Physics 4A Winter 2016 Test 2

Name:

Feb 16, 2016

Please show your work! Answer as many questions as you can, in any order. Calculatorsare allowed. Books and notes are not allowed. Use any blank space to answer questions, butplease make sure it is clear which question your answer refers to.

For all questions that require answers that are algebraic expressions, the expressions mustbe only in terms of the variables given in the question, and universal constants, such as g.

g = 9.8 ms−2 (If you like you can use g = 10 ms−2, but make your choice clear.)

Stokes Drag:R = −bvv(t) = vT (1− e−t/τ ), τ = m

b

The Drag Equation:R = −1

2DρAv2 v̂

v(t) = vT tanh(

gvTt)

sin2 θ + cos2 θ = 1

sin(2θ) = 2 sin(θ) cos(θ)

cos(2θ) = cos2 θ − sin2 θ

sin(α± β) = sinα cos β ± cosα sin β

cos(α± β) = cosα cos β ∓ sinα sin β

cosα cos β = 12[cos(α− β) + cos(α + β)]

sinα sin β = 12[cos(α− β)− cos(α + β)]

sinα cos β = 12[sin(α + β) + sin(α− β)]

sin(θ + π

2

)= cos θ

cos(θ + π

2

)= − sin θ

sec θ := 1cos θ

csc θ := 1sin θ

cot θ := 1tan θ

1

1. Two objects are connected by a light string that passes over a frictionless pulley asshown. The incline has friction and the coefficient of kinetic friction between the blockand the slope is µk. Also assume m1 > m2 and the angle of the incline is θ. The blocksstart from rest and begin sliding.

142 Chapter 5 The Laws of Motion

diagrams of both objects. Find (b) the magnitude of the acceleration of the objects and (c) the tension in the string.

41. Figure P5.41 shows the speed of a person’s body as he does a chin-up. Assume the motion is vertical and the mass of the person’s body is 64.0 kg. Determine the force exerted by the chin-up bar on his body at (a) t 5 0, (b) t 5 0.5 s, (c) t 5 1.1 s, and (d) t 5 1.6 s.

time (s)1.0 1.50

10

20

30

0.5 2.0sp

eed

(cm

/s)

Figure P5.41

42. Two objects are connected by a light string that passes over a frictionless pulley as shown in Figure P5.42. Assume the incline is frictionless and take m1 5 2.00 kg, m2  5 6.00  kg, and u 5 55.0°. (a)  Draw free-body dia-grams of both objects. Find (b)  the magnitude of the accel-eration of the objects, (c) the ten-sion in the string, and (d) the speed of each object 2.00 s after it is released from rest.

43. Two blocks, each of mass m 5 3.50 kg, are hung from the ceiling of an elevator as in Figure P5.43. (a) If the elevator moves with an upward acceler ation aS of magni-tude 1.60 m/s2, find the tensions T1 and T2 in the upper and lower strings. (b) If the strings can withstand a maximum tension of 85.0 N, what maximum accelera-tion can the elevator have before a string breaks?

44. Two blocks, each of mass m, are hung from the ceiling of an eleva-tor as in Figure P5.43. The elevator has an upward accel-eration a. The strings have negligible mass. (a) Find the tensions T1 and T2 in the upper and lower strings in terms of m, a, and g. (b) Compare the two tensions and determine which string would break first if a is made sufficiently large. (c) What are the tensions if the cable supporting the elevator breaks?

45. In the system shown in Figure P5.45, a horizontal force FS

x acts on an object of mass m2 5 8.00 kg. The hori-

m1 m2

u

Figure P5.42

T1

T2

aS m

m

Figure P5.43 Problems 43 and 44.S

M

36. Figure P5.36 shows loads hanging from the ceiling of an elevator that is moving at constant velocity. Find the tension in each of the three strands of cord supporting each load.

50.0°40.0°

T1 T2

T3

5.00 kg

60.0°

T1

T3

10.0 kg

T2

a b

Figure P5.36

37. An object of mass m 5 1.00 kg is observed to have an accel-eration aS with a magnitude of 10.0 m/s2 in a direction 60.0° east of north. Figure P5.37 shows a view of the object from above. The force F

S2 act-

ing on the object has a magni-tude of 5.00 N and is directed north. Determine the magnitude and direction of the one other horizontal force F

S1 acting on the object.

