Question

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

A) $a=0.7x$

B) $a=-200x^2$

C) $a=-10x$

D) $a=100x^3$

Answer 1

A) A motion represents simple harmonic motion if it
obeys the force law,
$F=-kx$
So, acceleration
$a=-\frac{k}{m}x$
Here $m$ is the mass of the body, $x$ is displacement
and $k$ is a positive constant.
$a=0.7x$
This is not in the form of $a=-\frac{k}{m}x$
Hence, this is not SHM.


B) A motion represents simple harmonic motion if it
obeys the force law,
$F=-kx$
So, acceleration
$a=-\frac{k}{m}x$
Here $m$ is the mass of the body, $x$ is displacement and
$k$ is a positive constant.
$a=-200x^2$
This is not in the form of $a=-\frac{k}{m}x$
Hence, this is not $SHM$.


C) A motion represents simple harmonic motion if
it obeys the force law,
$F=-kx$
So, acceleration
$a=-\frac{k}{m}x$
Here $m$ is the mass of the body, $x$ is displacement and $k$ is a positive constant.
$a=-10x$
This is in the form of $a=-\frac{k}{m}x$
Hence, this is $SHM$.


D) A motion represents simple harmonic motion if
it obeys the force law,
$F=-kx$
So, acceleration
$a=-\frac{k}{m}x$
Here $m$ is the mass of the body, $x$ is displacement and $k$ is a positive constant.
$a=100x^3$
This is not in the form of $a=-\frac{k}{m}x$
Hence, this is not $SHM$.