Correct Answer - Option 3 : Square of the amplitude
Concept:
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Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
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Example: Motion of an undamped pendulum, undamped spring-mass system.
- The potential energy (U) of a particle in simple harmonic motion is given by the formula:
\(\Rightarrow {\rm{U}} = \frac{1}{2}{\rm{k}}{{\rm{x}}^2}\)
Where x = Distance from its mean position and k = spring constant.
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Total Energy: Total energy of a particle is the sum of K.E. and P.E.
Total Energy = K.E. + P.E.
\(Total.Energy. = {1\over 2}mω^2(A^2-x^2)+{1\over 2}mω^2x^2\)
\(Total.Energy. = {1\over 2}mω^2A^2\)
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Total mechanical energy (TE) of a particle executing simple harmonic motion is
\(\Rightarrow TE = \frac{1}{2}{\rm{k}}{{\rm{A}}^2}\)
Explanation:
- The total energy of simple harmonic motion is
\(Total.Energy. = {1\over 2}mω^2A^2\)
The total energy of a simple harmonic oscillator is proportional to the square of Amplitude.
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Velocity in simple harmonic motion: The relation between velocity and displacement can be given as:
\(\Rightarrow {\rm{V}} = {\rm{\omega }}\sqrt {{A^2} - {y^2}} \)
Where V = velocity, ω = angular velocity, A = amplitude and y = displacement.
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Kinetic energy (KE) of a particle executing simple harmonic motion is
\(\Rightarrow KE = \frac{1}{2}mv^2\)