38. A setup similar to the one shown in Figure P5.38 is often used in hospitals to support and apply a horizontal trac-tion force to an injured leg. (a) Determine the force of tension in the rope supporting the leg. (b) What is the traction force exerted to the right on the leg?

70!

8.00 kg

Figure P5.38

39. A simple accelerometer is constructed inside a car by suspending an object of mass m from a string of length L that is tied to the car’s ceiling. As the car accelerates the string–object system makes a constant angle of u with the vertical. (a) Assuming that the string mass is negligible compared with m, derive an expression for the car’s acceleration in terms of u and show that it is independent of the mass m and the length L. (b) Deter-mine the acceleration of the car when u 5 23.0°.

40. An object of mass m1 5 5.00 kg placed on a frictionless, horizon-tal table is connected to a string that passes over a pulley and then is fastened to a hanging object of mass m2  5 9.00 kg as shown in Figure P5.40. (a) Draw free-body

60.0!

m

aS F2S

F1S

Figure P5.37

BIO

m1

m2

Figure P5.40 Problems 40, 63, and 87.

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(a) Draw free-body diagrams of both objects. [4 pts]

(b) Find an expression for the magnitude of the acceleration of the objects. [4 pts]

(c) When the mass m1 has descended a distance d, what is the change in the gravi-tational potential energy of the masses-and-Earth system? [3 pts]

(d) What is the change in the mechanical energy of the masses-and-Earth system? [2 pts]

(e) What is the speed of mass m1? [3 pts]

2

2. A submarine cuts its engine when its speed is vi and then coasts to rest. The equationdescribing the motion of the submarine during this period is v = vie

−ct, where v isthe speed at time t, vi is the initial speed at t = 0, and c is a constant (real, positivenumber).

(a) Find an expression for the acceleration in terms of the speed. Is the accelerationproportional to the speed or the speed squared? [4 pts]

(b) How far does the submarine travel as it comes to rest? (Let t → ∞, so thatv → 0.) [5 pts]

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3. An object of mass m is suspended from the ceiling of an accelerating truck as shown.The magnitude of the constant acceleration of the truck is a.

Part (a): Part (b):

Problems 171

of kinetic friction mk between the backpack and the elevator floor.

25. A small container of water is placed on a turntable inside a microwave oven, at a radius of 12.0 cm from the center. The turntable rotates steadily, turning one revolution in each 7.25 s. What angle does the water surface make with the horizontal?

Section 6.4 Motion in the Presence of Resistive Forces 26. Review. (a) Estimate the terminal speed of a wooden

sphere (density 0.830 g/cm3) falling through air, tak-ing its radius as 8.00 cm and its drag coefficient as 0.500. (b)  From what height would a freely falling object reach this speed in the absence of air resistance?

27. The mass of a sports car is 1 200 kg. The shape of the body is such that the aerodynamic drag coefficient is 0.250 and the frontal area is 2.20 m2. Ignoring all other sources of friction, calculate the initial accelera-tion the car has if it has been traveling at 100 km/h and is now shifted into neutral and allowed to coast.

28. A skydiver of mass 80.0 kg jumps from a slow-moving aircraft and reaches a terminal speed of 50.0 m/s. (a) What is her acceleration when her speed is 30.0 m/s? What is the drag force on the skydiver when her speed is (b) 50.0 m/s and (c) 30.0 m/s?

29. Calculate the force required to pull a copper ball of radius 2.00 cm upward through a fluid at the con-stant speed 9.00 cm/s. Take the drag force to be pro-portional to the speed, with proportionality constant 0.950 kg/s. Ignore the buoyant force.

30. A small piece of Styrofoam packing material is dropped from a height of 2.00 m above the ground. Until it reaches terminal speed, the magnitude of its accelera-tion is given by a 5 g 2 Bv. After falling 0.500 m, the Styrofoam effectively reaches terminal speed and then takes 5.00 s more to reach the ground. (a) What is the value of the constant B? (b) What is the acceleration at t 5 0? (c) What is the acceleration when the speed is 0.150 m/s?

31. A small, spherical bead of mass 3.00 g is released from rest at t 5 0 from a point under the surface of a vis-cous liquid. The terminal speed is observed to be vT 5 2.00 cm/s. Find (a) the value of the constant b that appears in Equation 6.2, (b) the time t at which the bead reaches 0.632vT, and (c) the value of the resistive force when the bead reaches terminal speed.

