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A babysitter is pushing a child on a swing. At the point where the swing reaches x size 12{x} {} , where would the corresponding point on a wave of this motion be located?

x size 12{x} {} is the maximum deformation, which corresponds to the amplitude of the wave. The point on the wave would either be at the very top or the very bottom of the curve.

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Test prep for ap courses

Use these figures to answer the following questions.

The image shows two graphs with oscillating lines. The top graph has amplitude from negative A to positive A on the y axis, and time 0 to 8 pi on the x axis. The line starts at 0, A, curves down to cross amplitude 0, and then meet pi, negative A. It curves up, and passes through amplitude 0 again, and then meets 2 pi, A. It curves down, passing through amplitude 0 again, and then meets 3 pi, negative A. It continues this way, finally passing through 8 pi, A. The bottom graph has displacement from negative 3 A over 2 to positive 3 A over 2 on the y-axis, and time 0 to 24 pi on the x-axis. The line starts at 0, 3 A over two curves down through amplitude 0, and meets negative 3 A over 2 at an unmarked x-value. It curves up, passes through amplitude 0 again, and meets 3 pi, 3 A over 2. It goes on in this way, finally meeting 24 pi, 3 A over 2.

  1. Which of the two pendulums oscillates with larger amplitude?
  2. Which of the two pendulums oscillates at a higher frequency?
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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?

  1. 10π
  2. 0.5π
  3. 0.1π

(c)

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The graph shows the simple harmonic motion of a mass m attached to a spring with spring constant k .

The image shows a graph of an oscillating line. The y-axis has amplitude 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 amplitude 0, and meets pi, negative one. The line curves up and passes through amplitude equals 0 again, and then meets 2 pi, positive 1. It continues this way, finally meeting 8 pi, positive 1.

What is the displacement at time 8 π ?

  1. 1 m
  2. 0 m
  3. Not defined
  4. −1 m
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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?

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

E = 1 2 m v 2 = 1 2 × 0.2 kg × ( 5  m·s 1 ) 2 = 2.5 J

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Phet explorations: masses and springs

A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.

Masses and Springs

Section summary

  • Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke’s law. Such a system is also called a simple harmonic oscillator.
  • Maximum displacement is the amplitude X size 12{X} {} . The period T size 12{T} {} and frequency f size 12{f} {} of a simple harmonic oscillator are given by

    T = m k size 12{T=2π sqrt { { {m} over {k} } } } {} and f = 1 k m , where m size 12{m} {} is the mass of the system.

  • Displacement in simple harmonic motion as a function of time is given by x ( t ) = X cos t T . size 12{x \( t \) =X"cos" { {2π`t} over {T} } } {}
  • The velocity is given by v ( t ) = v max sin t T , where v max = k / m X .
  • The acceleration is found to be a ( t ) = kX m cos t T . size 12{a \( t \) = - { { ital "kX"} over {m} } " cos " { {2π t} over {T} } } {}

Conceptual questions

What conditions must be met to produce simple harmonic motion?

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(a) If frequency is not constant for some oscillation, can the oscillation be simple harmonic motion?

(b) Can you think of any examples of harmonic motion where the frequency may depend on the amplitude?

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Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.

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Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a spongy material.

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As you pass a freight truck with a trailer on a highway, you notice that its trailer is bouncing up and down slowly. Is it more likely that the trailer is heavily loaded or nearly empty? Explain your answer.

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Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer springs? Explain your answer.

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Problems&Exercises

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?

2 . 37 N/m size 12{2 "." "37"`"N/m"} {}

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If the spring constant of a simple harmonic oscillator is doubled, by what factor will the mass of the system need to change in order for the frequency of the motion to remain the same?

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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?

0.389 kg

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By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?

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Suppose you attach the object with mass m size 12{"m"} {} to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring’s original rest length. (a) Show that the spring exerts an upward force of 2.00 mg on the object at its lowest point. (b) If the spring has a force constant of 10 . 0 N/m size 12{"10" "." 0`"N/m"} {} and a 0.25-kg-mass object is set in motion as described, find the amplitude of the oscillations. (c) Find the maximum velocity.

