A particle in linear simple harmonic motion has a velocity of 4 ms^-1 at 3m at 3 ms^-1 at 4 m from mean position.

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A particle in linear simple harmonic motion has a velocity of 4 ms-1 at 3m at 3 ms-1 at 4 m from mean position. What is the time taken to travel half the amplitude from its positive extreme position?

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Step 1. $v=\omega\sqrt{a^2\,-\,y^2},$

$v_2=\omega\sqrt{a^2\,-\,y_1^2}$

And $v_2 =\omega\sqrt{a^2\,-y^2_2}$

4 = $\omega\sqrt{a^2\,-\,3^2}$ ...(i)

And  3 = $\omega\sqrt{a^2\,-\,4^2}$ ...(ii)

Dividing eqn (i) by (ii) :

$\frac{4}{3}=\frac{\sqrt{a^2-9}}{\sqrt{a^2-16}}$

i.e., $\frac{16}{9}=\frac{\sqrt{a^2-9}}{\sqrt{a^2-16}}$

Or 16a2 − 256 = 9a2 − 81

Or a2= 25

Or a = 5 m.

Step 2. Putting a = 5 in eqn (i),

4 = 4ω

Or ω = 1 rad s-1

Step 3. Time taken to travel half amplitude from positive extreme positive extreme position is given by displacement,

x =acos ωt

i.e.,  $\frac{5}{2}= 5cos(1\times t) \,or\,cos\, t=\frac{1}{2}$

Or  t = cos-1$\frac{1}{2}$ = 60º = $\frac{\pi}{3}$

Or  t = $\frac{3.142}{3}$ = 1.047 s.