Write expressions for simple harmonic motion

[RedBarchetta]

Homework Statement

Write expressions for simple harmonic motion (a) with amplitude 10 cm, frequency 5.0 Hz, and maximum displacement at t=0; and (b) with amplitude 2.5 cm, angular frequency 5.0 1/s, and maximum velocity at t=0.

Homework Equations

$$\begin{gathered}<br /> x(t) = A\cos (\omega t + \varphi ) \hfill \\<br /> \omega = 2\pi f \hfill \\<br /> f = \frac{1}<br /> {T} \hfill \\ <br /> \end{gathered}$$

The Attempt at a Solution

(a)
A=10 cm
f=5.0 Hz

Since the amplitude equals the max displacement at a given t(in this instance t=0), this tells us that the phase angle is zero. So our equation should be...?

$$x(t) = (10cm)\cos \left[ {(10\pi s^{ - 1} )t} \right]$$

(b)
A=2.5 cm
w=5.0 s^-1

$$\begin{gathered}<br /> V(x) = - A\omega \sin (\omega t + \varphi ) \hfill \\<br /> V(0) = A\omega = V_{\max } \hfill \\<br /> V(0) = - A\omega \sin (\varphi ) \hfill \\<br /> A\omega = - A\omega \sin (\varphi ) \hfill \\<br /> - 1 = \sin (\varphi ) \hfill \\<br /> \varphi = \tfrac{{3\pi }}<br /> {2} \hfill \\ <br /> \end{gathered}$$

So...?

$$x(t) = (2.5cm)\cos \left[ {(5.0s^{ - 1} )t + \tfrac{{3\pi }}<br /> {2}} \right]$$

Do these look right? Here is what my answer book gives:

$$\begin{gathered}<br /> (a):x(t) = (10cm)\cos \left[ {(\pi s^{ - 1} )t} \right] \hfill \\<br /> (b):x(t) = (2.5cm)\sin \left[ {(5s^{ - 1} )t} \right] \hfill \\ <br /> \end{gathered}$$

Any help is appreciated, thank you.

 

[Coto]

At a glance, your part (a) should be right considering the frequency given. Your part (b) is equivalent to the answer they state.

A phase shift corresponds to a shift in where the peaks occur. You can consider then, that the phase shift can shift a cosine into a sine or a sine into a cosine. In this sense,

$$cos(x + \frac{3\pi}{2}) = sin (x)$$

 

[RedBarchetta]

At a glance, your part (a) should be right considering the frequency given. Your part (b) is equivalent to the answer they state.

A phase shift corresponds to a shift in where the peaks occur. You can consider then, that the phase shift can shift a cosine into a sine or a sine into a cosine. In this sense,

$$cos(x + \frac{3\pi}{2}) = sin (x)$$

Thanks Coto. I had a feeling that the part a solution key was incorrect.