Question

Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle implies simple harmonic motion?

• A. $a=0.7x$
• B. $a=-200x^2$
• C. $a=-10x$
• D. $a=100x^3$

Answer 1

The correct option is C $a=-10x$

Condition for a particle to be executing SHM is given by
$a=-{\omega }^{2}x$ or $a\propto -x.......\left(1\right)$
Comparing option (a) with $\left(1\right)$, we get
${\omega }^2=-0.7\Rightarrow \omega =\pm i\sqrt{0.7}$
Since angular frequency cannot be a complex number, option (a) does not represent SHM.
Comparing option (b) with $\left(1\right)$,
$a=-200x^2$, does not represent SHM.
Comparing option (c) with $\left(1\right)$,
${\omega }^2=10\Rightarrow \omega =\pm \sqrt{10}$
Since angular frequency is a real number, option (a) i.e $a=-10x$, represents SHM.
Comparing option (d) with $\left(1\right)$,
$a=100x^3$, does not represent SHM.
Thus, option (c) is the correct answer.