First slide

Simple harmonic motion

Question

A particle is subjected to two SHM’s $x_{1} = A_{1} S i n ω t a n d x_{2} = A_{2} S i n (ω t + π / 4)$ . The resultant SHM will have an amplitude of

Moderate

A

$(A_{1} + A_{2}) / 2$
B

$\sqrt{A_{1}^{2} + A_{2}^{2}}$
C

$\sqrt{A_{1}^{2} + A_{2}^{2} + \sqrt{2} A_{1} A_{2}}$
D

$A_{1} A_{2}$

Solution

$\emptyset$ = $\frac{π}{4}$
$T h e r e s u l \tan t S H M w i l l h a v e a n a m p l i t u d e = A = \sqrt{{A_{1}}^{2} + {A^{2}}_{2} + 2 A_{1} A_{2} \cos \emptyset} g i v e n p h a s e d i f f e r e n c e \emptyset = \frac{π}{4} = \sqrt{{A_{1}}^{2} + {A^{2}}_{2} + 2 A_{1} A_{2} \cos \frac{π}{4}} = \sqrt{{A_{1}}^{2} + {A^{2}}_{2} + 2 A_{1} A_{2} \frac{1}{\sqrt{2}}} r e s u l \tan t a m p l i t u d e = \sqrt{{A_{1}}^{2} + {A^{2}}_{2} + \sqrt{2} A_{1} A_{2}}$

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