examples

Sample problems from last year.

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sinq = b/c, cosq = a/c. = ma, 1 N = 1 Kg×m/s2,

1 gallon = 3.8 liters = 3800 cm3 = 0.14 ft3, 1 lb = 4.45 N,

1 mile = 1.6 Km, 1 ft = 30.5 cm, 1 yd = 91.4 cm, 1 in = 2.54 cm,

g = 9.8 m/s2, K = 103, c(centi) = 10-2, m(mili) = 10-3,

x(t) = xo + vo (t - to)+ a(t - to)2, v(t) = vo+ a(t - to), a = v2/r v(t)2 = vo2+ 2a(x(t)- xo). Ff = mkN, Ff £ msN, W = , K.E. = mv2, P.E.spring = kx2 P.E.grav = mgh, p = mv. t = r¥F, = Ia, I = , L = Iw = r¥p, I = ICM + Md2, I = MR2(ring, cylindrical shell), I = MR2(disk, cylinder), I = MR2(spherical shell), I = MR2(sphere), I = ML2(rod). Periodic motion: When = bx, x(t) = Asin(wt+d), w = , For spring-mass, w=, For a pendulum, w=. E=kA2, For damped oscillation: = -bx-a implies that x(t) = Aeat/2sin(wt+d), where w2 = b-a2/4. When the right-hand side becomes negative, it is said to be overdamped, and the system will not oscillate. v = , f' = fdQ = CdT, 1 cal = 4.2 J, 1 Pa = 1 N/m2, for ideal gas cp cv = R, where R = 8.31 J/mol×_K, PV = nRT, PVg = constant, where g = cp/cv, for adiabatic change. dW = PdV, or W = , dU = dQ-dW,
= ln(Vf/Vi), =

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Problem Test 1

1. You've all blocked out of your mind as a horrible past memory the video for George Thoroughgood's "Bad to the Bone," so I'll remind you of it. GT is shooting pool and at the end of the video makes a finesse shot, so that the 8-ball barely rolls into the corner pocket. The set-up is pictured to the right. The 8-ball is 0.85 meters from the pocket, and the pocket is 56° from the line connecting the white and 8-balls. The white ball is 0.3 meters from the 8-ball. The coefficient of kinetic friction of the balls on the table is mk=0.3. How fast must the white ball be going so that the 8-ball barely rolls into the hole? Assume that the collision is elastic.

Problem Test 2

1 You are helping to design a windmill for a start-up company. You need to calculate the moment of inertia of the blades around their rotation center. There will be 30 blades and each blade has a spoke connecting the blade to the hub. The blade is 2 m long and 30 cm wide and its mass is 10 Kg, whereas the spoke is 3 cm diameter and 3 m long and its mass is 5 Kg. Find the moment of inertia.

2. A yo-yo of mass 100 g, outer radius 3 cm is wound with a string on an inner radius 0.3 cm. The yo-yo is "dropped" from rest. However, it is not allowed to accelerate since the person holding the yo-yo pulls on the string upward, unwinding the string just fast enough to keep the yo-yo suspended with zero velocity. How much string will be pulled after 2 seconds?

3. Last weekend while skiing at Lutsen you fell at the top of the main hill which has an elevation of 1.5 km and a slope of 20 degrees. You lost the skis and started rolling down the hill. The snowy surface has a coefficient of friction of 0.1, so you were rolling without slipping. Your stomach can only tolerate 10 revolutions in one second before you begin to lose your breakfast. If your mass is 68 Kg, your height is 180 cm and your waist circumference is 86 cm, will you throw up? If so, how far will you roll before it happens? Discuss all assumptions clearly.

Problem Test 3

1 Sometimes, you need a spring of a certain spring constant, but you do not have it. Instead you have springs of different spring constants. Combining those springs can give you a right kind. Suppose you have three springs with spring constants 1 N/m, 2 N/m and 3 N/m at your disposal. Show how you would combine them to make a spring with spring constant:(a) 5 N/m.(b) 1.5 N/m.(c) 2/3 N/m.Note that it may not be necessary to use all three springs to achieve any of the above. Remember to show explicitly that your combination will yield the desired results.

2. You want to build a grandfather clock using a pendulum made of a 1-m long uniform-density bar (a meter stick for those at home). The mass of the bar is 300 g. Where along the bar's length should you drill the hole for the rotation axis so that the oscillation period is 2 s.

3. In nature, there are many destructive forces such as wind and earthquakes which contain much energy in 1 Hz to 10 Hz range. Suppose you are designing a suspension bridge whose mass is 2¥106 Kg. (a) What is the range of values for the spring constant of the wire which supports the bridge so that the bridge would have a resonant frequency outside this dangerous range?(b) If your design requires that the wire stretch no more than 1 m, what is the allowed range for the spring constant?