32. At major league baseball games, it is commonplace to flash on the scoreboard a speed for each pitch. This speed is determined with a radar gun aimed by an operator positioned behind home plate. The gun uses the Doppler shift of microwaves reflected from the baseball, an effect we will study in Chapter 39. The gun determines the speed at some particular point on the baseball’s path, depending on when the operator pulls the trigger. Because the ball is subject to a drag force due to air proportional to the square of its speed given by R 5 kmv2, it slows as it travels 18.3 m toward the

W

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lowest point instead of swinging up? (e) Explain your answer to part (d).

19. An adventurous archeologist (m 5 85.0 kg) tries to cross a river by swinging from a vine. The vine is 10.0 m long, and his speed at the bottom of the swing is 8.00 m/s. The archeologist doesn’t know that the vine has a breaking strength of 1 000 N. Does he make it across the river without falling in?

Section 6.3 Motion in Accelerated Frames 20. An object of mass m 5

5.00  kg, attached to a spring scale, rests on a frictionless, horizontal surface as shown in Fig-ure P6.20. The spring scale, attached to the front end of a boxcar, reads zero when the car is at rest. (a) Determine the acceleration of the car if the spring scale has a constant reading of 18.0 N when the car is in motion. (b) What constant reading will the spring scale show if the car moves with con-stant velocity? Describe the forces on the object as observed (c) by someone in the car and (d) by some-one at rest outside the car.

21. An object of mass m 5 0.500 kg is suspended from the ceiling of an accelerating truck as shown in Figure P6.21. Taking a 5 3.00 m/s2, find (a) the angle u that the string makes with the vertical and (b) the tension T in the string.

22. A child lying on her back experiences 55.0 N tension in the muscles on both sides of her neck when she raises her head to look past her toes. Later, sliding feet first down a water slide at terminal speed 5.70 m/s and rid-ing high on the outside wall of a horizontal curve of radius 2.40 m, she raises her head again to look for-ward past her toes. Find the tension in the muscles on both sides of her neck while she is sliding.

23. A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of 591 N. As the elevator later stops, the scale reading is 391 N. Assum-ing the magnitude of the acceleration is the same during starting and stopping, determine (a) the weight of the person, (b) the person’s mass, and (c) the accel-eration of the elevator.

24. Review. A student, along with her backpack on the floor next to her, are in an elevator that is accelerat-ing upward with acceleration a. The student gives her backpack a quick kick at t 5 0, imparting to it speed v and causing it to slide across the elevator floor. At time t, the backpack hits the opposite wall a dis-tance L away from the student. Find the coefficient

m

Figure P6.20

um

aS

Figure P6.21

M

M

S

174 Chapter 6 Circular Motion and Other Applications of Newton’s Laws

in part (d) depend on the numerical values given in this problem, or is it true in general? Explain.

54. A puck of mass m1 is tied to a string and allowed to revolve in a circle of radius R on a friction-less, horizontal table. The other end of the string passes through a small hole in the cen-ter of the table, and an object of mass m2 is tied to it (Fig. P6.54). The suspended object remains in equilibrium while the puck on the tabletop revolves. Find symbolic expressions for (a) the tension in the string, (b) the radial force acting on the puck, and (c) the speed of the puck. (d) Qualitatively describe what will happen in the motion of the puck if the value of m2 is increased by placing a small additional load on the puck. (e) Qualitatively describe what will happen in the motion of the puck if the value of m2 is instead decreased by removing a part of the hanging load.

55. Because the Earth rotates about its axis, a point on the equator experiences a centripetal acceleration of 0.033 7 m/s2, whereas a point at the poles experiences no centripetal acceleration. If a person at the equator has a mass of 75.0  kg, calculate (a) the gravitational force (true weight) on the person and (b) the normal force (apparent weight) on the person. (c) Which force is greater? Assume the Earth is a uniform sphere and take g 5 9.800 m/s2.

