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7.6 Modeling with trigonometric equations  (Page 4/14)

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Simple harmonic motion

A type of motion described as simple harmonic motion    involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When   t = 0 , d = 0.

Simple harmonic motion

We see that simple harmonic motion    equations are given in terms of displacement:

d = a   cos ( ω t )   or   d = a   sin ( ω t )

where   | a |   is the amplitude,   2 π ω   is the period, and   ω 2 π   is the frequency, or the number of cycles per unit of time.

Finding the displacement, period, and frequency, and graphing a function

For the given functions,

  1. Find the maximum displacement of an object.
  2. Find the period or the time required for one vibration.
  3. Find the frequency.
  4. Sketch the graph.
    1. y = 5   sin ( 3 t )
    2. y = 6   cos ( π t )
    3. y = 5   cos ( π 2 t )
  1. y = 5   sin ( 3 t )
    1. The maximum displacement is equal to the amplitude,   | a | , which is 5.
    2. The period is   2 π ω = 2 π 3 .
    3. The frequency is given as   ω 2 π = 3 2 π .
    4. See [link] . The graph indicates the five key points.
      Graph of the function y=5sin(3t) from 0 to 2pi/3. The five key points are (0,0), (pi/6, 5), (pi/3, 0), (pi/2, -5), (2pi/3, 0).
  2. y = 6   cos ( π t )
    1. The maximum displacement is   6.
    2. The period is   2 π ω = 2 π π = 2.
    3. The frequency is   ω 2 π = π 2 π = 1 2 .
    4. See [link] .
      Graph of the function y=6cos(pi t) from 0 to 3.
  3. y = 5   cos ( π 2 )   t
    1. The maximum displacement is   5.
    2. The period is   2 π ω = 2 π π 2 = 4.
    3. The frequency is   1 4 .
    4. See [link] .
      Graph of the function y=5cos(pi/2 t) from 0 to 4.
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Damped harmonic motion

In reality, a pendulum does not swing back and forth forever, nor does an object on a spring bounce up and down forever. Eventually, the pendulum stops swinging and the object stops bouncing and both return to equilibrium. Periodic motion in which an energy-dissipating force, or damping factor, acts is known as damped harmonic motion    . Friction is typically the damping factor.

In physics, various formulas are used to account for the damping factor on the moving object. Some of these are calculus-based formulas that involve derivatives. For our purposes, we will use formulas for basic damped harmonic motion models.

Damped harmonic motion

In damped harmonic motion    , the displacement of an oscillating object from its rest position at time   t   is given as

f ( t ) = a e − c t sin ( ω t )   or     f ( t ) = a e − c t cos ( ω t )

where   c   is a damping factor,   | a |   is the initial displacement and   2 π ω   is the period.

Modeling damped harmonic motion

Model the equations that fit the two scenarios and use a graphing utility to graph the functions: Two mass-spring systems exhibit damped harmonic motion at a frequency of   0.5   cycles per second. Both have an initial displacement of 10 cm. The first has a damping factor of   0.5   and the second has a damping factor of   0.1.

At time   t = 0 , the displacement is the maximum of 10 cm, which calls for the cosine function. The cosine function will apply to both models.

We are given the frequency   f = ω 2 π   of 0.5 cycles per second. Thus,

   ω 2 π = 0.5      ω = ( 0.5 ) 2 π         = π

The first spring system has a damping factor of   c = 0.5.   Following the general model for damped harmonic motion, we have

f ( t ) = 10 e − 0.5 t cos ( π t )

[link] models the motion of the first spring system.

Graph of the first spring system, f(t) = 10(e^(-.5t))cos(pi*t), which begins with a high amplitude and quickly decreases.

The second spring system has a damping factor of   c = 0.1   and can be modeled as

f ( t ) = 10 e − 0.1 t cos ( π t )

[link] models the motion of the second spring system.

Graph of f(t) = 10(e^(-.1t))cos(pi*t), which begins with a high amplitude and slowly decreases (but has a high frequency).
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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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