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Oscilações Harmônicas Simples, Exercícios de Fundamentos de Física

Universidade Federal de Minas Gerais (UFMG)Fundamentos de Física

Este documento aborda os conceitos fundamentais de oscilações harmônicas simples, incluindo a relação entre a aceleração, velocidade e deslocamento, as características do movimento oscilatório, como amplitude, período e frequência, bem como a energia cinética e potencial envolvidas nesse tipo de movimento. O documento também discute tópicos avançados, como oscilações amortecidas e forçadas, e suas implicações práticas. Com uma abordagem teórica e resolução de problemas, este material pode ser útil para estudantes de física, engenharia e áreas afins, tanto em nível de graduação quanto de pós-graduação, que buscam compreender profundamente os princípios e aplicações das oscilações harmônicas simples.

Tipologia: Exercícios

2024

Compartilhado em 16/07/2024

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Chapter 15: OSCILLATIONS
1. An oscillatory motion must be simple harmonic if:
A. the amplitude is small
B. the potential energy is equal to the kinetic energy
C. the motion is along the arc of a circle
D. the acceleration varies sinusoidally with time
E. the derivative, dU=dx,ofthepotentialenergyisnegative
ans: D
Section: 15{2; Di±culty: E
2. In simple harmonic motion, the magnitude of the acceleration is:
A. constant
B. proportional to the displacement
C. inversely proportional to the displacement
D. greatest when the velocity is greatest
E. never greater than g
ans: B
Section: 15{2; Di±culty: E
3. A particle is in simple harmonic motion with period T.Attimet= 0 it is at the equilibrium
point. Of the following times, at which time is it furthest from the equilibrium point?
A. 0:5T
B. 0:7T
C. T
D. 1:4T
E. 1:5T
ans: B
Section: 15{2; Di±culty: E
4. A particle moves back and forth along the xaxis from x=¡xmto x=+xm,insimple
harmonic motion with period T.Attimet= 0 it is at x=+xm.Whent=0:75T:
A. it is at x= 0 and is traveling toward x=+xm
B. it is at x= 0 and is traveling toward x=¡xm
C. it at x=+xmand is at rest
D. it is between x=0andx=+xmand is traveling toward x=¡xm
E. it is between x=0andx=¡xmand is traveling toward x=¡xm
ans: A
Section: 15{2; Di±culty: E
5. A particle oscillating in simple harmonic motion is:
A. never in equilibrium because it is in motion
B. never in equilibrium because there is always a force
C. in equilibrium at the ends of its path because its velocity is zero there
D. in equilibrium at the center of its path because the acceleration is zero there
E. in equilibrium at the ends of its path because the acceleration is zero there
ans: D
Section: 15{2; Di±culty: E
Chapter 15: OSCILLATIONS 255
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Chapter 15: OSCILLATIONS

1. An oscillatory motion must be simple harmonic if:

A. the amplitude is small

B. the potential energy is equal to the kinetic energy

C. the motion is along the arc of a circle

D. the acceleration varies sinusoidally with time

E. the derivative, dU=dx, of the potential energy is negative

ans: D

Section: 15{2; Di±culty: E

2. In simple harmonic motion, the magnitude of the acceleration is:

A. constant

B. proportional to the displacement

C. inversely proportional to the displacement

D. greatest when the velocity is greatest

E. never greater than g

ans: B

Section: 15{2; Di±culty: E

3. A particle is in simple harmonic motion with period T. At time t = 0 it is at the equilibrium

point. Of the following times, at which time is it furthest from the equilibrium point?

A. 0 : 5 T

B. 0 : 7 T

C. T

D. 1 : 4 T

E. 1 : 5 T

ans: B

Section: 15{2; Di±culty: E

4. A particle moves back and forth along the x axis from x = ¡x

m

to x = +x

m

, in simple

harmonic motion with period T. At time t = 0 it is at x = +x

m

. When t = 0: 75 T :

