Ncert Exercises & Solutions

A body of mass m is situated in a potential field U(x) $= U_{0} (1 - \cos α x)$ where $U_{0}$ and $α$ are constants. Find the time period of small oscillations.

Hint: Use the relation between the force and the potential energy.

Step 1: Find the force equation.

Given potential energy associated with the field,

U(x)

= U_{0} (1 - \cos α x)

...(i)

Now, Force, F

= - \frac{d U (x)}{d x}

[

∵

for a conservatine force F, we can write

F = \frac{- dU}{dx}

]

[We have assumed the field to be conservative]

F = - \frac{d}{d x} (U_{0} - U_{0} \cos α x) = - U_{0} α \sin α x

F = - U_{0} α^{2} x . . . (i i)

\Rightarrow F \propto (- x)

U_{0}, α

are constant,

∴

Motion is SHM for small oscillations.

Step 2: Find the time period of oscillations.

Standard equation for SHM,

F = - m ω^{2} x . . . (i i i)

Comparing Eqs. (ii) and (iii), we get,

m ω^{2} = U_{0} α^{2}

ω^{2} = \frac{U_{0} α^{2}}{m} o r ω = \sqrt{\frac{U_{0} α^{2}}{m}}

∴

Time period,

T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{U_{0} α^{2}}}

NEET Information

courses

Company

Botany Questions

Chemistry Questions

Physics Questions

Zoology Questions