Physics 4A Fall 2017 Test 2

Name:

Nov 3, 2017

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 = mb

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 θsin(α± β) = sinα cos β ± cosα sin βcos(α± β) = cosα cos β ∓ sinα sin βcosα cos β = 1

2[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. A child’s toy consists of a small wedge that has an acute angle θ (see figure). An objectof mass m on it remains at constant height on the sloping surface of the wedge. Thewedge is spun by rotating, as an axis, a vertical rod that is firmly attached to thewedge at the bottom end.

The surface of the slope has a coefficient of static friction µs. The object sits at rest ata point at distance L up along the wedge. The object moves with a constant speed v(into the page in the moment shown in the diagram), which is the maximum speed itcan travel at without slipping up the slope. Find an expression for µs in terms of v, θ,and L. [11 pts]

(Hint: the static friction force points parallel to the slope to help hold the object inplace.)

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

r1 5 25 m. Find the force that a seat in the roller-coaster car exerts on a 50-kg passenger at the lowest point.

39. A string under a ten-sion of 50.0 N is used to whirl a rock in a horizontal circle of radius 2.50 m at a speed of 20.4 m/s on a frictionless surface as shown in Figure P6.39. As the string is pulled in, the speed of the rock increases. When the string on the table is 1.00 m long and the speed of the rock is 51.0 m/s, the string breaks. What is the breaking strength, in newtons, of the string?

40. Disturbed by speeding cars outside his workplace, Nobel laureate Arthur Holly Compton designed a speed bump (called the “Holly hump”) and had it in stalled. Suppose a 1 800-kg car passes over a hump in a roadway that follows the arc of a circle of radius 20.4 m as shown in Figure P6.40. (a) If the car travels at 30.0 km/h, what force does the road exert on the car as the car passes the high-est point of the hump? (b) What If? What is the maximum speed the car can have with-out losing contact with the road as it passes this highest point?

41. A car of mass m passes over a hump in a road that fol-lows the arc of a circle of radius R as shown in Figure P6.40. (a) If the car travels at a speed v, what force does the road exert on the car as the car passes the highest point of the hump? (b) What If? What is the maximum speed the car can have without losing contact with the road as it passes this highest point?

42. A child’s toy consists of a small wedge that has an acute angle u (Fig. P6.42). The sloping side of the wedge is frictionless, and an object of mass m on it remains at constant height if the wedge is spun at a certain constant speed. The wedge is spun by rotating, as an axis, a vertical rod that is firmly attached to the wedge at the bottom end. Show that, when the object sits at rest at a point at distance L up along the wedge, the speed of the object must be v 5 (gL sin u)1/2.

43. A seaplane of total mass m lands on a lake with initial speed vi î. The only horizontal force on it is a resistive force on its pontoons from the water. The resistive force is proportional to the velocity of the seaplane: RS

5 2bvS. Newton’s second law applied to the plane is 2bv î 5 m 1dv/dt 2 î. From the fundamental theorem

mR

Figure P6.39

vS

Figure P6.40 Problems 40 and 41.

S

L

m

u

Figure P6.42

S

S

plate according to the formula v 5 vie2kx. Suppose the ball leaves the pitcher’s hand at 90.0 mi/h 5 40.2 m/s. Ignore its vertical motion. Use the calculation of R for baseballs from Example 6.11 to determine the speed of the pitch when the ball crosses the plate.

33. Assume the resistive force acting on a speed skater is proportional to the square of the skater’s speed v and is given by f 5 2kmv 2, where k is a constant and m is the skater’s mass. The skater crosses the finish line of a straight-line race with speed vi and then slows down by coasting on his skates. Show that the skater’s speed at any time t after crossing the finish line is v(t) 5 vi/(1 1 ktvi).

