Deriving position function for object in SHM

In summary, the conversation is about solving a problem using initial state equations, and the importance of recognizing and eliminating dead-end solutions. The expert suggests writing out the equations and finding ways to eliminate unknowns, instead of getting stuck on irrelevant solutions. The problem was eventually solved and the conversation ends with gratitude and an understanding of the expert's advice.
  • #1
ChiralSuperfields
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Homework Statement
Please see the image below
Relevant Equations
x(t) = Acos(wt + ϕ)
For this problem,
1670306945988.png

1670306964083.png

How did they get that formula shown?

My working is,
1670307096649.png

1670307174104.png

1670307305108.png


All the solutions wrote was,
1670307369422.png


Many thanks!
 

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  • #2
Your working headed off into a blind alley. You know that you need to be eliminating A, so tucking it inside an arccos function isn’t going to get you there.
Start by writing all three initial state equations: position, velocity and acceleration. Then see how you can combine them to eliminate one of the unknowns.
 
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  • #3
Please type out your work instead of attaching pictures. The pictures make it impossible to quote particular sections.

Edit: I suggest you change your unknowns from A and phi to something more direct.
 
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  • #4
haruspex said:
Your working headed off into a blind alley. You know that you need to be eliminating A, so tucking it inside an arccos function isn’t going to get you there.
Start by writing all three initial state equations: position, velocity and acceleration. Then see how you can combine them to eliminate one of the unknowns.
Thank you @haruspex and @Orodruin! I have solved the problem now.

Many thanks!
 
  • #5
Callumnc1 said:
Thank you @haruspex and @Orodruin! I have solved the problem now.

Many thanks!
You are welcome. But did you understand my comment about heading off into a blind alley? Recognising which directions cannot lead anywhere is a useful skill to develop.
 
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  • #6
haruspex said:
You are welcome. But did you understand my comment about heading off into a blind alley? Recognising which directions cannot lead anywhere is a useful skill to develop.
Thanks @haruspex! I did understand your comment, which I agree, would be very useful skill to have!

Many thanks,
Callum
 

1. What is SHM and how is it related to the position function of an object?

SHM stands for Simple Harmonic Motion, which is a type of periodic motion where an object oscillates back and forth around an equilibrium point. The position function of an object in SHM describes the object's position at any given time during its motion.

2. How do you derive the position function for an object in SHM?

The position function for an object in SHM can be derived using the equation x(t) = A*cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant. This equation can be derived from the equation for displacement in SHM, x(t) = A*sin(ωt + φ), by using trigonometric identities.

3. What is the significance of the amplitude in the position function for an object in SHM?

The amplitude in the position function represents the maximum displacement of the object from its equilibrium point. It is a measure of the object's maximum displacement during its motion and is directly related to its energy.

4. How does the angular frequency affect the position function for an object in SHM?

The angular frequency, ω, determines the rate at which the object oscillates back and forth. A higher angular frequency results in a shorter period and a faster oscillation, while a lower angular frequency results in a longer period and a slower oscillation.

5. Can the position function for an object in SHM be used to determine its velocity and acceleration?

Yes, the position function can be used to determine the velocity and acceleration of an object in SHM. The velocity function, v(t), can be found by taking the derivative of the position function, and the acceleration function, a(t), can be found by taking the second derivative of the position function.

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