Question

A particle of mass $m$ is moving in a field where the potential energy is given by $U\left(x\right)=U_0(1-\cos ax)$, where $U_0$ and $a$ are positive constants and $x$ is the displacement from mean position. Then (for small oscillations):

• A. the time period is $T=2\pi \sqrt{\frac{m}{a{U}_0}}$
• B. the speed of the particle is maximum at $x=0$
• C. the amplitude of oscillations is $\frac{\pi }{a}$
• D. the time period is $T=2\pi \sqrt{\frac{m}{a^2{U}_0}}$

Answer 1

Answer: (B), (C), (D)

The correct options are
B the time period is $T=2\pi \sqrt{\frac{m}{a^2{U}_0}}$
C the speed of the particle is maximum at $x=0$
D the amplitude of oscillations is $\frac{\pi }{a}$

$U\left(x\right)=U_0(1-cosax)$
$\frac{dU}{dx}=U_{0}asinax$
$F=-dU/dx$
$F=-U_{0}asinax$
for small value of "x" we will write, $F=-U_0{a}^{2}x$
$a=-U_0{a}^{2}x/m$
$T=2\pi \sqrt{\frac{m}{U_0{a}^2}}$
at x = 0, speed of particle will be maximum