# Underdamped harmonic oscillator with a sinusoidal driving force

## Homework Statement

Consider an underdamped harmonic oscillator (Q > 1/2) with a sinusoidal driving
force Focos(ωdt).
(a) (5 pts) By using differential calculus find ωd that maximizes the displacement amplitude.
(b) (7 pts) By using differential calculus find ωd that maximizes the velocity amplitude.

## Homework Equations

x(t)= (fo/m)/sqrt([wo^2-wd^2]^2+(gama(wd)^2)

## The Attempt at a Solution

I know what the end solution will be and the general theory behind this but I am unsure how to proceed. I am unsure how to prove the point at which it is the maximum.

The clue is pretty much in "differential calculus". What tool have you learnt in calculus for finding the maximum of a function?

## Homework Statement

Consider an underdamped harmonic oscillator (Q > 1/2) with a sinusoidal driving
force Focos(ωdt).
(a) (5 pts) By using differential calculus find ωd that maximizes the displacement amplitude.
(b) (7 pts) By using differential calculus find ωd that maximizes the velocity amplitude.

## Homework Equations

x(t)= (fo/m)/sqrt([wo^2-wd^2]^2+(gama(wd)^2)

## The Attempt at a Solution

I know what the end solution will be and the general theory behind this but I am unsure how to proceed. I am unsure how to prove the point at which it is the maximum.
Let me give you a hint on the general theory behind this:

your relevant equation seems flat bizarre

first of all, why is there no 't' in x(t)

and what is gama?
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to set you in the direction i would go:

you have your driving force F

F=-d/dx(V(x)

then V(x) = potential energy

maximum value of V(x)= total energy = 1/2*mass*velocity^2+V(x)

then solve for velocity, and integrate with respect to time to get x(t).
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or just say (1/m)*F= acceleration and integrate twice with respect to time to get x(t)

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