First slide

Simple hormonic motion

Question

The displacement of a particle executing simple harmonic motion is given by
x = a sin $ω$ t + a cos $ω$ t
The total energy of the particle is

Moderate

A

$\frac{1}{2} m a^{2} ω^{2}$
B

$m a^{2} ω^{2}$
C

$\frac{1}{4} m a^{2} ω^{2}$
D

$2 m a^{2} ω^{2}$

Solution

Given x = a sin $ω$ t + a cos $ω$ t …….. (1)
The displacement equation of a simple harmonic motion is
x = A sin ( $ω$ t + $ϕ$ )
where A = amplitude and $ϕ$ = phase constant.
x = A sin $ω$ t cos $ϕ$ + A cos $ω$ t sin $ϕ$ …….. (2)
Comparing (1) with (2) we get
A cos $ϕ$ = a (3)
and A sin $ϕ$ = a (4)
Squaring and adding Eqs. (3) and (4) we get
$A^{2} = a^{2} + a^{2} \Rightarrow A^{2} = 2 a^{2}$
$∴$ Energy = $\frac{1}{2} m ω^{2} A^{2} = m ω^{2} a^{2}$
So the correct choice is (b).

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