Problem Test 4

1 You have been asked to design a new experiment to demonstrate the Doppler effect. You have decided to drop a 500-Hz tuning fork off the roof of the physics building, which is 15 m high. One stormy day, when the air pressure is 105 Pa, you carry out the experiment. What frequency do you hear just before the silence? Assume that the density of the air is 1.3 Kg/m3 and cv = (5/2)R.

2. In the middle of Minnesota winter, a car which was driven in -20 _C air is brought into an attached garage, whose air temperature was 5 _C. The mass of the car (except the engine) is 700 Kg and effectively all steel, and 300 Kg of the mass is an (steel) engine whose temperature is 85 _C. The air volume of the garage is 50 m3. When the thermal equilibrium is reached (temperature equalizes among garage air, the car and its engine), what would the temperature be provided that cFe = 0.11 cal/g×K, and cair = 0.17 cal/g×K, and the mass of 1 mol of air is 28.8 g? Ignore the effects of air leak, heat loss/gain through the walls and heat capacities of garage structure.

3. A simple-minded air conditioner uses an ideal-gas like refrigerant. The compressor compresses 0.5 mol of gas at 25 _C (298 _K) every second adiabatically from 105 Pa to 3¥105 Pa. The gas then goes through an efficient radiator at a constant pressure of 3¥105 Pa., where its temperature is brought to 30 _C. It will enter an expansion volume to reduce the pressure to 105 Pa adiabatically. Finally the hot room air gets cooled by the refrigerant while the temperature of the refrigerant is brought back to 25 _C while the pressure is kept at 105 Pa.
(a) What is the temperature of the refrigerant when it enters the radiator.
(b) Every second, how many liters of gas enters the compresser and how many liters comes out?
(c) How much heat does the refrigerant release to or absorb from the room air every second? Assume that the molar specific heat of the gas at constant volume is cv = (5/2)R.

MC 1

Two gliders are on a horizontal, frictionless track (air track). Glider A, with an initial velocity vo, collides with stationary glider B. They stick together and move to the right with a final velocity vf.

1. If the gliders have the same mass (MA = MB), then their final velocity is

(a) greater than vo.

(b) greater than vo /2, but less than vo.

(d) less than vo /2, but greater than zero.

(e) zero.

2. If glider A has a larger mass than glider B (MA > MB), then their final velocity is

(a) greater than vo.

(b) greater than vo /2, but less than vo.

(d) less than vo /2, but greater than zero.

(e) zero.

3. If glider A has a smaller mass than glider B (MA < MB), then their final velocity is

(a) greater than vo.

(b) greater than vo /2, but less than vo.

(d) less than vo /2, but greater than zero.

(e) zero.

4. In all of the above collisions,

(a) energy is conserved.

(b) energy is not conserved.

5. In all of the above collisions (1-3),

(a) momentum is conserved.

(b) momentum is not conserved.

6. In all of the above collisions (1-3),

(a) the force exerted by glider A on glider B has the same magnitude as the force exerted by glider B on glider A.

(b) the force exerted by glider A on glider B does not have the same magnitude as the force exerted by glider B on glider A.

7. In all of the above collisions (1-3),

(a) the acceleration of glider A has the same magnitude as the acceleration of glider B.

(b) the acceleration of glider A does not have to have the same magnitude as the acceleration of glider B.

8. A fireworks package is fired to the right from a gun at an angle of 60o from the horizontal. At the top of its trajectory the package explodes and breaks into two equal parts. After the explosion, one part stays at rest momentarily, and falls straight down. What is the direction of the motion of the other part just after the explosion?

(a) Straight up

(b) Up and to the left

(d) Straight down

(e) Down and to the right

9. A boy and girl on ice skates face each other. The girl has a mass of 20 kg and the boy has a mass of 30 kg. The boy pushes the girl backwards at a speed of 3.0 m/s. As a result of the push, the speed of the boy is about

(a) zero.

(b) 2.0 m/s

(d) 4.5 m/s

(e) 9.0 m/s

10. Five equal mass balls made of different substances are dropped from the same height onto a board. Four of the balls bounce up to the maximum height shown on the diagram below. Ball E sticks to the board. For which ball was the transfer of momentum to the board the greatest?

(a) Ball A

(b) Ball B

(d) Ball D

(e) Ball E

11. A teenager has a mass 200 times that of a baseball. She throws the baseball horizontally while standing on roller skates. What is her maximum backward speed?

(a) about 200 times that of the baseball.

(b) about 1/200 that of the baseball.

(d) about 14 times that of the baseball.

