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# The amplitude of simple harmonic motion represented by the displacement equation $y\left( {cm} \right) = 4\left( {\operatorname{Sin} \,5\pi t + \sqrt 2 \cos \,5\pi t} \right)$is:A. $4\,cm$B. $4\sqrt 2 \,cm$C. $4\sqrt {3\,} cm$D. $4\left( {\sqrt 2 - 1} \right)\,cm$

Last updated date: 14th Sep 2024
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Hint: In the question, we have to determine the amplitude of the simple harmonic motion. The simple harmonic motion is represented by the displacement equation $y\left( {cm} \right) = 4\left( {\operatorname{Sin} \,5\pi t + \sqrt 2 \cos \,5\pi t} \right)$. For the simple harmonic motion, we will compare the given expression with the general equation of the simple harmonic motion and then we get the value of the amplitude of the simple harmonic motion.

Given that the equation of the simple harmonic motion
$y\left( {cm} \right) = 4\left( {\operatorname{Sin} \,5\pi t + \sqrt 2 \cos \,5\pi t} \right)$
Where,
$y$be the displacement in the simple harmonic motion.
The simple harmonic motion can be written as,
$y\left( {cm} \right) = 4\operatorname{Sin} \,5\pi t + 4\sqrt 2 \cos \,5\pi t..........\left( 1 \right)$
We know that the general equation of the simple harmonic equation is given by the formula,
$y\left( {cm} \right) = A\operatorname{Sin} \,\left( {\omega t} \right) + B\,\cos \,\left( {\omega t} \right)..........\left( 2 \right)$
Comparing the above given two equations, we get the values of the parameters of the simple harmonic motion, we get
$A = 4$
$B = 4\sqrt 2$
Now, we also know that the amplitude for a simple harmonic equation is given as
${\text{Amplitude}}\,{\text{ = }}\,\sqrt {{A^2} + {B^2}}$
Now, we substitute the values of A and B in the above amplitude expression, we get
${\text{Amplitude}}\,{\text{ = }}\,\sqrt {{{\left( 4 \right)}^2} + {{\left( {4\sqrt 2 } \right)}^2}}$
Performing the arithmetic operations in the above equation, we get
${\text{Amplitude}}\,{\text{ = }}\,\sqrt {16 + 32}$
${\text{Amplitude}}\,{\text{ = }}\,\sqrt {48}$
Simplify the equation of the amplitude, we get
${\text{Amplitude}}\,{\text{ = }}\,4\sqrt 3$
Therefore, the amplitude of the simple harmonic equation is $4\sqrt 3$.

Hence, from the above options, option C is correct.

Note: A special type of the periodic motion where the restoring force of the moving object is directly proportional to its magnitude of the displacement and which is acting towards the objects equilibrium position is called the simple harmonic motion.