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Question
Write down the equation of the time period for the linear harmonic oscillator.
Solution
From Newton’s second law, we can write the equation for the particle executing simple harmonic motion
`"m"("d"^2"x")/("dt"^2)` = −kx ............(1)
`("d"^2"x")/("dt"^2) = -"k"/"m""x"` .............(2)
Comparing the equation with simple harmonic motion equation, we get
`ω^2 = "k"/"m"` .............(3)
which means the angular frequency or natural frequency of the oscillator is
`ω = sqrt("k"/"m")` rad s−1 ............(4)
The frequency of the oscillation is
f = `ω/(2π) = 1/(2π) sqrt("k"/"m")` Hertz ...........(5)
and the time period of the oscillation is
T = `1/"f" = 2π sqrt("m"/"k")` seconds ........(6)
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