(e) about 1/14 that of the baseball.

12. A runner who speeds up from 2 m/s to 8 m/s. The runner's momentum increases by a factor of

(a) 4.

(b) 2

(d) .

(e) 1.

13. The center of mass of the three masses depicted on the right is located at

(a) A.

(b) B, 1/3 of the way from A to E.

(d) D, 2/3 of the way from A to E.

(e) E.

14. Two cars of equal mass travel in opposite directions with equal speeds on an icy patch of road. They lose control on the essentially frictionless surface, have a head-on collision, and then bounce apart, suffering extensive damage. Just after the collision, their velocities are

(a) zero.

(b) equal to their original velocities.

(d) less in magnitude and in the same direction as their original velocities.

(e) less in magnitude and opposite in direction to their original velocities.

15. In the type of collision described above,

(a) only kinetic energy is conserved.

(b) only momentum is conserved.

(d) neither momentum nor kinetic energy are conserved.

(e) the extent to which momentum and kinetic energy are conserved depends on how much energy is lost (to thermal energy) during the collision.

Questions 16 and 17 refer to the diagram at right. A golf ball is thrown at a bowling ball so that it hits head on and elastically bounces back. Ignore frictional effects.

16. Just after the collision, which of the following statements is (are) true about the momenta of the golf and bowling balls?.

(a) The bowling ball has the larger momentum.

(b) The golf ball has the larger momentum.

(d) It is impossible to tell without knowing the initial velocity of the golf ball.

(e) It is impossible to tell without knowing the mass of the balls.

17. Just after the collision, which of the following statements is (are) true about the kinetic energies of the golf and bowling balls?

(a) The bowling ball has the larger kinetic energy.

(b) The golf ball has the larger kinetic energy.

(d) It is impossible to tell without knowing the initial velocity of the golf ball.

(e) It is impossible to tell without knowing the mass of the balls.

18. The figure depicts two pucks on a frictionless table. Puck I is three times more massive than puck II. Starting from rest, the same constant and uniform force is applied to push the two pucks across the table. Which puck will have the greater momentum when the first puck reaches the finish line?

(a) puck I

(b) puck II

(d) Too little information to answer.

19. In the above problem, which puck will have the greater momentum when each puck reaches the finish line?

(a) puck I

(b) puck II

(d) Too little information to answer.

20. The center of mass of the four masses depicted on the right is located at

(a) A.

(b) B, 1/2 of the way from A to C.

(d) D, 1/2 of the way from C to E.

(e) E.

21. Two bars have the same uniform density, width and thickness and the horizontal bar is twice as long as the vertical one, as depicted above, The center of mass of these bars together is located at

(a) A.

(b) D, half way from A to F.

(d) C, 2/3 of the way from A to D.

(e) E, 1/2 of the way from D to F.

MC 2

1. A ring is rolling without slipping across the floor. The ring's rotational kinetic energy is(a) larger than its translational kinetic energy.

(b) smaller than its translational kinetic energy.

(d) is independent of its mass.

(e) is independent of its radius.

2. A child jumps from a rotating merry-go-round in the direction opposite to the direction of its rotation. Just after the child leaves the merry-go-round, which of the following is true about the child - merry-go-round system?(a) The total mechanical energy (kinetic + potential) of the system remains the same.

(b) The kinetic energy of the system is conserved.

3. A solid sphere rolls without slipping across the floor with a speed v. The sphere's rotational kinetic energy is(a) larger than its translational kinetic energy.

(b) smaller than its translational kinetic energy.

(d) independent of its mass.

(e) independent of its radius.

4. The figure depicts a record on a turntable which is rotating in the clockwise direction. When the power is turned off, the turntable gradually slows down and stops.Which figure below shows the directions of the angular velocity vector (w) and the angular acceleration vector (a) during this slowing down process?

5. Four beads of mass m are arranged in different ways on four identical rods, as shown in the diagram below. All rods are rotating with identical angular speeds. Which rod has the smallest kinetic energy?

(e) They are all the same.

6. There are cylindrical and spherical shells of the same radius, the same thickness and the same total mass. The moment of inertia (a) of the cylindrical shell is larger.

(b) of the spherical shell is larger.

(d) can be larger or smaller for either depending on the length of the cylindrical shell.

7. A ball starts from rest down the inside of a parabolic bowl and rolls without slipping. At the bottom the surface changes to a frictionless surface. The ball then moves up the other side of the bowl to a height(a) higher than the original height.

(b) lower than the original height.

(d) higher or lower than the original height depending on the moment of inertia of the ball.

8. Water is turning a waterwheel at a constant angular velociry. How much of the energy given to the wheel from the falling water is used to overcome friction of the wheel.(a) greater than 100%.

