.90 s d) None of above e)

4. 4 A1 A student raises one end of a board slowly, where a block lies. The block starts to move when the angle is 30º. The static friction coefficient between the block and the board surface should be: 30º 1. a) b) c) d) e)None of above An object falls freely from rest on a planet without atmosphere where the gravity acceleration is 10 m s -2. Its velocity when it crashes against the ground is 5 m s -1, therefore its initial height was: 2. a) b) c) d) e)None of above 3. The angle formed by the vectorsandis: a) b) c) d) e)None of above “An object at rest stays at rest unless acted on by an external force”. This is: 4. a) b) c) d) e) None of above Newton’s third law Law of inertia Law of motion along a curved path The work-energy theorem A punctual mass follows a circular trajectory with constant speed. As for its acceleration, it is true that a) b) c) d) e) None of above The punctual mass is not accelerated Its tangential acceleration is positive The modulus of its normal acceleration is constant There is no normal acceleration 5. A particle whose mass is (1.00±0.01)  10 -2 kg is moving along a straight path at (1.00±0.10) m s -1. Its momentum is 6. a) b) c) d) e)None of above PHYSICS FOR ENGINEERS. 2006-2007. MID-TERM EXAM A1. SOLUTION

5. 5 A1 A spring initially at rest, attached to a mass m, is stretched to distance x. As for the mass m, it is true that: 7. a) b) c) d) e) None of above The potential energy of m only varies in case the mass hangs vertically on the spring The potential energy of m varies if the mass hang vertically on the spring, but even if the spring and the mass lie on a horizontal surface. The potential energy of m never changes, only its kinetic energy undergoes some variation. Neither the potential energy of m nor its kinetic energy undergo any variation. Two blocks of masses m 1 and m 2 lie on a horizontal table. On m 1 we apply a horizontal force F 0 as shown in the picture. The kinetic friction coefficients for both masses are  1 y  2, respectively. Using the numerical values given below, answer the following questions: m1m1 m2m2 F0F0 Assuming the force F 0 is big enough to move the set of two blocks, find its acceleration. b) c) Find the force exerted by the first block (m 1 ) on the second one (m 2 ) and the force exerted by the second one on the first one.  1 = 0.075  2 = 0.040 m 1 = 8 kgF 0 = 2,50 kpm 2 = 6 kg Separately draw the free body diagram for the set of two blocks, for m 1 and for m 2. a) PROBLEM GRADING: PROBLEM: 6 POINTS QUESTIONS: 4 POINTS EACH CORRECT ANSWER: +0.500 EACH WRONG ANSWER: -0.125 A harmonic oscillator obeys the equationwhere every quantity is given in S.I. units. The frequency and the period are: 8. 4.5 Hz and 0.22 sa)0.5 Hz and 2.0 s b) 6.3 Hz and 0.16 sc)5 Hz and 9.90 s d) None of above SOLUTION(CONTINUED) e)

6. 6 A1 m1m1 m2m2 F0F0 F 21 F R1 F0F0 F 12 F R2 N2N2 F R1 F R2 N1N1 N2N2 Newton’s 2 nd law Friction forces Newton’s 2 nd law  1 = 0.075  2 = 0.040 m 1 = 8 kgF 0 = 2,50 kpm 2 = 6 kg Numerical result a = 1.162 m/s 2 Newton’s 2 nd law F 12 and F 21 have to be equal (action and reaction) N1N1 SOLUTION(CONTINUED)

7. 7 A2 a) The sphere rolls from the beginning on the horizontal surface. b) The velocity of the center of mass remains constant, the angular velocity increases. c) The velocity of the center of mass drops, the angular velocity remains null. d) The sphere slides, but cannot roll because the friction is not zero. e) None of the above. P2. A skater on ice has her arms extended while she spins at 2 rps. When she cross her arms on her body, her moment of inertia halves. As a result, we’ll see that: a) Her angular velocity halves. b) Her angular velocity duplicates. c) Her angular velocity reduces to a fourth. d) Her angular velocity increases four times. e) Her angular velocity remains constant, however her angular momentum duplicates. P3. The figure depicts a ring of radious R and mass M, where the mass is homogenously shared out along its rim. The momenta of inertia about the different axis depicted in the figure are: a) b) c) d) e) None of the above X Y Z P4. The variation of the angular momentum with time quantifies: a) The sum of external forces b) The sum of the momenta of inertia c) The sum of the torques of external forces d) The variation of the angular velocity e) None of the above Warning: to answer questions P1 to P5 you have to use the answer form, only the answers including there will be considered. For the choosen right answer, use the symbol. To invalidate a previously marked cell, use the symbol P1. A homogeneous sphere is thrown on a rough horizontal floor. Its initial center of mass velocity is v, meanwhile its initial angular velocity is zero. What happens is: X X For questions P1 to P5, each correct answer adds +1; each wrong answer takes off -0.25 PHYSICS FOR ENGINEERS. 2006-2007. MID-TERM EXAM A2E QUESTIONS AGR FOR SURNAMES FIRST NAME

