Damped oscillation and time between displacement maximums

[NihalRi]

Homework Statement

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Question 3.9

Homework Equations

equation for dampened ocillation[/B]

The Attempt at a Solution

In case this might appear confusing, I derived(with respect to t) the equation for dampened oscillation given above and tried to solve for when it equaled zero expecting this to happen for periodic values of t.
The constant A is removed in my derivation because it is unnecessary to find the maximum points.
I got stuck when trying to solve the derived equation for zero and I do not think I am heading in the right direction. I would greatly appreciate any any all help.

Thank you in advance.
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[NihalRi]

My attempt at a solution did not show up. Here it is

 

[Andrew Wildridge]

You can also get rid of phi as that is just a phase shift and does not change the period between maxima. That may help you figure out what you need to do next.

 

[MarkFL]

Look at where you have:

$\omega\sin\left(\omega t+\phi\right)+\dfrac{\gamma}{2}\cos\left(\omega t+\phi\right)=0$

Now, apply a linear combination identity for sine and cosine. What do you have?

 

[NihalRi]

You can also get rid of phi as that is just a phase shift and does not change the period between maxima. That may help you figure out what you need to do next.

Look at where you have:

$\omega\sin\left(\omega t+\phi\right)+\dfrac{\gamma}{2}\cos\left(\omega t+\phi\right)=0$

Now, apply a linear combination identity for sine and cosine. What do you have?

I tried to take what both of you said into account and it seemed to work. I was able to solve for t but it is not what I expected as is seems to still depend on the dampening constant $\Upsilon$. Did I make a mistake somewhere?

 

[MarkFL]

If you apply a linear combination identity to:

$\omega\sin\left(\omega t+\phi\right)+\dfrac{\gamma}{2}\cos\left(\omega t+\phi\right)=0$

you get a sinusoid of the form:

$\sqrt{\omega^2+\left(\dfrac{\gamma}{2}\right)^2}\sin\left(\omega t+\phi+\varphi\right)=0$

What is the period of this function?

 

[NihalRi]

If you apply a linear combination identity to:

$\omega\sin\left(\omega t+\phi\right)+\dfrac{\gamma}{2}\cos\left(\omega t+\phi\right)=0$

you get a sinusoid of the form:

$\sqrt{\omega^2+\left(\dfrac{\gamma}{2}\right)^2}\sin\left(\omega t+\phi+\varphi\right)=0$

What is the period of this function?

Oh I see. The period is, $\dfrac{2\pi}{\omega}$, which means that the equation becomes zero at this period meaning that the peaks occur at this period which is independent of $\varphi$.

Thank you all.