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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?

94.7 kg

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Suppose a diving board with no one on it bounces up and down in a simple harmonic motion with a frequency of 4.00 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the simple harmonic motion of a 75.0-kg diver on the board?

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The figure shows a little kid, about ten to twelve months old, standing in a toy jolly jumper, which is tied to the ceiling hook by its four spring belts.
This child’s toy relies on springs to keep infants entertained. (credit: By Humboldthead, Flickr)

The device pictured in [link] entertains infants while keeping them from wandering. The child bounces in a harness suspended from a door frame by a spring constant.

(a) If the spring stretches 0.250 m while supporting an 8.0-kg child, what is its spring constant?

(b) What is the time for one complete bounce of this child? (c) What is the child’s maximum velocity if the amplitude of her bounce is 0.200 m?

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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 [link] .

The figure shows two skydivers midway through the air, with both with open having their parachutes open.
The oscillations of one skydiver are about to be affected by a second skydiver. (credit: U.S. Army, www.army.mil)

1.94 s

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Questions & Answers

Determine the total force and the absolute pressure on the bottom of a swimming pool 28.0m by 8.5m whose uniform depth is 1 .8m.
Henny Reply
how solve this problem?
Foday
P(pressure)=density ×depth×acceleration due to gravity Force =P×Area(28.0x8.5)
Fomukom
for the answer to complete, the units need specified why
muqaddas Reply
That's just how the AP grades. Otherwise, you could be talking about m/s when the answer requires m/s^2. They need to know what you are referring to.
Kyle
Suppose a speck of dust in an electrostatic precipitator has 1.0000×1012 protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have?
Alexia Reply
how would I work this problem
Alexia
how can you have not an integer number of protons? If, on the other hand it supposed to be 1e12, then 1.6e-19C/proton • 1e12 protons=1.6e-7 C is the charge of the protons in the speck, so the difference between this and 5e-9C is made up by electrons
Igor
what is angular velocity
Obaapa Reply
angular velocity can be defined as the rate of change in radian over seconds.
Fidelis
Why does earth exert only a tiny downward pull?
Mya Reply
hello
Islam
Why is light bright?
Abraham Reply
what is radioactive element
Attah Reply
an 8.0 capacitor is connected by to the terminals of 60Hz whoes rms voltage is 150v. a.find the capacity reactance and rms to the circuit
Aisha Reply
thanks so much. i undersooth well
Valdes Reply
what is physics
Nwafor Reply
is the study of matter in relation to energy
Kintu
physics can be defined as the natural science that deals with the study of motion through space,time along with its related concepts which are energy and force
Fidelis
a submersible pump is dropped a borehole and hits the level of water at the bottom of the borehole 5 seconds later.determine the level of water in the borehole
Obrian Reply
what is power?
aron Reply
power P = Work done per second W/ t. It means the more power, the stronger machine
Sphere
e.g. heart Uses 2 W per beat.
Rohit
A spherica, concave shaving mirror has a radius of curvature of 32 cm .what is the magnification of a persons face. when it is 12cm to the left of the vertex of the mirror
Alona Reply
did you solve?
Shii
1.75cm
Ridwan
my name is Abu m.konnek I am a student of a electrical engineer and I want you to help me
Abu
the magnification k = f/(f-d) with focus f = R/2 =16 cm; d =12 cm k = 16/4 =4
Sphere
what do we call velocity
Kings
A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows pa
Kavita Reply
hi
Godfred
what about the wind vane
Godfred
If a prism is fully imersed in water then the ray of light will normally dispersed or their is any difference?
Anurag Reply
the same behavior thru the prism out or in water bud abbot
Ju
If this will experimented with a hollow(vaccum) prism in water then what will be result ?
Anurag
Practice Key Terms 3

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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