Question

Define oscillatory motion.

Answer 1

Hint: A motion is said to be oscillatory if its diagram in displacement-time graph is either
A. Sinusoidal with constant amplitude. (undamped)
B. Sinusoidal with decreasing amplitude. (damped)

Complete step-by-step answer:
We first need to know about the simple harmonic motion where the force on a body is proportional to its displacement from a mean position. Its equation of motion can be given by,
$$m{\displaystyle \frac{d^{2}y}{d{t}^2}}+R{\displaystyle \frac{dy}{dt}}+ky=0$$
Where R is the damping force per unit velocity of the object. k is called the force constant. The equation is modified as,
$${\displaystyle \frac{d^{2}y}{d{t}^2}}+2b{\displaystyle \frac{dy}{dt}}+{\omega }_0^{2}y=0$$
Here, $$2b={\displaystyle \frac{R}{m}}$$ and $${\omega }_0^2={\displaystyle \frac{k}{m}}$$
Undamped oscillatory motion- In this case, R=0 $$\Rightarrow$$ b=0. Hence the equation reduces to,
$${\displaystyle \frac{d^{2}y}{d{t}^2}}+{\omega }_0^{2}y=0$$
The solution will be, $$y=A.cos({\omega }_{0}t-\theta )$$
Where, A is the amplitude of oscillation and $$\theta$$ depends on the initial conditions. Its diagram looks like…..

Its energy is given by,
$$\begin{gathered} E=E_k+E_p={\displaystyle \frac{1}{2}}m({\displaystyle \frac{dy}{dt}}{)}^2+{\displaystyle \frac{1}{2}}m{\omega }_0^2{y}^2 \\ ={\displaystyle \frac{1}{2}}m{\omega }_0^2{A}^2 \end{gathered}$$
Damped oscillation- In case $$b\ne 0$$, oscillation can only be observed if $$b\le {\omega }_0$$ and in all the other cases no oscillation will be observed.
If $$b\ge {\omega }_0$$ ,the motion is called over damped.
If $$b={\omega }_0$$, the motion is said to be critically damped.
In these two cases there is no oscillation. For oscillation (damped), $$b\le {\omega }_0$$
In this case, the solution is given by, $$y=A.e^{-bt}.cos(\omega t-\theta )$$
Here, $$\omega =\sqrt{{\omega }_0^2-b^2}$$

The diagram will look like this.
In this case, the amplitude decreases as , $$A_1=A{e}^{-bt}$$
Its energy can be shown to be $$E={\displaystyle \frac{1}{2}}m{\omega }_0^2{A}^2{e}^{-2bt}$$

Note: There is another type of oscillation that is called forced oscillation. In this case, the system is forced to move in oscillation by an oscillatory external force.