chapter (15)

Chapter 15

Oscillatory Motion

Multiple Choice 1. A body of mass 5.0 kg is suspended by a spring which stretches 10 cm when the

mass is attached. It is then displaced downward an additional 5.0 cm and released. Its position as a function of time is approximately

a. y = .10 sin 9.9t b. y = .10 cos 9.9t c. y = .10 cos (9.9t + .1) d. y = .10 sin (9.9t + 5) e. y = .05 cos 9.9t

2. A body oscillates with simple harmonic motion along the x-axis. Its displacement varies with time according to the equation x = 5.0 sin (πt). The acceleration (in m/s2) of the body at t = 1.0 s is approximately

a. 3.5 b. 49 c. 14 d. 43 e. 4.3

3. A body oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x = 5 sin (πt + π/3). The phase (in rad) of the motion at t = 2 s is

a. 7π/3 b. π/3 c. π d. 5π/3 e. 2π

4. A body oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x = 5.0 sin (πt + π/3). The velocity (in m/s) of the body at t = 1.0 s is

a. +8.0 b. –8.0 c. –14 d. +14 e. –5.0

277

278 CHAPTER 15

5. The motion of a particle connected to a spring is described by x = 10 sin (πt). At what time (in s) is the potential energy equal to the kinetic energy?

a. 0 b. 0.25 c. 0.50 d. 0.79 e. 1.0

6. The amplitude of a system moving with simple harmonic motion is doubled. The total energy will then be

a. 4 times larger b. 3 times larger c. 2 times larger d. the same as it was e. half as much

7. A mass m = 2.0 kg is attached to a spring having a force constant k = 290 N/m as in the figure. The mass is displaced from its equilibrium position and released. Its frequency of oscillation (in Hz) is approximately

k

mmm

a. 12 b. 0.50 c. 0.01 d. 1.9 e. 0.08

8. The mass in the figure slides on a frictionless surface. If m = 2 kg, k1 = 800 N/m and k2 = 500 N/m, the frequency of oscillation (in Hz) is approximately

k1 k2

mmm

a. 6 b. 2 c. 4 d. 8 e. 10

Oscillatory Motion 279

9. Two circus clowns (each having a mass of 50 kg) swing on two flying trapezes (negligible mass, length 25 m) shown in the figure. At the peak of the swing, one grabs the other, and the two swing back to one platform. The time for the forward and return motion is

a. 10 s b. 50 s c. 15 s d. 20 s e. 25 s

10. A uniform rod (mass m = 1.0 kg and length L = 2.0 m) pivoted at one end oscillates in a vertical plane as shown below. The period of oscillation (in s) is approximately

a. 4.0 b. 1.6 c. 3.2 d. 2.3 e. 2.0

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11. A horizontal plank (m = 2.0 kg, L = 1.0 m) is pivoted at one end. A spring (k = 1.0 × 103 N/m) is attached at the other end, as shown in the figure. Find the angular frequency (in rad/s) for small oscillations.

tθ

a. 39 b. 44 c. 55 d. 66 e. 25

12. The figure shows a uniform rod (length L = 1.0 m, mass = 2.0 kg) suspended from a pivot a distance d = 0.25 m above its center of mass. The angular frequency (in rad/s) for small oscillations is approximately

axisaxis

a. 1.0 b. 2.5 c. 1.5 d. 4.1 e. 3.5

13. In the figure below, a disk (radius R = 1.0 m, mass = 2.0 kg) is suspended from a pivot a distance d = 0.25 m above its center of mass. The angular frequency (in rad/s) for small oscillations is approximately

axisaxis

a. 4.2 b. 2.1 c. 1.5 d. 1.0 e. 3.8


14. In the figure below, a hoop (radius R = 1.0 m, mass = 2.0 kg) having four spokes of negligible mass is suspended from a pivot a distance d = .25 m above its center of mass. The angular frequency (in rad/s) for small oscillations is approximately

axisaxis

a. 4.0 b. 2.5 c. 1.5 d. 1.0 e. 0.5

15. A torsional pendulum consists of a solid disk (mass = 2.0 kg, radius = 1.0 m) suspended by a wire attached to a rigid support. The body oscillates about the support wire. If the torsion constant is 16 N ⋅ m. What is the angular frequency (in rad/s)?

a. 2 b. 4 c. 6 d. 8 e. 7

16. The mass in the figure below slides on a frictionless surface. When the mass is pulled out, spring 1 is stretched a distance x1 from its equilibrium position and spring 2 is stretched a distance x2. The spring constants are k1 and k2 respectively. The force pulling back on the mass is:

k1 k2

mmm

a. –k2x1. b. –k2x2. c. –(k1x1 + k2x2).

d. – )(2 21

21 xxkk

++

.

e. – )( 2121

21 xxkkkk

++

.

