Content check — College Physics (AP), Chapter 16: Oscillatory Motion and Waves

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 16: Oscillatory Motion and Waves

55 problems · 2 figures

Fish are hung on a spring scale to determine their mass (most fishermen feel no obligation to truthfully report the mass).

(a) What is the force constant of the spring in such a scale if it the spring stretches 8.00 cm for a 10.0 kg load?

(b) What is the mass of a fish that stretches the spring 5.50 cm?

Answer

(a) $1 . 23 \times 10^{3} N/m$

(b) $6 . 88 kg$

16-3 §16.1 source ↗

One type of BB gun uses a spring-driven plunger to blow the BB from its barrel. (a) Calculate the force constant of its plunger’s spring if you must compress it 0.150 m to drive the 0.0500-kg plunger to a top speed of 20.0 m/s. (b) What force must be exerted to compress the spring?

Answer

(a) 889 N/m

(b) 133 N

16-5 §16.1 source ↗

When an 80.0-kg man stands on a pogo stick, the spring is compressed 0.120 m.

(a) What is the force constant of the spring? (b) Will the spring be compressed more when he hops down the road?

Answer

(a) $6 . 53 \times 10^{3} N/m$

(b) Yes

16-7 §16.2 source ↗

What is the period of

60.0 Hz

electrical power?

Answer

16.7 ms

16-8 §16.2 source ↗

If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds?

Answer

0.400 s / beats

16-9 §16.2 source ↗

Find the frequency of a tuning fork that takes

2 . 50 \times 10^{- 3} s

to complete one oscillation.

Answer

400 Hz

16-10 §16.2 source ↗

A stroboscope is set to flash every

8 . 00 \times 10^{- 5} s

. What is the frequency of the flashes?

Answer

12,500 Hz

16-11 §16.2 source ↗

A tire has a tread pattern with a crevice every 2.00 cm. Each crevice makes a single vibration as the tire moves. What is the frequency of these vibrations if the car moves at 30.0 m/s?

Answer

1.50 kHz

16-12 §16.2 source ↗

Engineering Application

Each piston of an engine makes a sharp sound every other revolution of the engine. (a) How fast is a race car going if its eight-cylinder engine emits a sound of frequency 750 Hz, given that the engine makes 2000 revolutions per kilometer? (b) At how many revolutions per minute is the engine rotating?

Answer

(a) 93.8 m/s

(b) $11 . 3 \times 10^{3} rev/min$

16-13 §16.3 source ↗

A type of cuckoo clock keeps time by having a mass bouncing on a spring, usually something cute like a cherub in a chair. What force constant is needed to produce a period of 0.500 s for a 0.0150-kg mass?

Answer

2.37 N/m

16-15 §16.3 source ↗

A 0.500-kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to the object to change the period to 2.00 s?

Answer

0.389 kg

16-18 §16.3 source ↗

A diver on a diving board is undergoing simple harmonic motion. Her mass is 55.0 kg and the period of her motion is 0.800 s. The next diver is a male whose period of simple harmonic oscillation is 1.05 s. What is his mass if the mass of the board is negligible?

Answer

94.7 kg

16-21 §16.3 source ↗

A 90.0-kg skydiver hanging from a parachute bounces up and down with a period of 1.50 s. What is the new period of oscillation when a second skydiver, whose mass is 60.0 kg, hangs from the legs of the first, as seen in Figure 16.43.

The figure shows two skydivers midway through the air, with both with open having their parachutes open. — Figure 16.43 in the book ↗

Answer

1.94 s

16-22 §16.4 source ↗

As usual, the acceleration due to gravity in these problems is taken to be $g = 9.80 m / s^{2}$ , unless otherwise specified.

What is the length of a pendulum that has a period of 0.500 s?

Answer

6.21 cm

16-24 §16.4 source ↗

As usual, the acceleration due to gravity in these problems is taken to be $g = 9.80 m / s^{2}$ , unless otherwise specified.

What is the period of a 1.00-m-long pendulum?

Answer

2.01 s

16-26 §16.4 source ↗

As usual, the acceleration due to gravity in these problems is taken to be $g = 9.80 m / s^{2}$ , unless otherwise specified.

The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?

Answer

2.23 Hz

16-28 §16.4 source ↗

As usual, the acceleration due to gravity in these problems is taken to be $g = 9.80 m / s^{2}$ , unless otherwise specified.

(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is $9.79 {m/s}^{2}$ is moved to a location where the acceleration due to gravity is $9.82 {m/s}^{2}$ . What is its new period? (b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.

