Which of the following statements is/ are true for a simple harmo… - Vidyadip

MCQ

Which of the following statements is/ are true for a simple harmonic oscillator?

A
Force acting is directly proportional to displacement from the mean position and opposite to it.
B
Motion is periodic.
C
The velocity is periodic.
✓
All of the above

✓

Answer

Correct option: D.

All of the above

Let us write the equation for the $\text{SHM} \text{x}=\text{a}\sin(\omega\text{t}+\phi)$
Clearly, it is a periodic motion as it involves sine function.
Let us find velocity of the particle, $\text{v}=\frac{\text{dx}}{\text{dt}}=\frac{\text{d}}{\text{dt}}(\text{a}\sin(\omega\text{t}+6\phi))$
Velocity is also periodic because it is a cosine function.
Now let us find acceleration, $\text{A}=\frac{\text{dv}}{\text{dt}}=\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\text{a}\omega^2\sin(\omega\text{t}+\phi)$
The acceleration is a sine function, hence cannot be constant.
$\Rightarrow\text{A}=-(\omega^2\text{a})\sin(\omega\text{t}+\phi)=-\omega^2\text{x}$
Force, $F =$ mass $\times$ acceleration.
$=\text{mA}=-\text{m}\omega^2\text{x}$
Hence, force acting is directly proportional to displacement from the mean position and opposite to it.

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