Two waves are simultaneously passing through a string and their e… - Vidyadip

MCQ

Two waves are simultaneously passing through a string and their equations are :

${y}_{1}={A}_{1} \sin {k}({x}-v {t}), {y}_{2}={A}_{2} \sin {k}\left({x}-{vt}+{x}_{0}\right) .$ Given amplitudes ${A}_{1}=12\, {mm}$ and ${A}_{2}=5\, {mm}$ ${x}_{0}=3.5\, {cm}$ and wave number ${k}=6.28\, {cm}^{-1}$. The amplitude of resulting wave will be $......\,{mm}$

✓
$7$
B
$10$
C
$25$
D
$49$

✓

Answer

Correct option: A.

$7$

a
${y}_{1}={A}_{1} \operatorname{sink}({x}-{vt})$

${y}_{1}=12 \sin 6.28({x}-{vt})$

${y}_{2}=5 \sin 6.28({x}-{vt}+3.5)$

$\Delta \phi=\frac{2 \pi}{\lambda}(\Delta {x})$

$={K}(\Delta {x})$

$=6.28 \times 3.5=\frac{7}{2} \times 2 \pi=7 \pi$

${A}_{{DE}}=\sqrt{{A}_{1}^{2}+{A}_{2}^{2}+2 {A}_{1} {A}_{2} \cos \phi}$

${A}_{{DA}}=\sqrt{(12)^{2}+(5)^{2}+2(12)(5) \cos (7 \pi)}$

$=\sqrt{144+25-120}$

$=\,7$

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