Answer

(a) 2.99541 s

(b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by $(0.01)^{2} =$ $0 . 01%$ so it is necessary to have at least 4 digits after the decimal to see the changes.

16-30 §16.4 source ↗

As usual, the acceleration due to gravity in these problems is taken to be $g = 9.80 m / s^{2}$ , unless otherwise specified.

(a) What is the effect on the period of a pendulum if you double its length?

(b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?

Answer

(a) Period increases by a factor of 1.41 ( $\sqrt{2}$ )

(b) Period decreases to 97.5% of old period

16-32 §16.4 source ↗

As usual, the acceleration due to gravity in these problems is taken to be $g = 9.80 m / s^{2}$ , unless otherwise specified.

At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is $1.63 {m/s}^{2}$ , if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock’s minute hand, which on Earth makes one revolution per hour, to make one revolution on the Moon.

Answer

Slow by a factor of 2.45

16-34 §16.4 source ↗

As usual, the acceleration due to gravity in these problems is taken to be $g = 9.80 m / s^{2}$ , unless otherwise specified.

If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?

Answer

length must increase by 0.0116%.

16-35 §16.5 source ↗

The length of nylon rope from which a mountain climber is suspended has a force constant of $1.40 \times 10^{4} N/m$

(a) What is the frequency at which he bounces, given his mass plus and the mass of his equipment are 90.0 kg? Ignore the change in gravitational potential energy after the cord begins to stretch.

(b) How much would this rope stretch to break the climber’s fall if he free-falls 2.00 m before the rope runs out of slack? Hint: Use conservation of energy.

Answer

(a) $1.99 Hz$

(b) 50.2 cm

16-36 §16.5 source ↗

Engineering Application

Near the top of the Citigroup Center building in New York City, there is an object with mass of $4 . 00 \times 10^{5} kg$ on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven—the driving force is transferred to the object, which oscillates instead of the entire building. (a) What effective force constant should the springs have to make the object oscillate with a period of 2.00 s? (b) What energy is stored in the springs for a 2.00-m displacement from equilibrium?

Answer

(a) $3 . 95 \times 10^{6} N/m$

(b) $7 . 90 \times 10^{6} J$

16-37 §16.6 source ↗

(a)What is the maximum velocity of an 85.0-kg person bouncing on a bathroom scale having a force constant of

1 . 50 \times 10^{6} N/m

, if the amplitude of the bounce is 0.200 cm? (b)What is the maximum energy stored in the spring?

Answer

a). 0.266 m/s

b). 3.00 J

16-39 §16.6 source ↗

At what positions is the speed of a simple harmonic oscillator half its maximum? That is, what values of

x / X

give

v = \pm v_{max} / 2

, where $X$ is the amplitude of the motion?

Answer

\pm \frac{\sqrt{3}}{2}

16-42 §16.8 source ↗

How much energy must the shock absorbers of a 1200-kg car dissipate in order to damp a bounce that initially has a velocity of 0.800 m/s at the equilibrium position? Assume the car returns to its original vertical position.

Answer

384 J

16-44 §16.8 source ↗

(a) How much will a spring that has a force constant of 40.0 N/m be stretched by an object with a mass of 0.500 kg when hung motionless from the spring? (b) Calculate the decrease in gravitational potential energy of the 0.500-kg object when it descends this distance. (c) Part of this gravitational energy goes into the spring. Calculate the energy stored in the spring by this stretch, and compare it with the gravitational potential energy. Explain where the rest of the energy might go.

Answer

(a). 0.123 m

(b). −0.600 J

(c). 0.300 J. The rest of the energy may go into heat caused by friction and other damping forces.

16-46 §16.8 source ↗

Engineering Application: A suspension bridge oscillates with an effective force constant of

1 . 00 \times 10^{8} N/m

. (a) How much energy is needed to make it oscillate with an amplitude of 0.100 m? (b) If soldiers march across the bridge with a cadence equal to the bridge’s natural frequency and impart

1 . 00 \times 10^{4} J

of energy each second, how long does it take for the bridge’s oscillations to go from 0.100 m to 0.500 m amplitude?

Answer

(a) $5 . 00 \times 10^{5} J$

(b) $1.20 × 10^{3}$ s

16-47 §16.9 source ↗

Storms in the South Pacific can create waves that travel all the way to the California coast, which are 12,000 km away. How long does it take them if they travel at 15.0 m/s?

Answer

t = 9 . 26 d

16-49 §16.9 source ↗

Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency?