(b) exactly equal to 100%.

(d) less than 50%.

(e) It depends on what else the waterwheel is doing, for example, doing nothing or milling grain.

9. A flywheel is spinning at constant angular velocity about an axis through its center. The tangential velocity is (a) larger at a radius 1 m than at 2 m.

(b) larger at a radius 2 m than at 1 m.

10. A flywheel is spinning at constant angular velocity about an axis through its center. The angular velocity is (a) larger at a radius 1 m than at 2 m.

(b) larger at a radius 2 m than at 1 m.

11. A flywheel is spinning at constant angular velocity about an axis through its center. The tangential acceleration is (a) larger at a radius 1 m than at 2 m.

(b) larger at a radius 2 m than at 1 m.

(d) zero both at radii 1 m and 2 m.

12. A flywheel is spinning at constant angular velocity about an axis through its center. The radial acceleration is (a) larger at a radius 1 m than at 2 m.

(b) larger at a radius 2 m than at 1 m.

(d) zero both at radii 1 m and 2 m.

13. A 1-m rod, whose mass is 2 Kg, is free to rotate around one its end, and a 9-Kg mass is attached at the other end. Which one(s) of the five forces in the figure by itself can satisfy the net torque being zero.(a) A.

(b) B.

(d) D.

(e) E.

14. A flywheel slows uniformly from 10 to 5 rev/s in 4 s. The angular acceleration is about(a) 0.80 rad/s2.

(b) 1.25 rad/s².

(d) 7.85 rad/s².

(e) 23.6 rad/s².

15. The moment of inertia of a thin ring of radius R, and mass M is(a) MR2.

(b) (1/2)MR².

(d) (2/3)MR².

(e) (2/5)MR².

16. A pulley of radius 10 cm and moment of inertia 0.1 Kg×m2 is rotated by a rope with a force 9.8 N. Starting from rest, it will move in 4 s(a) 39.2 m.

(b) 78.4 m.

(d) 3920 m.

(e) 7840 m.

17. If the pulley in problem 16 is pulled instead by a 1-Kg mass, it will move (a) shorter distance than the above case.

(b) longer distance than the above case.

18. Bicycle racers sometimes use solid wheels in order to cut down the drag force between the air and the spokes of an ordinary wheel. This can be important effect because drag force rise quite rapidly as the speed of an object through the air increases. If the radius of a wheel is 35 cm and the speed of the bicycle relative to the ground is 30 m.p.h. (13.4 m/s), what is the speed relative to the ground of the end of the spoke closest to the rim for a spoke leading to the contact point with the ground, that is, a spoke pointing vertically down?(a) zero.

(b) 6.7 m/s.

(d) 26.8 m/s.

(e) 38.3 m/s.

19. What is the speed of the rim-end of a spoke pointing vertically up?(a) zero.

(b) 6.7 m/s.

(d) 26.8 m/s.

(e) 38.3 m/s.

20. What is the speed of the rim-end of a spoke that is horizontal and points forward?(a) zero.

(b) 6.7 m/s.

(d) 26.8 m/s.

(e) 38.3 m/s.

21. Door handles are located on the side away from the hinges(a) because the moment of inertia of the door would be minimized.

(b) because one can minimize the torque needed to open/close the doors.

(d) because one would need to do minimum work to open/close the doors.

(e) because the mass of the doors would be minimum.

MC 3

1. A spring is attached to a wall. The other end of the spring is attached to a physics book. The book is pulled a distance Æx from the spring's equilibrium position and released from rest. While the book is moving back to the equilibrium position, which of the following result in an energy input to the system defined as the physics book?(a) The force (pull) of the spring on the book.

(b) The frictional force of the floor on the book.

(d) The gravitational force of the earth on the book.

(e) Can't tell since energy is not conserved in this system.

Questions 2 to 4 refer to the energy versus time graph shown below for a girl on a swing. It may help to sketch the situation described in the question before you answer it.

2. If the curve represents the kinetic energy of the girl on a swing, then at time t0 she

(a) is as high as she will rise above the ground.

(b) is as close as she will come to the ground.

(d) is traveling her fastest speed.

(e) is at some point between her highest and lowest points traveling at an intermediate speed.

3. If the curve represents her kinetic energy, at what time will she return to the same place along her path and be traveling in the same direction as she was at time t0?(a) t1

(b) t2

(d) t4

(e) t5

4. Suppose instead of kinetic energy the curve represents the gravitational potential energy of the girl-earth system. Then at time t0 the girl(a) is as high as she will rise above the ground.

(b) is as close as she will come to the ground.

(d) is traveling her fastest speed.

