First slide

Simple hormonic motion

Question

Two particles P and Q start from origin at the same time and execute simple harmonic motion along x-axis with same amplitude but with periods 3 sec and 6 sec respectively. The ratio of the speeds of P and Q when they meet for the first time after start is

Difficult

A

1 : 2
B

2 : 1
C

2 : 3
D

3 : 2

Solution

(2)
$\begin{array}{l} x_{P} = A \sin \frac{2 π}{3} t a n d x_{a} = A \sin \frac{2 π}{6} t = A \sin \frac{π}{3} t . \\ V_{P} = \frac{d x_{P}}{d t} = \frac{2 π}{3} A . \cos \frac{2 π}{3} t a n d V_{Q} = \frac{d x_{Q}}{d t} = \frac{π}{3} A . \cos \frac{π}{3} t \\ W h e n x_{P} = x_{Q}, A \sin \frac{2 π}{3} t = A \sin \frac{π}{3} t \\ \Rightarrow \sin \frac{π}{3} t (2 \cos \frac{π}{3} - 1) = 0 \\ w h e n \sin \frac{π}{3} t = 0 \Rightarrow t = 0 \\ w h e n 2 \cos \frac{π}{3} - 1 = 0 \Rightarrow \cos \frac{π}{3} t = \frac{1}{2} = \cos \frac{π}{3} \Rightarrow t = 1 S \\ ∴ \frac{|V_{P}|}{V_{Q}} = \frac{\frac{2 π A}{3} . |\cos \frac{π}{3} .1|}{\frac{π A}{3} |\cos \frac{π}{3}|} = \frac{2. \frac{1}{2}}{1 / 2} = 2. \end{array}$

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