34. Review. A window washer pulls a rubber squeegee down a very tall vertical window. The squeegee has mass 160 g and is mounted on the end of a light rod. The coefficient of kinetic friction between the squee-gee and the dry glass is 0.900. The window washer presses it against the window with a force having a horizontal component of 4.00 N. (a) If she pulls the squeegee down the window at constant velocity, what vertical force component must she exert? (b) The win-dow washer increases the downward force component by 25.0%, while all other forces remain the same. Find the squeegee’s acceleration in this situation. (c) The squeegee is moved into a wet portion of the window, where its motion is resisted by a fluid drag force R pro-portional to its velocity according to R 5 220.0v, where R is in newtons and v is in meters per second. Find the terminal velocity that the squeegee approaches, assum-ing the window washer exerts the same force described in part (b).

35. A motorboat cuts its engine when its speed is 10.0 m/s and then coasts to rest. The equation describing the motion of the motorboat during this period is v 5 vie2ct, where v is the speed at time t, vi is the initial speed at t 5 0, and c is a constant. At t 5 20.0 s, the speed is 5.00 m/s. (a) Find the constant c. (b) What is the speed at t 5 40.0 s? (c) Differentiate the expression for v(t) and thus show that the acceleration of the boat is proportional to the speed at any time.

36. You can feel a force of air drag on your hand if you stretch your arm out of the open window of a speeding car. Note: Do not endanger yourself. What is the order of magnitude of this force? In your solution, state the quantities you measure or estimate and their values.

Additional Problems 37. A car travels clockwise at con-

stant speed around a circular section of a horizontal road as shown in the aerial view of Fig-ure P6.37. Find the directions of its velocity and acceleration at (a) position ! and (b) position ".

38. The mass of a roller-coaster car, including its passengers, is 500 kg. Its speed at the bottom of the track in Figure P6.16 is 19 m/s. The radius of this section of the track is

S

AMT

E

N

S

W

!

"

Figure P6.37

2

2. A particle has a potential energy described by the function:

U(x) = 2(x− 1) − (x− 2)3

where U is measured in Joules and x in meters. It is plotted in the figure.

x

(a) Find an expression for the force Fx corresponding to this potential energy as afunction of x. [3 pts]

(b) Is Fx a conservative force? [1 pt]

(c) What are the values x1 and x2? [4 pts]

(d) What values of x (if any) correspond to stable equilibria? Unstable equilibria?Neutral equilibria? [3 pts]

(e) Imagine that the force Fx is the only force that acts on a particle. If the particleis released from rest at x = 0, what is its kinetic energy when it passes throughthe point x1? [4 pts]

3

3. A pendulum, comprising a light string of length L and a small sphere (bob), swings inthe vertical plane. The bob is released from rest at an angle θ (shown in the diagram).The string hits a peg located a distance d = 3L/4 below the point of suspension (seefigure). In this problem, neglect any air resistance or frictions.

(a) Imagine that we would like the bob to swing in a complete circle centered on thepeg. What is the minimum value of the bob’s speed at the top of the circle forthis to happen? [4 pts]

(b) Find the minimum value of the angle θ such that the bob will swing in a completecircle centered on the peg. [7 pts]

Problems 243

The line from the center of curvature of the cap to the pumpkin makes an angle ui 5 08 with the vertical. While you happen to be standing nearby in the middle of a rainy night, a breath of wind makes the pumpkin start sliding downward from rest. It loses contact with the cap when the line from the center of the hemi-sphere to the pumpkin makes a certain angle with the vertical. What is this angle?

67. Review. The mass of a car is 1 500 kg. The shape of the car’s body is such that its aerodynamic drag coefficient is D 5 0.330 and its frontal area is 2.50 m2. Assuming the drag force is proportional to v2 and ignoring other sources of friction, calculate the power required to maintain a speed of 100 km/h as the car climbs a long hill sloping at 3.208.

68. A pendulum, comprising a light string of length L and a small sphere, swings in the vertical plane. The string hits a peg located a distance d below the point of suspension (Fig. P8.68). (a) Show that if the sphere is released from a height below that of the peg, it will return to this height after the string strikes the peg. (b) Show that if the pendulum is released from rest at the horizontal position (u 5 908) and is to swing in a complete circle centered on the peg, the minimum value of d must be 3L/5.

69. A block of mass M rests on a table. It is fastened to the lower end of a light, vertical spring. The upper end of the spring is fastened to a block of mass m. The upper block is pushed down by an additional force 3mg, so the spring compression is 4mg/k. In this configuration, the upper block is released from rest. The spring lifts the lower block off the table. In terms of m, what is the greatest possible value for M?

