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

[Lchan1]

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.

 

[aim1732]

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.

 

[Lchan1]

I am not sure how to make a substitution for dx.

 

[aim1732]

Well dx= vdt. Turns out a pretty lengthy problem after all. Will be happy to help further, if needed.

 

[Lchan1]

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.

 

[aim1732]

Well that is how you presented the problem. All textbooks use F=-kx to prove the result.