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17. A hoop, a solid cylinder, and a solid sphere all have the same mass m and the same radius R. Each is mounted to oscillate about an axis a distance 0.5 R from the center. The axis is perpendicular to the circular plane of the hoop and the cylinder and to an equatorial plane of the sphere as shown below. Which is the correct ranking in order of increasing angular frequency ω?

a. hoop, cylinder, sphere b. cylinder, sphere, hoop c. sphere, cylinder, hoop d. hoop, sphere, cylinder e. sphere, hoop, cylinder

18. Three pendulums with strings of the same length and bobs of the same mass are pulled out to angles θ1, θ2 and θ3 respectively and released. The approximation sin θ = θ holds for all three angles, with θ3 > θ2 > θ1. How do the angular frequencies of the three pendulums compare?

a. ω3 > ω2 > ω1

b. Need to know amplitudes to answer this question. c. Need to know Lg/ to answer this question.

d. ω1 > ω2 > ω3

e. ω1 = ω2 = ω3

19. A weight of mass m is at rest at O when suspended from a spring, as shown. When it is pulled down and released, it oscillates between positions A and B. Which statement about the system consisting of the spring and the mass is correct?

mm

mm

m

m

O

B

A

x

a. The gravitational potential energy of the system is greatest at A. b. The elastic potential energy of the system is greatest at O. c. The rate of change of momentum has its greatest magnitude at A and B. d. The rate of change of gravitational potential energy is smallest at O. e. The rate of change of gravitational potential energy has its greatest

magnitude at A and B.


20. An object of mass m is attached to string of length L. When it is released from point A, the object oscillates between points A and B. Which statement about the system consisting of the pendulum and the Earth is correct?

A

O

B

a. The gravitational potential energy of the system is greatest at A and B. b. The kinetic energy of mass m is greatest at point O. c. The greatest rate of change of momentum occurs at A and B. d. All of the above are correct. e. Only (a) and (b) above are correct.

21. A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below. A point or points at which the object has positive velocity and zero acceleration is(are)

–4 –3 –2 –1

–1

1

1 2 3 4

x (m)

t (s)A

B

C

D

E

a. B b. C c. D d. B or D e. A or E

22. A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below. The point at which the object has negative velocity and zero acceleration is

–4 –3 –2 –1

–1

1

1 2 3 4

x (m)

t (s)A

B

C

D

E

a. A b. B c. C d. D e. E

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23. A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below. The point at which the object has zero velocity and positive acceleration is

–4 –3 –2 –1

–1

1

1 2 3 4

x (m)

t (s)A

B

C

D

E


24. A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below. The point at which the object has zero velocity and negative acceleration is

–4 –3 –2 –1

–1

1

1 2 3 4

x (m)

t (s)A

B

C

D

E


25. In an inertia balance, a body supported against gravity executes simple harmonic oscillations in a horizontal plane under the action of a set of springs. If a 1.00 kg body vibrates at 1.00 Hz, a 2.00 kg body will vibrate at

a. 0.500 Hz. b. 0.707 Hz. c. 1.00 Hz. d. 1.41 Hz. e. 2.00 Hz.


26. At sea level, at a latitude where g = 9.80 ms2 , a pendulum that takes 2.00 s for a

complete swing back and forth has a length of 0.993 m. What is the value of g in m/s2 at a location where the length of such a pendulum is 0.970 m?

a. 0.0983 b. 3.05 c. 9.28 d. 10.0 e. 38.3

27. Suppose it were possible to drill a frictionless cylindrical channel along a diameter of the Earth from one side of the Earth to another. A body dropped into such a channel will only feel the gravitational pull of mass within a sphere of radius equal to the distance of the mass from the center of the Earth . The density

of the Earth is 5.52×103 kgm3 and G = 6.67 ×10−11 N ⋅m2

kg2 . The mass will oscillate

with a period of

a. 84.4 min. b. 169 min. c. 24.0 h. d. 1130 h. e. 27.2 d.

28. A 2.00 m-long 6.00 kg ladder pivoted at the top hangs down from a platform at the circus. A 42.0 kg trapeze artist climbs to a point where her center of mass is at the center of the ladder and swings at the systems natural frequency. The angular frequency (in ) of the system of ladder and woman is s−1

a. 1.01. b. 2.01. c. 4.03. d. 8.05. e. 16.2.

