Question

A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is $U\left(x\right)=K|x{|}^3,$ where K is a positive constant. If the amplitude of oscillation is a, then its time period T is:

• A. Proportional to $\frac{1}{\sqrt{a}}$
• B. Independent of a
• C. Proportional to $\sqrt{a}$
• D. Proportional to $a^{3/2}$
  

Answer 1

The correct option is A Proportional to $\frac{1}{\sqrt{a}}$

$U\left(x\right)=K|x^3|$
$\therefore \left[K\right]=\frac{\left[U\right]}{[x{]}^3}=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[L^3\right]}=\left[M{L}^{-1}{T}^{-2}\right]$
Now time period may depend on $T\propto \left(mass{)}^x\right(amplitude{)}^y(K{)}^z$
$=\left[M^{x+z}{L}^{y-z}{T}^{-2z}\right]$
Equating the powers, we get
$-2z=1orz=-1/2$
$y-z=0ory=z=-1/2$
$\therefore T\propto (amplitude{)}^{-1/2}$
Or $T\propto (a{)}^{-1/2}$
Or $T\propto \frac{1}{\sqrt{a}}$.