A. it is at x = 0 and is traveling toward x = +x

m

B. it is at x = 0 and is traveling toward x = ¡x

m

C. it at x = +x

m

and is at rest

D. it is between x = 0 and x = +x

m

and is traveling toward x = ¡x

m

E. it is between x = 0 and x = ¡x

m

and is traveling toward x = ¡x

m

ans: A

Section: 15{2; Di±culty: E

5. A particle oscillating in simple harmonic motion is:

A. never in equilibrium because it is in motion

B. never in equilibrium because there is always a force

C. in equilibrium at the ends of its path because its velocity is zero there

D. in equilibrium at the center of its path because the acceleration is zero there

E. in equilibrium at the ends of its path because the acceleration is zero there

ans: D

Section: 15{2; Di±culty: E

6. An object is undergoing simple harmonic motion. Throughout a complete cycle it:

A. has constant speed

B. has varying amplitude

C. has varying period

D. has varying acceleration

E. has varying mass

ans: D

Section: 15{2; Di±culty: E

7. When a body executes simple harmonic motion, its acceleration at the ends of its path must

be:

A. zero

B. less than g

C. more than g

D. suddenly changing in sign

E. none of these

ans: E

Section: 15{2; Di±culty: E

8. A particle is in simple harmonic motion with period T. At time t = 0 it is halfway between

the equilibrium point and an end point of its motion, traveling toward the end point. The next

time it is at the same place is:

A. t = T

B. t = T = 2

C. t = T = 4

D. t = T = 8

E. none of the above

ans: E

Section: 15{2; Di±culty: E

9. An object attached to one end of a spring makes 20 complete oscillations in 10 s. Its period is:

A. 2 Hz

B. 10 s

C. 0 :5 Hz

D. 2 s

E. 0 :50 s

ans: E

Section: 15{2; Di±culty: E

10. An object attached to one end of a spring makes 20 vibrations in 10 s. Its frequency is:

A. 2 Hz

B. 10 s

C. 0 :05 Hz

D. 2 s

E. 0 :50 s

ans: A

Section: 15{2; Di±culty: E

16. In simple harmonic motion, the displacement is maximum when the:

A. acceleration is zero

B. velocity is maximum

C. velocity is zero

D. kinetic energy is maximum

E. momentum is maximum

ans: C

Section: 15{2; Di±culty: E

17. In simple harmonic motion:

A. the acceleration is greatest at the maximum displacement

B. the velocity is greatest at the maximum displacement

C. the period depends on the amplitude

D. the acceleration is constant

E. the acceleration is greatest at zero displacement

ans: A

Section: 15{2; Di±culty: E

18. The amplitude and phase constant of an oscillator are determined by:

A. the frequency

B. the angular frequency

C. the initial displacement alone

D. the initial velocity alone

E. both the initial displacement and velocity

ans: E

Section: 15{2; Di±culty: E

19. Two identical undamped oscillators have the same amplitude of oscillation only if:

A. they are started with the same displacement x

B. they are started with the same velocity v

C. they are started with the same phase

D. they are started so the combination!

x

+ v

is the same

E. they are started so the combination x

v

is the same

ans: D

Section: 15{2; Di±culty: M

20. The amplitude of any oscillator can be doubled by:

A. doubling only the initial displacement

B. doubling only the initial speed

C. doubling the initial displacement and halving the initial speed

D. doubling the initial speed and halving the initial displacement

E. doubling both the initial displacement and the initial speed

ans: E

Section: 15{2; Di±culty: M

21. It is impossible for two particles, each executing simple harmonic motion, to remain in phase

with each other if they have di®erent:

A. masses

B. periods

C. amplitudes

D. spring constants

E. kinetic energies

ans: B

Section: 15{2; Di±culty: E

22. The acceleration of a body executing simple harmonic motion leads the velocity by what phase?

A. 0

B. ¼=8 rad

C. ¼=4 rad

D. ¼=2 rad

E. ¼ rad

ans: D

Section: 15{2; Di±culty: E

23. The displacement of an object oscillating on a spring is given by x(t) = x

m

cos(!t + Á). If the

initial displacement is zero and the initial velocity is in the negative x direction, then the phase

constant Á is:

A. 0

B. ¼=2 rad

C. ¼ rad

D. 3 ¼=2 rad

E. 2 ¼ rad

ans: B

Section: 15{2; Di±culty: M

24. The displacement of an object oscillating on a spring is given by x(t) = x

m

cos(!t + Á). If

the object is initially displaced in the negative x direction and given a negative initial velocity,

then the phase constant Á is between:

A. 0 and ¼=2 rad

B. ¼=2 and ¼ rad

C. ¼ and 3¼=2 rad

D. 3 ¼=2 and 2¼ rad

E. none of the above (Á is exactly 0, ¼=2, ¼, or 3¼=2 rad)

ans: B

Section: 15{2; Di±culty: M

29. A simple harmonic oscillator consists of an particle of mass m and an ideal spring with spring

constant k. The particle oscillates as shown in (i) with period T. If the spring is cut in half

and used with the same particle, as shown in (ii), the period will be:

m

(i)

m

(ii)

A. 2 T

B.

p

2 T

C. T =

p

D. T

E. T = 2

ans: C

Section: 15{3; Di±culty: M

30. An object of mass m, oscillating on the end of a spring with spring constant k, has amplitude

A. Its maximum speed is:

A. A

p

k=m

B. A

k=m

C. A

p

m=k

D. Am=k

E. A

m=k

ans: A

Section: 15{2, 3; Di±culty: M

31. A 0:20-kg object attached to a spring whose spring constant is 500 N=m executes simple har-

monic motion. If its maximum speed is 5:0 m=s, the amplitude of its oscillation is:

A. 0 :0020 m

B. 0 :10 m

C. 0 :20 m

D. 25 m

E. 250 m

ans: B

Section: 15{2, 3; Di±culty: M

32. Let U be the potential energy (with the zero at zero displacement) and K be the kinetic energy

of a simple harmonic oscillator. U

avg

and K

avg

are the average values over a cycle. Then:

A. K

avg

> U

avg

B. K

avg

< U

avg

C. K

avg

= U

avg

D. K = 0 when U = 0

E. K + U = 0

ans: C

Section: 15{4; Di±culty: M

33. A particle is in simple harmonic motion along the x axis. The amplitude of the motion is x

m

At one point in its motion its kinetic energy is K = 5 J and its potential energy (measured

with U = 0 at x = 0) is U = 3 J. When it is at x = x

m

, the kinetic and potential energies are:

A. K = 5 J and U = 3 J

B. K = 5 J and U = ¡3 J

C. K = 8 J and U = 0

D. K = 0 and U = 8 J

E. K = 0 and U = ¡8 J

ans: D

Section: 15{4; Di±culty: M

34. A particle is in simple harmonic motion along the x axis. The amplitude of the motion is x

m

When it is at x = x

, its kinetic energy is K = 5 J and its potential energy (measured with

U = 0 at x = 0) is U = 3 J. When it is at x = ¡

x

, the kinetic and potential energies are:

A. K = 5 J and U = 3 J

B. K = 5 J and U = ¡3 J

C. K = 8 J and U = 0

D. K = 0 and U = 8 J

E. K = 0 and U = ¡8 J

ans: A

Section: 15{4; Di±culty: M

35. A 0:25-kg block oscillates on the end of the spring with a spring constant of 200 N=m. If the

system has an energy of 6:0 J, then the amplitude of the oscillation is:

A. 0 :06 m

B. 0 :17 m

C. 0 :24 m

D. 4 :9 m

E. 6 :9 m

ans: C

Section: 15{4; Di±culty: M

40. A block attached to a spring undergoes simple harmonic motion on a horizontal frictionless

surface. Its total energy is 50 J. When the displacement is half the amplitude, the kinetic

energy is:

A. zero

B. 12 :5 J

C. 25 J

D. 37 :5 J

E. 50 J

ans: D

Section: 15{4; Di±culty: M

41. A mass-spring system is oscillating with amplitude A. The kinetic energy will equal the po-

tential energy only when the displacement is:

A. zero

B. §A= 4

C. §A=

p

D. §A= 2

E. anywhere between ¡A and +A

ans: C

Section: 15{4; Di±culty: M

42. If the length of a simple pendulum is doubled, its period will:

A. halve

B. be greater by a factor of

p

C. be less by a factor of

p

D. double

E. remain the same

ans: B

Section: 15{6; Di±culty: E

43. The period of a simple pendulum is 1 s on Earth. When brought to a planet where g is one-tenth

that on Earth, its period becomes:

A. 1 s

B. 1 =

p

10 s

C. 1 =10 s

D.