Answer

f = 40 . 0 Hz

16-51 §16.9 source ↗

Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 8.00 m apart. If they shake the bridge twice per second, what is the propagation speed of the waves?

Answer

v_{w} = 16 . 0 m/s

16-53 §16.9 source ↗

What is the wavelength of an earthquake that shakes you with a frequency of 10.0 Hz and gets to another city 84.0 km away in 12.0 s?

Answer

λ = 700 m

16-55 §16.9 source ↗

Your ear is capable of differentiating sounds that arrive at the ear just 1.00 ms apart. What is the minimum distance between two speakers that produce sounds that arrive at noticeably different times on a day when the speed of sound is 340 m/s?

Answer

d = 34 . 0 cm

16-57 §16.10 source ↗

A car has two horns, one emitting a frequency of 199 Hz and the other emitting a frequency of 203 Hz. What beat frequency do they produce?

Answer

f = 4 Hz

16-59 §16.10 source ↗

Two tuning forks having frequencies of 460 and 464 Hz are struck simultaneously. What average frequency will you hear, and what will the beat frequency be?

Answer

462 Hz,

4 Hz

16-61 §16.10 source ↗

A wave traveling on a Slinky® that is stretched to 4 m takes 2.4 s to travel the length of the Slinky and back again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?

Answer

(a) 3.33 m/s

(b) 1.25 Hz

16-63 §16.11 source ↗

Medical Application

Ultrasound of intensity $1 . 50 \times 10^{2} {W/m}^{2}$ is produced by the rectangular head of a medical imaging device measuring 3.00 by 5.00 cm. What is its power output?

Answer

0.225 W

16-65 §16.11 source ↗

To increase intensity of a wave by a factor of 50, by what factor should the amplitude be increased?

Answer

7.07

16-67 §16.11 source ↗

Astronomy Application

Energy from the Sun arrives at the top of the Earth’s atmosphere with an intensity of $1.30 {kW/m}^{2} .$ How long does it take for $1.8 \times 10^{9} J$ to arrive on an area of $1.00 m^{2}$ ?

Answer

16.0 d

16-68 §16.11 source ↗

Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity. If the device produces 10.0 kW of power on a day when the breakers are 1.20 m high, how much will it produce when they are 0.600 m high?

Answer

2.50 kW

16-70 §16.11 source ↗

A microphone receiving a pure sound tone feeds an oscilloscope, producing a wave on its screen. If the sound intensity is originally

2.00 \times 10^{–5} {W/m}^{2},

but is turned up until the amplitude increases by 30.0%, what is the new intensity?

Answer

3.38 \times 10^{–5} {W/m}^{2}

AP Test Prep 16.1 §16.1 source ↗

Which of the following represents the distance (how much ground the particle covers) moved by a particle in a simple harmonic motion in one time period? (Here, A represents the amplitude of the oscillation.)

a.
0 cm
b.
A cm
c.
2A cm
d.
4A cm

Answer

D. 4A cm

AP Test Prep 16.3 §16.1 source ↗

In the formula

F = - k x

, what does the minus sign indicate?

a.
It indicates that the restoring force is in the direction of the displacement.
b.
It indicates that the restoring force is in the direction opposite the displacement.
c.
It indicates that mechanical energy in the system decreases when a system undergoes oscillation.
d.
None of the above

Answer

B. It indicates that the restoring force is in the direction opposite the displacement.

AP Test Prep 16.5 §16.2 source ↗

A mass attached to a spring oscillates and completes 50 full cycles in 30 s. What is the time period and frequency of this system?

Answer

The frequency is given by

$f = \frac{1}{T} = \frac{50 c y c l e s}{30 s} = 1.66 H z$

Time period is:

$T = \frac{1}{f} = \frac{1}{1.66} = 0.6 s$

AP Test Prep 16.7 §16.3 source ↗

A particle of mass 100 g undergoes a simple harmonic motion. The restoring force is provided by a spring with a spring constant of 40 N∙m⁻¹. What is the period of oscillation?

a.
10π
b.
0.5π
c.
0.1π
d.
1π

Answer

C. 0.1π

AP Test Prep 16.9 §16.3 source ↗

A pendulum of mass 200 g undergoes simple harmonic motion when acted upon by a force of 15 N. The pendulum crosses the point of equilibrium at a speed of 5 m∙s⁻¹. What is the energy of the pendulum at the center of the oscillation?

