Question

Which of the following expressions corresponds to simple harmonic motion along a straight line, where $x$ is the displacement and $a,b,c$ are positive constants?
$\begin{aligned} & A)a+bx-c{{x}^{2}} \\\ & B)b{{x}^{2}} \\\ & C)a-bx+c{{x}^{2}} \\\ & D)-bx \\\ \end{aligned}$

Answer 1

Simple harmonic motion is a special type of oscillatory motion in which the object executing simple harmonic motion is always acted upon by a restoring force, which is directly proportional to the displacement of the object from its equilibrium position. Also, this restoring force acts towards the equilibrium position of the object. An expression for simple harmonic motion along a straight line, say, for example, a spring-mass system, is given by Hooke’s law.

Complete answer:
If an object is said to execute simple harmonic motion, then the object is acted upon by a restoring force, whose magnitude is directly proportional to the displacement of the object from the equilibrium position and direction is towards the equilibrium position. One of the best examples to understand simple harmonic motion is the linear expansion of a spring in a spring-mass system.
Suppose a mass is attached to a spring, which is fixed on a heavy surface like a wall on the other side. The mass will cause the spring to expand from its equilibrium position, along a straight line. After expanding for some time, the mass will retrace its path along the straight line, back to its equilibrium position. The process continues until there is a permanent deformation in the spring, due to the weight of the mass. The motion executed by the mass in this spring-mass system is nothing but simple harmonic motion, in which a restoring force acts on the mass, to bring it back to the equilibrium position, every time the mass expands the spring from its equilibrium position. The restoring force acting on the mass has a magnitude which is proportional to the displacement of the mass from its equilibrium position. It also has a direction acting towards the equilibrium position of the spring. An expression for simple harmonic motion along a straight line, as in the above case, is given by Hooke’s law, which states that
$F\propto -x\Rightarrow F=-kx$
where
$F$ is the restoring force acting on the mass
$x$ is the displacement of the mass from its equilibrium position
$k$ is any constant which describes the stiffness of the spring attached to a heavy surface like wall
Let this be equation 1.
The negative sign in equation 1 suggests that the restoring force acting along a straight line on the mass in a spring-mass system, is towards the equilibrium position.

Hence, from equation 1, we can conclude that option $D$ is the correct answer.

Note:
We know that force acting on an object is given by
$F=ma$
where
$m$ is the mass of an object
$a$ is the acceleration of the object
Substituting this equation in equation 1, we have
$F=ma=-kx\Rightarrow a\propto -x\Rightarrow a=-{{\omega }^{2}}x$
where
$F$ is the restoring force acting on the mass of a spring-mass system
$m$ is the mass
$a$ is the acceleration of the mass
$\omega$ is the angular velocity of the mass
This is additional information, which can be used if questions similar to the above question, are asked.