p

10 s

E. 10 s

ans: D

Section: 15{6; Di±culty: E

44. The amplitude of oscillation of a simple pendulum is increased from 1

to 4

. Its maximum

acceleration changes by a factor of:

A. 1 = 4

B. 1 = 2

C. 2

D. 4

E. 16

ans: D

Section: 15{6; Di±culty: M

45. A simple pendulum of length L and mass M has frequency f. To increase its frequency to 2f :

A. increase its length to 4L

B. increase its length to 2L

C. decrease its length to L= 2

D. decrease its length to L= 4

E. decrease its mass to < M= 4

ans: D

Section: 15{6; Di±culty: M

46. A simple pendulum consists of a small ball tied to a string and set in oscillation. As the

pendulum swings the tension force of the string is:

A. constant

B. a sinusoidal function of time

C. the square of a sinusoidal function of time

D. the reciprocal of a sinusoidal function of time

E. none of the above

ans: E

Section: 15{6; Di±culty: E

47. A simple pendulum has length L and period T. As it passes through its equilibrium position,

the string is suddenly clamped at its midpoint. The period then becomes:

A. 2 T

B. T

C. T = 2

D. T = 4

E. none of these

ans: E

Section: 15{6; Di±culty: M

52. The rotational inertia of a uniform thin rod about its end is M L

=3, where M is the mass

and L is the length. Such a rod is hung vertically from one end and set into small amplitude

oscillation. If L = 1:0 m this rod will have the same period as a simple pendulum of length:

A. 33 cm

B. 50 cm

C. 67 cm

D. 100 cm

E. 150 cm

ans: C

Section: 15{6; Di±culty: M

53. Two uniform spheres are pivoted on horizontal axes that are tangent to their surfaces. The

one with the longer period of oscillation is the one with:

A. the larger mass

B. the smaller mass

C. the larger rotational inertia

D. the smaller rotational inertia

E. the larger radius

ans: E

Section: 15{6; Di±culty: M

54. The x and y coordinates of a point each execute simple harmonic motion. The result might be

a circular orbit if:

A. the amplitudes are the same but the frequencies are di®erent

B. the amplitudes and frequencies are both the same

C. the amplitudes and frequencies are both di®erent

D. the phase constants are the same but the amplitudes are di®erent

E. the amplitudes and the phase constants are both di®erent

ans: B

Section: 15{7; Di±culty: E

55. The x and y coordinates of a point each execute simple harmonic motion. The frequencies are

the same but the amplitudes are di®erent. The resulting orbit might be:

A. an ellipse

B. a circle

C. a parabola

D. a hyperbola

E. a square

ans: A

Section: 15{7; Di±culty: E

56. For an oscillator subjected to a damping force proportional to its velocity:

A. the displacement is a sinusoidal function of time.

B. the velocity is a sinusoidal function of time.

C. the frequency is a decreasing function of time.

D. the mechanical energy is constant.

E. none of the above is true.

ans: E

Section: 15{8; Di±culty: E

57. Five particles undergo damped harmonic motion. Values for the spring constant k, the damping

constant b, and the mass m are given below. Which leads to the smallest rate of loss of

mechanical energy?

A. k = 100 N=m, m = 50 g, b = 8 g=s

B. k = 150 N=m, m = 50 g, b = 5 g=s

C. k = 150 N=m, m = 10 g, b = 8 g=s

D. k = 200 N=m, m = 8 g, b = 6 g=s

E. k = 100 N=m, m = 2 g, b = 4 g=s

ans: B

Section: 15{8; Di±culty: M

58. A sinusoidal force with a given amplitude is applied to an oscillator. To maintain the largest

amplitude oscillation the frequency of the applied force should be:

A. half the natural frequency of the oscillator

B. the same as the natural frequency of the oscillator

C. twice the natural frequency of the oscillator

D. unrelated to the natural frequency of the oscillator

E. determined from the maximum speed desired

ans: B

Section: 15{9; Di±culty: E

59. A sinusoidal force with a given amplitude is applied to an oscillator. At resonance the amplitude

of the oscillation is limited by:

A. the damping force

B. the initial amplitude

C. the initial velocity

D. the force of gravity

E. none of the above

ans: A

Section: 15{9; Di±culty: E