Question

An object is in simple harmonic motion. Find an equation for the motion given that the period is pi/4 and, at time t=o, y=1, y'=0. What is the amplitude? What is the frequency?

Answer 1

you will need the formulas for this

Answer 2

Period is basically the reciprocal of freqiency

Answer 3

yes, i believe we will use

\[y=Asin(\omega t+\phi _{0})\]

Answer 4

and amplitude = -angular frequency^2 times x

Answer 5

\[\omega = 8\]

Answer 6

I'll list the formulas you will need. remember them

Answer 7

\[\omega = 2 \pi f\]
\[a = -\omega ^{2} x\]
\[f = \frac{ 1 }{ T }\]

Answer 8

yes and I have f = 4pi and i have omega as 8.. but how to i find the rest.

Answer 9

thr amplitude has to be infered

Answer 10

y'=0 means a minima or a maxima (i.e the amplitude)

Answer 11

therefore, the question stated the amplitude

Answer 12

which is 1

Answer 13

so you have the amplitude and frequency. Find the angular frequency and use the SHM formula to get the equation

Answer 14

so you have the amplitude and frequency. Find the angular frequency and use the SHM formula to get the equation\[x = x _{o} \sin (\omega t)\]

Answer 15

okay so then i have

\[y=\sin(8t+\phi _{0})\]

how do i find phi

Answer 16

\( t=0, y=1, y'=0\), it says that \(t=0\) is a critical point meaning \(y\) is at its maximum so amplitude is \(1\).

Answer 17

Now \(\sin(0) = 0\) and \(\cos(0)=1\) which tells you that you want to use \(\cos\) rather than \(\sin\), in which case your initial angle is just \(0\).

Answer 18

the answer key says phi is pi/2, i do not know how they got that... so the equation is

\[y=\sin(8t+\frac{ \Pi }{ 2 })\]

Answer 19

They likely used co-function identities.
\[
\cos(t) = \sin(\pi/2-t)
\]

Answer 20

It is also known that \(\sin(\pi/2\))=1

Answer 21

well the class is differential equation, so i am certain that i have to use the derivative of the general equation somehow.

Answer 22

Meaning if you start your initial angle at \(\pi/2\) then you'll start at your amplitude.