(e) is at some point between her highest and lowest points traveling at an intermediate speed.

5. A pendulum has simple harmonic motion provided that(a) its bob is not too heavy.

(b) the string is not too long.

(d) the arc through which it swings is not too large.

Questions 6 - 8 refer to the diagram on the right of an object in circular motion with angular speed w. At t=0, y=0.

6. The y component of the object's position is given by(a) A cos(wt).

(b) A sin(wt).

(d) A tan(wt).

(e) A cos(vt).

7. The y component of the object's velocity is given by(a) A wcos(wt).

(b) A wsin(wt).

(d) A wtan(wt).

(e) A vcos(vt).

8. The y component of the object's acceleration is given by(a) -A w2cos(wt).

(b) -A w2sin(wt).

(d) -A w2tan(wt).

(e) A v2cos(vt).

9. A mass-spring system and a simple pendulum each have a period of one second on the earth. They are both taken to the moon for an experiment. On the surface of the moon,(a) the pendulum has a period longer than one second.

(b) the mass-spring system has a period longer than one second.

(d) the mass-spring system has a period shorter than one second.

(e) both periods are unchanged.

10. There is a mass hanging at the end of a vertically hanging spring and oscillating. Its position is plotted in A of the graph to the right. When two of such springs are used side by side (parallel) as pictured below in the middle, the position graph should look like(a) A.

(b) B.

(d) D.

(e) E.

11. Instead of above, the two springs are used in tandem (series) and the mass is doubled, as in the picture above right The position graph should look like(a) A.

(b) B.

(d) D.

(e) E.

12. A pendulum is swinging. Which of the following is possible?(a) The momentum of the pendulum bob is constant throughout one cycle.

(b) The potential energy of the pendulum bob-earth system is constant throughout one cycle.

(d) The angular momentum of the pendulum bob is constant throughout one cycle.

(e) None of the above.

13. Two identical springs are attached to two blocks which oscillate without friction.
Block II has a mass m and block I has a mass of 2m. The largest displacement of block II from its equilibrium position is three times the largest displacement of block I. The total energy of block-spring system II is(a) one-half the total energy of block-spring system I.

(b) one and a half times the total energy of block-spring system I.

(d) four and a half times the total energy of block-spring system I.

(e) nine times the total energy of block-spring system I.

14. When a pendulum clock is moved from sea level to the top of a mountain, it will (a) run slower.

(b) run faster.

(d) gain or lose time depending on the length of the pendulum.

(e) gain or lose time depending on the mass of the pendulum.

15. An object undergoes simple harmonic oscillation with its oscillation amplitude A . The distance (not displacement) it travels in a time equal to half of its period is(a) (1/2)A.

(b) A.

(d) 4A.

(e) dependent on what phase the time interval starts.

16. An object is oscillating and its oscillation amplitude is A . The distance (not displacement) it travels in a time equal to quarter of its period is(a) (1/2)A.

(b) A.

(d) 4A.

(e) dependent on what phase the time interval starts.

17. Two objects hang on springs of the same spring constant. One has mass m and the other, 2m. TWhen they undergo simple harmonic oscillation with the same total energy. Compared to 2m-mass, the amplitude of m is (a) half as much.

(b) 1/.

(d) .

(e) twice.

18. The position of an object undergoing simple harmonic oscillation is illustrated to the right. When it is written in a form x(t) = Asin(wt+d), A>0, d is(a) between 0 and p/2.

(b) between p/2 and p.

(d) between (3/2)p and 2p.

19. The position of an object undergoing simple harmonic oscillation is illustrated to the right. The velocity of the object corresponds to (a) A.

(b) B

(d) D

(e) E

20. The position of an object undergoing simple harmonic oscillation is illustrated above. The acceleration of the object corresponds to (a) A.

(b) B

(d) D

(e) E

Final

1 You are planning to make iced tea for a summer(!) picnic. Using 10 tea bags, you can make V liters of tea of just the right strength. Your procedure is to make a strong tea with boiling water, and then put in some ice to bring down the temperature to the desired value, T.
(a) How much boiling water should be used to make this happen. Assume that the densities of water and ice are both r g/cm3, the specific heat of water is cw cal/g×K, that of ice is ci cal/g×K and the latent heat of fusion of ice is Qm cal/g. Define any additional quantities you need to find the answer. (b) Numerically, how much boiling water should you use? Assume that V = 5 liters, r = 1 g/cm3, cw = 1 cal/g×K, ci = 0.5 cal/g×K and Qm = 80 cal/g.

