Simple Harmonic Motion oscillation amplitude

In summary: K/m)K=.5mAw^2cos^2(wt)=.5mAw^2(1-sin^2(wt))=.5mA^2w^2-.5mAw^2sin^2(wt)2K/m = Aw^2 -A^2w^2sin^2(wt)2K/m-Aw^2 = -A^2w^2sin^2(wt)A^2w^2 = (2K/m-Aw^2)/sin^2(wt)A=sqrt( (2K/m-Aw^2)/sin^2(wt) )In summary, to find the oscillation amplitude of
  • #1
sugz
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Homework Statement


A mass on a spring has an angular oscillation frequency of 2.56 rad/s. The spring constant is 27.2 N/m, and the system's kinetic energy is 4.16 J when t = 1.56 s. What is the oscillation amplitude? Assume that the mass is at its equilibrium position when t = 0.a. 63.1 cm
b. 47.7 cm
c. 73.4 cm
d. 55.3 cm
e. 84.0 cm

Homework Equations


[/B]

The Attempt at a Solution


w^2=k/m => m= 27.2/6.5536 = 4.15kg mg= - kx => x=1.497

K=(1/2)mv^2 => v=1.416 m/s
x= Acos(2.56t + φ)
v(1.56) = 1.416 = -2.56Asin(3.994+φ)
 
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  • #2
sugz said:
x= Acos(2.56t + φ)
I would use x=Asin( ... ) that way φ = 0 because the function sin(ct) starts out at zero (equilibrium) at t=0

Anyway that's not important.

At 1.56 seconds, a certain fraction of the total energy will be kinetic. This fraction is said to be equal to 4.16 joules, so you can then find the total energy. Then you can use that total energy to find the displacement amplitude, because they told you the spring constant.
 
  • #3
But how would you propose I find the total energy because at that time, I don't know what position it is in.
 
  • #4
You don't know the position, but can you solve for the speed at that time in terms of A? Then try to find the ratio of kinetic energy at that time to the kinetic energy when it is maximum. The mass and the amplitude should cancel.
 
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Likes Jediknight
  • #5
Now I get it, thank you so much for answering all of my questions! I really appreciate it!
 
  • #6
good question, nice explanation, it was e right?

get v at time t in terms of A from the first derrivative, then solve for A with the Kinetic energy at time t given

v=Awcos(wt) or Awsin(wt+pi/2) if you prefer

K=.5mv^2
 
Last edited:

1. What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion (SHM) is a type of oscillatory motion in which an object moves back and forth around an equilibrium point, with a restoring force that is directly proportional to the displacement from the equilibrium point. This results in a sinusoidal motion where the object moves with a constant frequency and amplitude.

2. What is the equation for calculating the amplitude of an oscillation in SHM?

The equation for calculating the amplitude of an oscillation in SHM is A = xmax, where A is the amplitude and xmax is the maximum displacement from the equilibrium point. This means that the amplitude is equal to half of the total distance the object moves in one complete oscillation.

3. How does the amplitude affect the period and frequency of an oscillation in SHM?

The amplitude of an oscillation in SHM does not affect the period or frequency of the motion. The period and frequency are only dependent on the mass and spring constant of the system. However, a larger amplitude will result in a greater amount of energy being transferred during each oscillation.

4. What factors can affect the amplitude of an oscillation in SHM?

The amplitude of an oscillation in SHM can be affected by the initial displacement of the object, the mass of the object, and the spring constant of the system. A larger initial displacement or a larger mass will result in a larger amplitude, while a larger spring constant will result in a smaller amplitude.

5. How does the amplitude of an oscillation change over time in SHM?

In SHM, the amplitude remains constant over time as long as there is no external force acting on the system. This is because the restoring force is always proportional to the displacement, resulting in a constant amplitude. However, external forces can cause the amplitude to decrease over time due to energy loss through friction or other forms of damping.

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