Answer

The energy of the particle at the center of the oscillation is given by

$E = \frac{1}{2} m v^{2}$ $= \frac{1}{2} \times 0.2 kg \times {(5 {m·s}^{- 1})}^{2}$ $= 2.5 J$

AP Test Prep 16.11 §16.4 source ↗

Which of the following gives the correct relation between the acceleration due to gravity and period of a pendulum?

a.
$g = \frac{2 π L}{T^{2}}$
b.
$g = \frac{4 π^{2} L}{T^{2}}$
c.
$g = \frac{2 π L}{T}$
d.
$g = \frac{2 π^{2} L}{T}$

Answer

g = \frac{4 π^{2} L}{T^{2}}

AP Test Prep 16.13 §16.5 source ↗

A mass of 1 kg undergoes simple harmonic motion with amplitude of 1 m. If the period of the oscillation is 1 s, calculate the internal energy of the system.

Answer

19.7 J

AP Test Prep 16.15 §16.7 source ↗

The non-conservative damping force removes energy from a system in which form?

a.
Mechanical energy
b.
Electrical energy
c.
Thermal energy
d.
None of the above

Answer

C. Thermal energy

AP Test Prep 16.17 §16.7 source ↗

A 0.5-kg object is connected to a spring that undergoes oscillatory motion. There is friction between the object and the surface it is kept on given by coefficient of friction

μ_{k} = 0.06

. If the object is released 0.2 m from equilibrium, what is the distance that the object travels? Given that the force constant of the spring is 50 N m^-1 and the frictional force between the objects is 0.294 N.

Answer

d = \frac{k}{2 μ_{K} m g} {(X^{2} - \frac{μ_{K} m g}{k})}^{2}

w h e r e k = 50 N \cdot m^{- 1}

μ_{k} = 0.06

m = 0.5 kg

d = \frac{50 N \cdot m^{- 1}}{2 \times 0.06 \times 9.8 m \cdot s^{- 2}} ({(0.2)}^{2} - (\frac{{(0.06 \times 0.5 kg \times 9.8 m \cdot s^{- 2})}^{2}}{{(50 N \cdot m^{- 1})}^{2}}))

= 1.698 m

AP Test Prep 16.19 §16.9 source ↗

What is the difference between the waves coming from a tuning fork and electromagnetic waves?

Answer

The waves coming from a tuning fork are mechanical waves that are longitudinal in nature, whereas electromagnetic waves are transverse in nature.

AP Test Prep 16.21 §16.9 source ↗

Why is the sound produced by a tambourine different from that produced by drums?

Answer

The sound energy coming out of an instrument depends on its size. The sound waves produced are relative to the size of the musical instrument. A smaller instrument such as a tambourine will produce a high-pitched sound (higher frequency, shorter wavelength), whereas a larger instrument such as a drum will produce a deeper sound (lower frequency, longer wavelength).

AP Test Prep 16.23 §16.9 source ↗

The graph shows propagation of a mechanical wave. What is the wavelength of this wave?

The image shows a graph of an oscillating line. The y-axis has displacement from negative 1 meter to positive 1 meter. The x-axis has time from 0 seconds to 8 pi seconds. The line starts at 0, positive 1, curves down to pass through displacement 0, and meets pi, negative one. The line curves up and passes through displacement equals 0 again, and then meets 2 pi, positive 1. It continues this way, finally meeting 8 pi, positive 1. — Figure 16.47 in the book ↗

Answer

2 π m

AP Test Prep 16.25 §16.10 source ↗

Explain the principle of superposition with figures that show the changes in the wave amplitude.

Answer

The student explains the principle of superposition and then shows two waves adding up to form a bigger wave when a crest adds with a crest and a trough with another trough. Also the student shows a wave getting cancelled out when a crest meets a trough and vice versa.

AP Test Prep 16.27 §16.10 source ↗

A string is fixed on both sides. It is snapped from both ends at the same time by applying an equal force. What happens to the shape of the waves generated in the string? Also, will you observe an overlap of waves?

Answer

The student must note that the shape of the wave remains the same and there is first an overlap and then receding of the waves.

AP Test Prep 16.29 §16.10 source ↗

Two sine waves travel in the same direction in a medium. The amplitude of each wave is A, and the phase difference between the two is 180°. What is the resultant amplitude?

a.
2A
b.
3A
c.
0
d.
9A

Answer

C. 0

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Problems and answers are adapted from College Physics for AP® Courses 2e by Gregg Wolfe, Erika Gasper, John Stoke, Julie Kretchman, David Anderson and others, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/college-physics-ap-courses-2e/pages/1-connection-for-ap-r-courses. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.