2. It is time to change the oil of your car, which weighs 2000 lb. From the manual, you find out that 60% of the weight is on the front wheels. If you were to lift the front of the car by the bumper, what is the minimum weight that the jack has to be able to lift. The wheel base (the distance between the front and rear wheels) measures 10 ft, and the front wheels are 2 ft from the bumper.

3. Your child is playing with a merry-go-round in a playground. Its radius is 3 m and it is going around once every 5 seconds. You decide to speed it up by running along its edge tangentially and jumping on it. You know you can still run at the top speed of 7 m/s. Once you are on, the merry-go-round goes around once every 4.5 seconds. What is the weight of the merry-go-round? You weigh 150 lbs, and your child weighing 30 lbs is riding near the edge of the merry-go-round.

4. The child of a friend has asked you to help with a school project. She wants to build a clock from common materials. She has found a meter stick which has a mass of 300g. She decides to attach a disk of mass 300 g at the bottom of the meter stick so that it would really look like an old clock, and asks you to determine where to drill a hole in the ruler so that when it is hung by a nail through that hole it will be a pendulum with a period of 2.0 seconds for small oscillations.

5. You want to make a special solid-disk yo-yo so that when you drop it from a third-story window, the speed it falls just before it reaches the ground will be no more than 1 m/s. What does the ratio of the radius of the yo-yo and the radius of the cylinder that the string is wound need to be? Assume that the window is 10 m high.

6. While driving on a highway at 55 m.p.h. (24.4 m/s), an ambulance traveling in the opposite direction speeds by you and the pitch which you hear from the siren drops by a "perfect fifth," which means the frequency you hear decreases by 33%. What is the speed of the ambulance? Assume that the sound velocity is 330 m/s.

7. Calculate the moment of inertia of a square shaped board about the axis perpendicular to the board passing through the center of mass by setting up and evaluating an integral. The sides of the board is a and its mass is m.. Divide the square into thin strips and estimate how much moment of inertia each strip will contribute. You may use the fact that the moment of inertia of a thin strip (about its center) is (1/12)ML2.

1. Four beads of mass m are arranged in different ways on four identical rods, as shown in the diagram to the right.

The rods start at rest, the same uniform torque is used and result in the same energy of rotation about an axis through the center of the rod, which rod would require the longest time that the torque is applied to reach the same energy?

(a) Rod #1.

(b) Rod #2.

(d) Rod #4.

(e) Since the final energy is the same, they would need an equal angle.

2. A uniform beam is balanced at its midpoint. Then two forces, F1 and F2, simultaneously act on the beam, as shown in the diagram below. The beam rotates clockwise on its pivot with an angular acceleration, a, directed into the page. Assume that q > f.

How do the magnitudes of the torques about the pivot caused by the two forces compare?(a) The torque caused by F2 is larger than the torque caused by F1.

(b) The torque caused by F2 is the same as the torque caused by F1.

(d) It is impossible to tell which torque is larger without knowing the magnitude of the forces and angles q and f.

(e) It is impossible to tell which torque is larger without knowing both the magnitude of the forces and the distance of each force from the pivot point.

3. Two identical disks have a common axis. Initially, one of the disks is spinning in one direction and the other in the opposite diretion. When the two disks are brought into contact, they stick together. Which of the following statements is true?

(a) The total angular momentum and the total kinetic energy are unchanged from their initial values.

(b) Both the total angular momentum and the total kinetic energy decrease.

(d) The total angular momentum is unchanged, but the total kinetic energy decreases.

(e) The total angular momentum is unchanged, but the total kinetic energy increases.

4. When the source of sound is traveling away from an observer

(a) the frequency appears higher and the wavelength appears shorter.

(b) the frequency appears lower and the wavelength appears longer.

(d) the wavelength and the velocity are unchanged, but the frequency is higher.

(e) the wavelength and the frequency are unchanged, but the velocity is decreased.

* Refer to the diagram below right when answering the next two questions.

The diagram depicts two pucks on a frictionless table. Puck II is four times as massive as puck I. The pucks start from rest and the same constant forces are applied to the pucks as they move along the table.

5. Which puck will have the greater momentum upon reaching the finish line?

(a) puck I.

(b) puck II.

(d) Too little information to answer.

6. Which puck will reach the finish line first?

(a) puck I

(b) puck II

(d) Too little information to answer.

7. A grinding wheel, just after the motor is turned off, rotates with a high angular velocity in the counterclockwise direction, as shown in the diagram at right. A wooden rod pushing toward the center on the outside edge of the grinder wheel causes the angular velocity to decrease.

A force diagram of the grinder wheel is shown at right. Which force(s) exert a torque on the grinding wheel about its center?(a) W and N1

(b) N2

(d) W and Fk

(e) N2 and Fk

* Refer to the diagram to the right when answering the next two questions.

