Question

Potential Energy(U) of a body of unit mass moving in a one-dimension conservative force field is given by $U=(x^2-4x+3)$. All units are in S.I.
(i) Find the equilibrium positions of the body.
(ii) Show that oscillations of the body about this equilibrium positions is simple harmonic motion & find its time period.
(iii) Find the amplitude of oscillations if speed of the body at equilibrium position is $2\sqrt{6}$ m/s.

Answer 1

Potential energy $\left(u\right)$ of a body of unit mass moving

Given, $U=x^2-4x+3mgh=x^2-4x+3$

$\left(i\right)$ equilibrium position of the body at equilibrium condition,

Potential energy$U=0$

$x^2-4x+3=0x^2-3x-x+3=0x(x-3)-(x-3)=0x=3,1$

$\left(ii\right)$Oscillations of the body about this equilibrium

$T=\frac{1}{2\pi }\sqrt{\frac{x}{g}}$

when $x=3$

$T=\frac{1}{2\pi }\sqrt{\frac{3}{g}}andT=\frac{1}{2\pi }\sqrt{\frac{1}{g}}$

$\left(iii\right)$ Amplitude of oscillation if speed of the body at equilibrium position.

The magnitude of the velocity of a particle performing S.H.M is

$v=\omega \sqrt{a^2-x^2}=\sqrt{3\times 8}=2\sqrt{6}m/s$