70. Review. Why is the follow-ing situation impossible? An athlete tests her hand strength by having an assistant hang weights from her belt as she hangs onto a horizontal bar with her hands. When the weights hanging on her belt have increased to 80% of her body weight, her hands can no longer support her and she drops to the floor. Frustrated at not meeting her hand-strength goal, she decides to swing on a tra-peze. The trapeze consists of a bar suspended by two parallel ropes, each of length ,, allowing performers to swing in a vertical circular arc (Fig. P8.70). The athlete holds the bar and steps off an elevated platform, start-ing from rest with the ropes at an angle ui 5 60.08 with respect to the vertical. As she swings several times back and forth in a circular arc, she forgets her frustration related to the hand-strength test. Assume the size of the

dL

Peg

u

Figure P8.68

S

S

!ui

Figure P8.70

63. A 10.0-kg block is released from rest at point ! in Fig-ure P8.63. The track is frictionless except for the por-tion between points " and #, which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2 250 N/m, and compresses the spring 0.300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between points " and #.

3.00 m6.00 m

!

" #

Figure P8.63

64. A block of mass m1 5 20.0 kg is connected to a block of mass m2  5 30.0 kg by a massless string that passes over a light, frictionless pulley. The 30.0-kg block is connected to a spring that has negligible mass and a force constant of k 5 250 N/m as shown in Figure P8.64. The spring is unstretched when the system is as shown in the figure, and the incline is frictionless. The 20.0-kg block is pulled a distance h 5 20.0 cm down the incline of angle u 5 40.08 and released from rest. Find the speed of each block when the spring is again unstretched.

65. A block of mass 0.500 kg is pushed against a horizon-tal spring of negligible mass until the spring is com-pressed a distance x (Fig. P8.65). The force constant of the spring is 450 N/m. When it is released, the block travels along a frictionless, horizontal surface to point !, the bottom of a vertical circular track of radius R 5 1.00 m, and continues to move up the track. The block’s speed at the bottom of the track is v! 5 12.0 m/s, and the block experiences an average friction force of 7.00 N while sliding up the track. (a) What is x? (b) If the block were to reach the top of the track, what would be its speed at that point? (c) Does the block actually reach the top of the track, or does it fall off before reaching the top?

km

x

!

R!v

S

Figure P8.65

66. Review. As a prank, someone has balanced a pumpkin at the highest point of a grain silo. The silo is topped with a hemispherical cap that is frictionless when wet.

M

u

m1

m2

k

Figure P8.64

MAMT

Q/C

4

4. The coefficient of friction between the block of mass m1 and the surface in the diagramis µk. The system starts from rest. (You may assume m2 > m1.)

(a) What is the speed of the ball of mass m2 when it has fallen a distance h? [6 pts]

(b) Once ball m2 has fallen a distance h it strikes the ground making a loud noise.How far does the block m1 travel in total (from the point where it started at rest)along the surface before coming to a stop? (You can assume it does not collidewith the pulley.) [5 pts]

238 Chapter 8 Conservation of Energy

of the cannon? (b) At what point does the ball have maximum speed? (c) What is this maximum speed?

22. The coefficient of friction between the block of mass m1 5 3.00 kg and the surface in Figure P8.22 is mk 5 0.400. The system starts from rest. What is the speed of the ball of mass m2 5 5.00 kg when it has fallen a distance h 5 1.50 m?

23. A 5.00-kg block is set into motion up an inclined plane with an initial speed of vi 5 8.00 m/s (Fig. P8.23). The block comes to rest after trav-eling d 5 3.00 m along the plane, which is inclined at an angle of u 5 30.08 to the horizontal. For this motion, determine (a) the change in the block’s kinetic energy, (b) the change in the potential energy of the block–Earth system, and (c) the friction force exerted on the block (assumed to be constant). (d) What is the coef-ficient of kinetic friction?

