First slide

Simple harmonic motion

Question

A particle is executing simple harmonic motion. The equation of motion is given by $\frac{d^{2} x}{d t^{2}} + 4 x = 0$ . Then their time period of oscillation is (in seconds)

Moderate

A

$8 π$
B

$4 π$
C

$2 π$
D

$π$

Solution

A simple harmonic motion can be represented by the equation $a = - ω^{2} x$ . where acceleration a can be written as $\frac{d^{2} x}{d t^{2}}$ .
Hence $\frac{d^{2} x}{d t^{2}} = - ω^{2} x$ .
Writing the given equation $\frac{d^{2} x}{d t^{2}} + 4 x = 0$ in the above form, we get $\frac{d^{2} x}{d t^{2}} = - 4 x$ .
Comparing this with the above equation, we get $ω^{2} = 4$ .
Taking square root on both sides $ω = 2$
Using the relation $ω = \frac{2 π}{T}$ , we get $2 = \frac{2 π}{T}$
Hence, time period of oscillation is $π$ s

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