Verifying Energy Conservation with x(t)=Acos(wt+phi)

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Homework Help Overview

The discussion revolves around verifying the conservation of total energy using the equation x(t) = A cos(wt + phi), which is related to harmonic motion. Participants are exploring the relationship between force, displacement, and energy in this context.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using the integral of force to verify energy conservation, with some suggesting differentiation and integration of the motion equation. Others raise questions about making substitutions and the complexity of the problem.

Discussion Status

The discussion is ongoing, with various approaches being explored. Some participants have provided insights into integrating force and using specific equations, while others express uncertainty about the steps involved. There is no explicit consensus on a single method yet.

Contextual Notes

Participants are working under the constraints of homework guidelines, which may limit the methods they can use. There is also mention of differing approaches found in textbooks, indicating a variety of interpretations of the problem.

Lchan1
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Homework Statement


Using x(t)= A cos(wt+phi) verify that the total energy is conserved.


Homework Equations


V(x)= Integral x to x1 F(x)dx


The Attempt at a Solution



I thought about using the aboved equation but have no idea where to start.
 
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F=ma so you can differentiate x[t] twice for a[t].Integrate F[t]=ma[t] w.r.t dx (you would have to make a substitution for x for that).You would also think K[t]=1/2m*v*v where v=v[t] at general instant t.
Add and you will have a quantity independent of t.
 
I am not sure how to make a substitution for dx.
 
Well dx= vdt. Turns out a pretty lengthy problem after all. Will be happy to help further, if needed.
 
Instead of your approach, I used F=-kx and integrate to get 1/2kx^2

x(t) is given so I have an equation.
w=sqrt (k/m)
so things actaully worked nicely. and It was only a page of work.
 
Well that is how you presented the problem. All textbooks use F=-kx to prove the result.
 

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