8. A ball is thrown straight up into the air from point xo. The diagram at right shows the position vector r while the ball is still on its way up.

What is the direction of the angular momentum of the ball about the origin?

(a) Down (-y)

(b) Up (+y)

(d) Directed into the page (-z)

(e) There is no angular momentum vector because the ball is not rotating around the origin.

9. While the ball is on the way up, which of the following statements about the magnitude of the ball's angular momentum is true?(a) The angular momentum of the ball decreases because its momentum is decreasing.

(b) The angular momentum of the ball is constant because gravitational force is constant.

(d) The angular momentum of the ball is zero because the ball is not rotating around the origin.

(e) The angular momentum could either increase or decrease; it is impossible to tell.

* Refer to the graph to the right when answering the next two questions.

The graph represents the motion of a mass attached to a horizontal spring undergoing simple harmonic motion.

10. Which picture above shows the position of the mass at time t = 0?

11. Which graph below is the velocity versus time graph for the mass?

12. Three equal forces are applied to the massless beam as shown to the right. The beam is mounted on a hinge and connected to a wall. What is the direction of the angular acceleration?

(a) Out of the page (+z).

(b) Into the page (-z).

(d) -y direction

(e) No angular acceleration.

13. To produce simple harmonic motion, which of the following statements must always be true?

(a) Acceleration is proportional to negative velocity.

(b) Velocity is proportional to positive displacement.

(d) Acceleration is proportional to negative displacement.

(e) Displacement is proportional to period.

14. Consider the massless beam which pivots about an axis through one end as shown below. Which forces cause torques directed out of the page (i.e. anti-clockwise rotation)?

(a) F1 and F4

(b) F4 and F5

(d) F1 and F5

(e) F2 and F4

15. Consider the following representations of a car and a person. Assume that when the car is moving, it is moving faster than the person.

The observer would hear a higher frequency from the car's horn in which situation(s)?

(a) 1.

(b) 2.

(d) 2 and 3.

(e) 2 and 5.

16. A uniform plank is supported by the backs of two chairs, as shown in the diagram at right. How do the magnitudes of the torques exerted by the two chairs compare?

(a) Chair A exerts a greater torque about the center of the plank than Chair B.

(b) Both chairs exert the same magnitude of torque about the center of the plank.

(d) It is impossible to tell which torque is larger without knowing the magnitude of forces each chair exerts on the plank.

(e) It is impossible to tell which torque is larger without knowing both the magnitude of the forces and the distance of each force from the center of the plank.

17. For the uniform plank supported by the backs of two chairs in problem 19, how do the magnitudes of the forces exerted by the two chairs compare?

(a) Chair A exerts a greater force on the plank than Chair B.

(b) Both chairs exert equal forces on the plank.

(d) It is impossible to tell which force is larger without knowing the weight of the plank.

(e) It is impossible to tell which force is larger without knowing more about the chairs.

18. Two objects hang on springs of the same spring constant. One has mass m and the other, 2m. When they undergo simple harmonic oscillation with the same amplitude. Compared to the 2m-object, the energy of m object is (a) 1/.

(b) the same.

(d) twice.

(e) four times.

19. You are boiling water in order to make a cup of tea following your physics final exam. You would like the tea to be as hot as possible. Which of the following scenarios will lead to the hottest cup of tea: (a) You turn the heat down to a simmer for ten minutes as soon as the water boils

(b) You turn the heat on high for ten minutes as soon as the water boils

(d) You turn the heat to a simmer as soon as the water boils, then pour right away

(e) All of the above are the same

20. The mobile at right hangs from a string. The beam has a weight of one unit. Determine the number of weight units that must be hung from the left side in order to balance the beam.

(a) 7/5.

(b) 2.

(d) 11.

(e) 13.

21. A gas confined in a cylinder by a piston expands from a volume of 1L to a volume of 3L at a constant pressure of 1 atmosphere. Which of the following is true?(a) The internal energy of the system goes up.

(b) The system does no net work.

(d) The temperature goes down under this transition.

(e) None of the above.

22. Three children, who weigh about the same, want to play on a see-saw. A see-saw is a rigid board which is placed over a sturdy bar which acts as a pivot. Two children get on one end while the third sits on the other end. Where should the pivot be placed to balance the see-saw?

(a) In the middle between the children.

(b) Closer to the two children.

(d) It is impossible to tell without knowing the length of the board.

(e) It is impossible to tell without knowing the masses of the children and the see-saw.

23. Consider the system to the right. A small cylinder is mounted on axis to the front of a larger cylinder so that the two must spin together. The two masses are then suspended from each cylinder as shown so that the cylinder does not spin. How do the two masses compare?