8. 8 A2 P5. A torque of 10 N  m is acting on a spinning disc, which turns around a normal fixed axis passing through its simmetry center. If the rotational work is 5.24 J, then we can say that the angle gone over is (within ±1º): a) 90º b) 30º c) 60º d) 45º e) 180º abcde P1 P2 P3 P4 P5 ANSWER FORM A B L L A thin rigid rod of lenght 2L = 80 cm is composed by two sections, each of lenght L, made on different materials whose linear densities are  A = 2 kg/m and  B = 5 kg/m. The rod is hung horizontally from both ends on two threads A and B, as shown in the figure. Find: a) The tension supported by each thread (1.5 p) c) Assume the thread B is cut. Find the angular acceleration and the angular velocity of the rod when it reaches the vertical position (2 p). b) The moment of inertia about end A (1.5 p) PROBLEM     

9. 9 A2 A B L L a)Both sections are homogeneous and have the same lenght, so the position of both center of masses will be Center of mass of the whole rod: Mass of each section: Total mass: Y PROBLEM SOLUTION

10. 10 A2 A L L A B b) Moment of inertia about A c) Angular acceleration and angular velocity PROBLEM SOLUTION

11. 11 A3 The pipe shown in figure 1 is used to supply water to a reservoir. It has two open tubes, each in different parts of the pipe having cross sections S 1 and S 2 (see numerical values below). 1. The difference of height between the water level in both open tubes is h (see numerical value below). Find the volume flux and the mass flux in the pipe. 2. There is a pump P (figure 2) which takes out from the reservoir M kg of water per minute and carries them up to the height H at c 0 m/s (see numerical values below). Find the power of the pump (neglect friction losses) S1S1 h 1 2 S2S2 Figure 1 Figure 2 PROBLEM Gravity acceleration g = 9.8 m/s 2 ; density of water  = 1.00 g/cm 3 PHYSICS FOR ENGINEERS. 2006-2007. MID-TERM EXAM A3E. Warning: to answer questions P1 to P5 you have to use the answer form, only the answers including there will be considered. For the choosen right answer, use the symbol. To invalidate a previously marked cell, use the symbol X X For questions P1 to P5, each correct answer adds +1; each wrong answer takes off -0.25 E

12. 12 A3 Bernoulli’s law S1S1 y1y1 h 1 2 S2S2 y2y2 c1c1 c2c2 z2z2 z1z1 Continuity equation PROBLEM SOLUTION

13. 13 A3 0 Bernoulli’s Law Pressures in S and 0 are the sameThe surface of the water is at rest Work done by the pump raising a water mass m to a height H Power: The pump raises M kg a minute: Figure 2 PROBLEM SOLUTION (CONTINUED) 2. There is a pump P (figure 2) which takes out from the reservoir M kg of water per minute and carries them up to the height H at c 0 m/s (see numerical values below). Find the power of the pump (neglect friction losses)

14. 14 A3 MODEL E PROBLEM SOLUTION (CONTINUED)

15. 15 A3 PHYSICS FOR ENGINEERS. 2006-2007. MID-TERM EXAM A3E. E P1. A student blows up the tyres of his car in a place where the atmospheric pressure is 980 mb. When the gauge indicates 2.2 kp/cm 2, the absolute pressure inside the tyres is: a) 284200 Pab) 3136 mbc) 3.05 kp/cm 2 e) None of aboved) 3.45 kp/cm 2 1 2 3 P2. Three identical glasses are filled up with water. The glass 1 contains just water; a cube of ice floats in the glass 2 (ice density 0.89 g/cm 3 ), and a piece of wood floats in the glass 3 (wood density 0,75 g/cm 3 ). We call W 1, W 2 y W 3 the weights of the three glasses including their contents, then it is true that: b) Ordered from more to less weight, the order is W 1 > W 2 > W 3 c) Ordered from more to less weight, the order is W 3 > W 2 > W 1 d) The weight of the three glasses is the same. e) The weight of the glass containing just water is smaller than the weights of the other two glasses. a) We cannot say which glass is more weighted without knowing the weights of the ice and the piece of wood. C 1 m 2 5 m 2 1 m 0.5 m 1 m A B P3. The initial level of the water in the system shown in the figure is indicated by A. The cylinder C is held in a fixed position (it cannot move). If we fill out the vertical thin tube up to the level B, what happens with the pressure P C exerted by the liquid against the cylinder C is: a) Pressure P C multiplies by 3 its initial value. b) Pressure P C keeps nearly exactly its initial value because the vertical tube is thin. c) Pressure P C multiplies by 5 the its inital value. d) Pressure P C duplicates. e) None of the above. QUESTIONS