24. A 1.50-kg object is held 1.20 m above a relaxed mass-less, vertical spring with a force constant of 320 N/m. The object is dropped onto the spring. (a) How far does the object compress the spring? (b) What If? Repeat part (a), but this time assume a constant air-resistance force of 0.700 N acts on the object during its motion. (c) What If? How far does the object compress the spring if the same experiment is performed on the Moon, where g 5 1.63 m/s2 and air resistance is neglected?

25. A 200-g block is pressed against a spring of force constant 1.40 kN/m until the block compresses the spring 10.0  cm. The spring rests at the bottom of a ramp inclined at 60.08 to the horizontal. Using energy considerations, determine how far up the incline the block moves from its initial position before it stops (a) if the ramp exerts no friction force on the block and (b) if the coefficient of kinetic friction is 0.400.

26. An 80.0-kg skydiver jumps out of a balloon at an alti-tude of 1 000 m and opens his parachute at an altitude of 200 m. (a) Assuming the total retarding force on the skydiver is constant at 50.0 N with the parachute closed and constant at 3 600 N with the parachute open, find the speed of the skydiver when he lands on the ground. (b) Do you think the skydiver will be injured? Explain. (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00 m/s? (d) How realistic is the assumption that the total retarding force is constant? Explain.

27. A child of mass m starts from rest and slides without friction from a height h along a slide next to a pool (Fig.  P8.27). She is launched from a height h/5 into the air over the pool. We wish to find the maximum height she reaches above the water in her projec-tile motion. (a) Is the child–Earth system isolated or

m1

m2

Figure P8.22

WAMT

d

u

vi

Figure P8.23

M

M

Q/C

Q/CS

GP

between box and floor is 0.300. Find (a) the work done by the applied force, (b) the increase in internal energy in the box–floor system as a result of friction, (c) the work done by the normal force, (d) the work done by the gravitational force, (e) the change in kinetic energy of the box, and (f) the final speed of the box.

17. A smooth circular hoop with a radius of 0.500 m is placed flat on the floor. A 0.400-kg particle slides around the inside edge of the hoop. The particle is given an initial speed of 8.00  m/s. After one revolu-tion, its speed has dropped to 6.00 m/s because of fric-tion with the floor. (a)  Find the energy transformed from mechanical to internal in the particle–hoop–floor system as a result of friction in one revolution. (b) What is the total number of revolutions the particle makes before stopping? Assume the friction force remains constant during the entire motion.

Section 8.4 Changes in Mechanical Energy for Nonconservative Forces 18. At time ti, the kinetic energy of a particle is 30.0 J and

the potential energy of the system to which it belongs is 10.0 J. At some later time tf , the kinetic energy of the particle is 18.0 J. (a) If only conservative forces act on the particle, what are the potential energy and the total energy of the system at time tf ? (b) If the poten-tial energy of the system at time tf is 5.00 J, are any non-conservative forces acting on the particle? (c) Explain your answer to part (b).

19. A boy in a wheelchair (total mass 47.0 kg) has speed 1.40 m/s at the crest of a slope 2.60 m high and 12.4 m long. At the bottom of the slope his speed is 6.20 m/s. Assume air resistance and rolling resistance can be modeled as a constant friction force of 41.0 N. Find the work he did in pushing forward on his wheels during the downhill ride.

20. As shown in Figure P8.20, a green bead of mass 25 g slides along a straight wire. The length of the wire from point ! to point " is 0.600 m, and point ! is 0.200 m higher than point ". A constant friction force of magnitude 0.025 0 N acts on the bead. (a) If the bead is released from rest at point !, what is its speed at point "? (b) A red bead of mass 25 g slides along a curved wire, subject to a friction force with the same constant magnitude as that on the green bead. If the green and red beads are released simultaneously from rest at point !, which bead reaches point " with a higher speed? Explain.

21. A toy cannon uses a spring to project a 5.30-g soft rub-ber ball. The spring is originally compressed by 5.00 cm and has a force constant of 8.00 N/m. When the cannon is fired, the ball moves 15.0 cm through the horizontal barrel of the cannon, and the barrel exerts a constant friction force of 0.032 0 N on the ball. (a) With what speed does the projectile leave the barrel

Q/C

!

"

Figure P8.20

Q/C

W

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