56. Galileo thought about whether acceleration should be defined as the rate of change of velocity over time or as the rate of change in velocity over distance. He chose the former, so let’s use the name “vroomosity” for the rate of change of velocity over distance. For motion of a particle on a straight line with constant acceleration, the equation v 5 vi 1 at gives its velocity v as a function of time. Similarly, for a particle’s linear motion with constant vroomosity k, the equation v 5 vi 1 kx gives the velocity as a function of the position x if the parti-cle’s speed is vi at x 5 0. (a) Find the law describing the total force acting on this object of mass m. (b) Describe an example of such a motion or explain why it is unre-alistic. Consider (c) the possibility of k positive and (d) the possibility of k negative.

57. Figure P6.57 shows a photo of a swing ride at an amusement park. The structure consists of a horizon-tal, rotating, circular platform of diameter D from which seats of mass m are sus-pended at the end of massless chains of length d. When the system rotates at

m1

m2

R

Figure P6.54

Q/CS

M

Q/CS

Figure P6.57

Stua

rt Gr

egor

y/Ge

tty I

mag

es

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separation from the line of best fit. Express this scatter as a percentage. (e) In a short paragraph, state what the graph demonstrates and compare it with the the-oretical prediction. You will need to make reference to the quantities plotted on the axes, to the shape of the graph line, to the data points, and to the results of parts (c) and (d).

50. A basin surrounding a drain has the shape of a circular cone opening upward, having everywhere an angle of 35.0° with the horizontal. A 25.0-g ice cube is set slid-ing around the cone without friction in a horizontal circle of radius R. (a) Find the speed the ice cube must have as a function of R. (b) Is any piece of data unnec-essary for the solution? Suppose R is made two times larger. (c) Will the required speed increase, decrease, or stay constant? If it changes, by what factor? (d) Will the time required for each revolution increase, decrease, or stay constant? If it changes, by what factor? (e) Do the answers to parts (c) and (d) seem contradic-tory? Explain.

51. A truck is moving with constant acceleration a up a hill that makes an angle f with the horizontal as in Figure P6.51. A small sphere of mass m is suspended from the ceiling of the truck by a light cord. If the pendulum makes a constant angle u with the perpendicular to the ceiling, what is a?

52. The pilot of an airplane executes a loop-the-loop maneuver in a vertical circle. The speed of the airplane is 300 mi/h at the top of the loop and 450 mi/h at the bottom, and the radius of the circle is 1 200 ft. (a) What is the pilot’s apparent weight at the lowest point if his true weight is 160 lb? (b) What is his apparent weight at the highest point? (c)  What If? Describe how the pilot could experience weightlessness if both the radius and the speed can be varied. Note: His apparent weight is equal to the magnitude of the force exerted by the seat on his body.

53. Review. While learning to drive, you are in a 1 200-kg car moving at 20.0 m/s across a large, vacant, level parking lot. Suddenly you realize you are heading straight toward the brick sidewall of a large supermar-ket and are in danger of running into it. The pavement can exert a maximum horizontal force of 7 000 N on the car. (a) Explain why you should expect the force to have a well-defined maximum value. (b) Suppose you apply the brakes and do not turn the steering wheel. Find the minimum distance you must be from the wall to avoid a collision. (c) If you do not brake but instead maintain constant speed and turn the steering wheel, what is the minimum distance you must be from the wall to avoid a collision? (d) Of the two methods in parts (b) and (c), which is better for avoiding a colli-sion? Or should you use both the brakes and the steer-ing wheel, or neither? Explain. (e) Does the conclusion

Q/C

um

f

aS

Figure P6.51

S

Q/C

(a) If the truck travels horizontally, Find an expression for the angle θ that the stringmakes with the vertical. [4 pts]

(b) Now suppose the truck is moving with constant acceleration a up a hill that makesan angle φ with the horizontal as in the second figure. Find an expression for theangle θ that the string supporting the mass makes with the perpendicular to theceiling. [6 pts]

Give your answers in terms of some or all of m, g, φ, and a.

4

4. An elevator of mass 1000 kg accelerates from rest moving upward. During the firstfew seconds of motion, the elevator’s acceleration increases with time according to theexpression

a = 2.00t− 0.300t2

where t is in seconds and a is in m/s2.

(a) What is the change in kinetic energy of the elevator during the interval from t = 0to t = 2.00 s? [5 pts]

(b) How far does the elevator rise in this time? [4 pts]

(c) What is the minimum average power output of the motor lifting the elevator overthis time interval? [3 pts]

(d) Why might the average power actually supplied be larger that the minimum valuecalculated in the previous part? [1 pt]

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