Question

A particle of mass m is located in a one-dimensional field where potential energy is given by:
$V\left( x \right) = A\left( {1 - \cos px} \right)$ where A and p are constants.
The period of small oscillations of the particle is
(A) $2\pi \sqrt {\dfrac{m}{{\left( {Ap} \right)}}}$
(B) $2\pi \sqrt {\dfrac{m}{{\left( {A{p^2}} \right)}}}$
(C) $2\pi \sqrt {\dfrac{m}{A}}$
(D) $\dfrac{1}{{2\pi }}\sqrt {\dfrac{{\left( {Ap} \right)}}{m}}$

Answer 1

This could be simply solved by breaking the diagrams into simple free body diagrams of both the blocks. Then we need to apply the frequency formula.

Formula used: Here, we will use the frequency formula:
${\omega _n} = \dfrac{g}{\vartriangle }$
Here, ${\omega _n}$ is the frequency
$\vartriangle$ is the deflection in the spring
Also,
${\omega _n} = \dfrac{{2\pi }}{T}$
Here, $T$ is the time period.

Complete step by step answer:
We are given that a particle of mass m is located in a one dimensional potential field and the potential energy is given by: $V\left( x \right) = A\left( {1 - \cos px} \right)$
So, we can find the force experienced by the particle as
$F = - \dfrac{{dV}}{{dx}} = - Ap\sin px$
For small oscillations, we have, $F \approx - A{p^2}x$
Hence, the acceleration would be given by, $a = \dfrac{F}{m} = - \dfrac{{A{p^2}}}{m}x$
Also, we know that, $a = \dfrac{F}{m} = - {\omega ^2}x$
So, $\omega = \sqrt {\dfrac{{A{p^2}}}{m}}$
$T = 2\pi /\omega = 2\pi \sqrt {\dfrac{m}{{A{p^2}}}}$
Then we need to match the correct option.

The correct option is B.

Note: It should be always kept in mind that T is defined as the time of one full oscillation. In this applet, the small hanging mass always swings from its rightmost position This can be used as a reference point or state for counting the number of oscillations. The time elapsed between every two consecutive states is the period, T.
We define periodic motion to be a motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by an object on a spring moving up and down. The time to complete one oscillation remains constant and is called the period T. Its units are usually seconds, but may be any convenient unit of time. The word period refers to the time for some event whether repetitive or not; but we shall be primarily interested in periodic motion, which is by definition repetitive. A concept closely related to period is the frequency of an event.