sinx derivative & integrate

[curicuri]

Hello, ran into a problem that I would appreciate if someone could help me out with.

Problem

Harmonic oscillation:

v=speed (cm/s)
t= seconds

Find the distance the object is moving between the turning points

Solution


My question
How do I really know that the time calculated in the solution is the time that passes by when the object moves between two turning points? When I first saw the solution I thought that t1 accounted for half of the time (assuming the full time would be between two turning points), which in that case would give t1*2. Hope that I make some sense to you?

 

[MarkFL]

I would write:

[MATH]v(t)=82\sin\left(\frac{36}{5}t\right)[/MATH]
Hence, via integration, we find:

[MATH]x(t)=-\frac{205}{18}\cos\left(\frac{36}{5}t\right)+C[/MATH]
Now, we then know the distance \(D\) moved (in cm) between two subsequent turning points is twice the amplitude:

[MATH]D=2\left|-\frac{205}{18}\right|=\frac{205}{9}\approx22.8[/MATH]
This is a simpler way to obtain the same result. Now, regarding your question, when given:

[MATH]\sin(\omega t)=0[/MATH]
Then we know this implies:

[MATH]\omega t=k\pi[/MATH] where \(k\in\mathbb{Z}\)

Hence:

[MATH]t=\frac{k\pi}{\omega}[/MATH]

 

[curicuri]

Awesome, great to see how you simplify this!
Super thanks