Question

A particle of mass m is located in a one dimensional potential field where potential energy is given by U(x) = A(1 - cos px), where A and p are constants. The period of small oscillations of the particle

• A. $2\pi \sqrt{\frac{m}{A{p}^2}}$
• B. $\pi \sqrt{\frac{m}{AP}}$
• C. $\pi \sqrt{\frac{m}{A}}$
• D. $\left(\frac{1}{2\pi }\right)\sqrt{\frac{Ap}{m}}$

Answer 1

Answer: (A)

$2\pi \sqrt{\frac{m}{A{p}^2}}$

$U\left(x\right)=A(1-\cos pX)$

On comparing it with the SHM of spring block system.

$U=\frac{1}{2}k{x}^2$

Here

$U=A(1-\cos pX)\Rightarrow U=A\left(2{\sin }^{2}pX\right)[\cos 2\theta =1-2{\sin }^2\theta ]\Rightarrow A\left(2{\sin }^{2}pX\right)=\frac{1}{2}k{x}^{2}x=\sin pXk=A{p}^{2}T=2\pi \sqrt{\frac{m}{k}}T=2\pi \sqrt{\frac{m}{A{p}^2}}$