(a) Mass 1 is greater than mass 2.

(b) The masses are equal.

(d) It is impossible to tell without knowing the radii of the cylinders

(e) It is impossible to tell without knowing the moment of inertia of the cylinders.

The remaining questions are for extra credits related to the Wave lab.

24. You are holding the end of a long wire which is 10 m long. You excite a pulse on this wire which has a period of 2 seconds and a wavelength of 30 cm. How long does it take for the pulse to travel to the end of the wire, reflect, and return to your end of the wire?(a) 30 s

(b) 43 s

(d) 120 s

(e) 133 s

* Refer to the diagrams below when answering the next three questions.

The following diagrams depict standing wave patterns on the same tight string.

25. Which diagram best represents the fundamental (lowest frequency) standing wave of a string that is fixed at one end and free to move at the other end?

26. Which diagram best represents the first harmonics (second lowest frequency) standing wave of a string that is fixed at both ends?

27. Which of the diagrams shows the standing wave pattern with the highest wave velocity? Choose from the answers given below.

(a) (a)

(b) None of these because the wave velocity of a standing wave is zero.

(d) (d)

(e) None of these because the wave velocity is the same for each standing wave pattern

Group practice problem 1

1. You are designing a bungee for a site 100 m deep. You can use an elastic cord, one meter of which has a spring constant k = 1000 N/m, to keep divers from hitting the bottom.

(a) Find the length of the cord you would need so that a person of 100 Kg of mass would barely reach the bottom.

(b) How long would it take the person to reach the bottom?

(c) In order for lay people not to get too sick, you want the maximum acceleration of the person to be less than 3g (30 m/s2). Does this design meet this criterion?

Group Test 1

1. You are involved in the design of an emergency sand bank, which will stop cars safely if they lose brakes on a downhill mountain road. Assuming that the mass of a typical car is 1500 Kg. Also assume that when the transmission is in neutral, the combined moment of inertia of the rolling elements is 10 Kg×m2. The radius of a tire is 0.3m. The slope of the hill is 20° and its length is 100m.

(a) For what maximum speed must you design the sand bank?

(b) When the driver uses "engine braking", the engine "resists" rolling, and the effective moment of inertia increases to 1000 Kg×m2. What effect will this have on the maximum speed? (Quantitative answers please)

Group Practice problem 2

1. A car moving at v = 150 Km/h (41.7 m/s) is approaching a stationary police car whose radar speed detector operates at a frequency of f = 15 GHz. The radar waves are emitted for Dt= 0.1 s. The speed to the radar waves is c = 3¥108 m/s.

(a) What is the time between the time when the car starts reflecting the radar and the time it finishes?

(b) What is the time between the time when the speed detector starts receiving the reflected waves and the time it finishes.

(d) When the effect of "relativity" is taken into account, the equation which gives the right frequency is f' = f. Show that your answer in (c) is very close to the right answer given by this equation.

Group Test 2

1. A physicist designed a very energy efficient underground train, which would connect Chicago and Minneapolis by a straight tunnel. The train runs using gravitational force of the earth. The two cities are 600 Km apart (along the earth's surface). Assume that the radius of the earth is 6400 Km. The gravitational acceleration when measured at a location underground r from the earth's center is given by gr/R, where g is the surface gravitational acceleration (=9.8 m/s2) and R is the earth's radius.

(a) Calculate how deep would the tunnel be in its middle.

(b) If you measure the location of the train with respect to the middle point, show that its motion can be described by x(t) = Asin(wt+p/2) by showing that the acceleration of the train is proportional to -x, and find the value of A and w , where w = , b being the ratio between the acceleration and -x.

(d) What is the maximum speed?

(e) What technical challenges would this physicist have to overcome for this idea to be realized? List at least two.

Group Test 3

1. The Carnot Cycle is the most important thermodynamic cycle for theoretical heat engine analysis. It is the simplest model of the "real" Otto cycle, the 4-stroke internal combustion cycle used in all gas-powered automobiles, and it is also a close approximation to reality. With this in mind we will describe the steps in the Carnot Cycle.

The compression stroke:

A. An isothermal (constant T1=300_K) compression from V1=500 cm3 and P1 = 105 Pa to V2=(1/2)V1.

B. An adiabatic compression from V2 to V3=(1/8)V1.

The expansion stroke:

C. An isothermal (constant T2) expansion from V3 to V4.

D. An adiabatic expansion from V4 back to V1.

Assume an ideal gas whose molar heat capacity, cv = (5/2)R, and mass/mole is 28.8 g,

(a) calculate P2 (pressure after step A) and P3 (pressure after step B).

(b) determine the work done by the gas during steps A and B?