Fast SHM question

**Misa-Campo** · August 4th 2008, 03:54 AM

thanks in advance

**flyingsquirrel** · August 4th 2008, 04:08 AM

Hi !

Originally Posted by Misa-Campo

Note that $\sin(4t)+\sqrt{3}\cos(4t)=2\left( \frac{1}{2}\sin(4t)+\frac{\sqrt{3}}{2}\cos(4t)\rig ht)<br /> =2\left( \cos\frac{\pi}{3}\sin(4t)+\sin\frac{\pi}{3}\cos(4t )\right)$ . As $\sin a \cos b+\sin b \cos a =\sin(a+b)$ , we get that $\sin(4t)+\sqrt{3}\cos(4t)=2\sin\left( 4t+\frac{\pi}{3}\right)$ hence $x(t)=3+2\sin\left( 4t+\frac{\pi}{3}\right)$ . Can you use this to answer the two questions?

**mr fantastic** · August 4th 2008, 04:13 AM

Originally Posted by Misa-Campo

thanks in advance

By re-writing as $x = 3 + 2 \sin(4t + \phi)$ it's simple to see that the centre of motion is x = 3.

i) Solve x = 3 for t: $\tan (4t) = -\sqrt{3}$ etc.

ii) $v = dx/dt = 8 \cos (4t + \phi)$ . Clearly the maximum value is v = 8.

**Misa-Campo** · August 4th 2008, 04:18 AM

Originally Posted by mr fantastic

By re-writing as $x = 3 + 2 \sin(4t + \phi)$ it's simple to see that the centre of motion is x = 3.

i) Solve x = 3 for t: $\tan (4t) = -\sqrt{3}$ etc.

ii) $v = dx/dt = 8 \cos (4t + \phi)$ . Clearly the maximum value is v = 8.

for i) im still stuck

what do you mean by solve x = 3 for t: $\tan (4t) = -\sqrt{3}$ etc. I've only started this topic so im not too sure

**mr fantastic** · August 4th 2008, 04:24 AM

Originally Posted by Misa-Campo

thanks in advance

Originally Posted by mr fantastic

By re-writing as $x = 3 + 2 \sin(4t + \phi)$ it's simple to see that the centre of motion is x = 3.

i) Solve x = 3 for t: $\tan (4t) = -\sqrt{3}$ etc.

[snip]

Originally Posted by Misa-Campo

for i) im still stuck

what do you mean by solve x = 3 for t: $\tan (4t) = -\sqrt{3}$ etc.

Since the centre of motion is x = 3, you want the times at which x = 3:

$3 = 3 + \sin (4t) + \sqrt{3} \cos (4t)$

$\Rightarrow 0 = \sin (4t) + \sqrt{3} \cos (4t)$

$\Rightarrow - \sin (4t) = \sqrt{3} \cos (4t)$

$\Rightarrow - \tan (4t) = \sqrt{3}$

$\Rightarrow \tan (4t) = - \sqrt{3}$ .

And you know how to solve this trig equation ....?

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