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

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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