16. 16 A3 F1 F2 F3 P4. The arrows depicted in the schemes F1, F2 and F3 indicate the velocities of the particles of a fluid moving to the right in a pipe. We can say about that: b) The three velocity schemes are characteristic of turbulent flow a) F1 represents laminar flow and F3 turbulent flow c) The three velocity schemes are characteristic of laminar flow d) The schemes F2 and F3 represent laminar flow; F1 represents turbulent flow e) None of above A B C P5. The figure shows a siphon. We try to transvase liquid from the left–hand to the right-hand vessel. The neccesary condition for the siphon to work is: e) The P5 heading is wrong, because it is impossible that a siphon setup such as the depicted one could work. a) It is necesary that h << d. b) It is necesary that d.= 0 d) The pressure in A must be smaller than the pressure in B. c) That siphon only works if the liquid is less dense than the water. abcde P1 P2 P3 P4 P5 ANSWER FORM PHYSICS FOR ENGINEERS. 2006-2007. MID-TERM EXAM A3E. E QUESTIONS     

17. 17 A3 QUESTION 1 SOLUTION Every floating body drives out of the glas a volume of liquid equal to its weight. Therefore, the weights of the three glasses are the same. QUESTION 2 SOLUTION QUESTION 3 SOLUTION The lower part of the cylinder C lies initially 0.5 m below the free surface of the water (level A). When the vertical thin tube is fillep up to level B (1 m besides level A), the new deep of the lower part of cylinder C is three times the initial one, so the pressure in C multiplies 3 times. QUESTION 4 SOLUTION See theory. QUESTION 5 SOLUTION See the siphon problem.

18. 18 A4 10 mol of an ideal gas (initial temperatue 20 ºC) expand reversibly following the law p = aV, where p, V are pressure and volume, whereas a is a constant. The final volume after the expansion is twice the initial one. Gasses constant R = 8.314 J/(K  mol). 1. Plot the expansion process in a p-V diagram (1 p). PROBLEM A 2. Find the final temperature (1.5 p). 3. Find the work in the expansion (1.5 p). P1. About the phase diagram depicted aside, it can be said that: a) 1 is the critical point; 2 is the triple point; A  B is a sublimation process. b) 1 is the triple point; 2 is the boiling point; A  B is a melting process. c) 1 is the melting point; 2 is the boiling point; A  B is an evaporation process. d) 1 is the triple point; 2 is the critical point; A  B is a sublimation process. e) None of the above. QUESTIONS T P T P 1 2 A B P2. An ideal gas expands reversibly without changing its temperature. The gas absorbs 200 Kcal. The internal energy variation in this process is: a) -200 kJb) +200 Kcalc) -200 Kcale) Zerod) +200 KJ ANSWER FORM PHYSICS FOR ENGINEERS. 2006-2007. MID-TERM EXAM A4E. Warning: to answer questions P1 to P6 you have to use the answer form, only the answers including there will be considered. For the choosen right answer, use the symbol. To invalidate a previously marked cell, use the symbol X X For questions P1 to P6, each correct answer adds +1; each wrong answer takes off -0.25

19. 19 A4 QUESTIONS a) The internal energy variation of the gas is positive if the cycle is described is clockwise direction. b) The entropy variation of the gas once the cycle is completed is positive if any of the cycle steps is irreversible. c) The heat absorbed by the gas is always the same than the work done by the gas. d) The internal energy variation of the gas once the cycle is completed is not well-defined, it depends on the way. e) None of the above. P3. An ideal gas complete a thermodynamics cycle composed by two adiabatics, one isochoric and one isobar. Indicate the correct option. P4. A 20-ton iceberg fall of from a coastal glacier. The temperature of the ice is -5º C. The specific heat of the ice is 0.50 Kcal/kg  K, ans its latent heat is 80 Kcal/kg. The energy needed to melt a half of the mass of the iceberg is: a) 8  10 5 Kcalb) 1.6  10 5 Kcalc) 82.5  10 4 Kcald) 25  10 5 Kcal e) None of the above P5. About the picture on the right, W is work, and about the temperatures, T a > T b. The energies transferred are Q a = 60 KJ and  Q b  = 50 kJ. Using those data, we can say that the picture is a representation of: a) A heat engine, whose yield is 20% b) A refrigerator, whose COP is 5. c) A heat engine, whose yield is 6/5. d) A refrigerator, whose COP is 5/6. e) None of the above P6. A system absorbs 300 Kcal from a source at 300 K during a reversible process. The variation of entropy of the universe, once the process is completed, is: a) Zerob) +1 Kcal/Kc) -1 Kcal/Ke) None of the aboved) +1 KJ/K A PHYSICS FOR ENGINEERS. 2006-2007. MID-TERM EXAM A4E.

20. 20 A4 PROBLEM PHYSICS FOR ENGINEERS. 2006-2007. EXAM A4. A p V 1 2 V1V1 V 2 =2V 1 We know T 1 = 20º C = 293 K We must find a relationship to calculate T 2 from T 1 and the volume rate V 2 /V 1. (in this case m = 2) Expansion work: 10 mol of an ideal gas (initial temperatue 20 ºC) expand reversibly following the law p = aV, where p, V are pressure and volume, whereas a is a constant. The final volume after the expansion is twice the initial one. Gasses constant R = 8.314 J/(K  mol). 1. Plot the expansion process in a p-V diagram (1 p). 2. Find the final temperature (1.5 p). 3. Find the work in the expansion (1.5 p).