29. Ellen says that whenever the acceleration is directly proportional to the displacement of an object from its equilibrium position, the motion of the object is simple harmonic motion. Mary says this is true only if the acceleration is opposite in direction to the displacement. Which one, if either, is correct?

a. Ellen, because ω 2 is directly proportional to the constant multiplying the displacement and to the mass.

b. Ellen, because ω 2 is directly proportional to the mass. c. Mary, , because ω 2 is directly proportional to the constant multiplying the

displacement and to the mass. d. Mary, because ω 2 is directly proportional to the mass. e. Mary, because the second derivative of an oscillatory function like sin(ωt)

or cos(ωt) always is the negative of the original function.

286 CHAPTER 15

30. John says that the value of the function cos[ω(t + T) +ϕ] , obtained one period T after time t, is greater than cos(ωt + ϕ) by 2π . Larry says that it is greater by the addition of 1.00 to cos(ωt + ϕ) . Which one, if either, is correct?

a. John, because ωT = 2π . b. John, because ωT =1 radian. c. Larry, because ωT = 2π . d. Larry, because ωT =1 radian. e. Neither, because cos(θ + 2π ) = cosθ .

31. Simple harmonic oscillations can be modeled by the projection of circular motion at constant angular velocity onto a diameter of the circle. When this is done, the analog along the diameter of the centripetal acceleration of the particle executing circular motion is

a. the displacement from the center of the diameter of the projection of the position of the particle on the circle.

b. the projection along the diameter of the velocity of the particle on the circle. c. the projection along the diameter of tangential acceleration of the particle on

the circle. d. the projection along the diameter of centripetal acceleration of the particle

on the circle. e. meaningful only when the particle moving in the circle also has a non-zero

tangential acceleration.

32. When a damping force is applied to a simple harmonic oscillator which has angular frequency ω0 in the absence of damping, the new angular frequency ω is such that

a. 0

ω ω< .

b. 0ω ω= .

c. 0

ω ω> .

d. . 0 0

T Tω ω<

e. . 0 0

T Tω ω>

33. When a damping force is applied to a simple harmonic oscillator which has period T in the absence of damping, the new period T is such that 0

a. . 0

T T<

b. . 0

T T=

c. . 0

T T>

d. . 0 0

T Tω ω<

e. . 0 0

T Tω ω>


34. To double the total energy of a mass oscillating at the end of a spring with

amplitude A, we need to

a. increase the angular frequency by 2 . b. increase the amplitude by 2 . c. increase the amplitude by 2. d. increase the angular frequency by 2.

e. increase the amplitude by 4 and decrease the angular frequency by 12

.

Open-Ended Problems

35. An automobile (m = 1.00 × 103 kg) is driven into a brick wall in a safety test. The bumper behaves like a spring (k = 5.00 × 106 N/m), and is observed to compress a distance of 3.16 cm as the car is brought to rest. What was the initial speed of the automobile?

36. The mat of a trampoline is held by 32 springs, each having a spring constant of 5000 N/m. A person with a mass of 40.0 kg jumps from a platform 1.93 m high onto the trampoline. Determine the stretch of each of the springs.

37. An archer pulls her bow string back 0.4 m by exerting a force that increases uniformly from zero to 240 N. What is the equivalent spring constant of the bow, and how much work is done in pulling the bow?

38. An ore car of mass 4000 kg starts from rest and rolls downhill on tracks from a mine. A spring with k = 400 000 N/m is located at the end of the tracks. At the spring’s maximum compression, the car is at an elevation 10 m lower than its elevation at the starting point. How much is the spring compressed in stopping the ore car? Ignore friction.

39. The motion of a piston in an auto engine is simple harmonic. If the piston travels back and forth over a distance of 10 cm, and the piston has a mass of 1.5 kg, what is the maximum speed of the piston and the maximum force acting on the piston when the engine is running at 4200 rpm?

288 CHAPTER 15


Chapter 15

Oscillatory Motion

1. e

2. d

3. a

4. b

5. b

6. a

7. d

8. c

9. a

10. d

11. a

12. d

13. b

14. c

15. b

16. b

17. a

18. e

19. c

20. d

21. e

22. c

23. d

24. b

25. b

26. c

27. a

28. c

29. e

30. e

31. d

32. a

33. c

34. b

35. 2.23 m/s (5 mph)

36. 9.97 cm

37. 600 N/m, 48 J

38. 1.4 m

39. 22 m/s, 14 500 N

290 CHAPTER 15