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Physics-

General

Easy

Question

A cylindrical piston of mass $M$ slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of gas. The cylinder is kept with its axis horizontal. If the piston is disturbed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will be

$T = 2 π \sqrt{(\frac{M h}{P A})}$
$T = 2 π \sqrt{(\frac{M A}{P h})}$
$T = 2 π \sqrt{(\frac{M}{P A h})}$
$T = 2 π \sqrt{M P h A}$

The correct answer is: $T equals 2 pi square root of open parentheses fraction numerator M h over denominator P A end fraction close parentheses end root$

Let the piston be displaced through distance $x$ towards left, then volume decreases, pressure increases. If $increment P$ is increase in pressure and $increment V$ is decreases in volume, then considering the process to take place gradually (i.e. isothermal)

$P subscript 1 end subscript V subscript 1 end subscript equals P subscript 2 end subscript V subscript 2 end subscript$
$rightwards double arrow P V equals left parenthesis P plus increment P right parenthesis left parenthesis V minus increment V right parenthesis$
$rightwards double arrow P V equals P V plus increment P V minus P increment V minus increment P increment V$
$rightwards double arrow increment P. V minus P. increment V equals 0 blank$ (neglecting $increment P. increment V right parenthesis$
$increment P open parentheses A h close parentheses equals P open parentheses A x close parentheses rightwards double arrow increment P equals fraction numerator P. x over denominator h end fraction$
This excess pressure is responsible for providing the restoring force $left parenthesis F right parenthesis$ to the piston of mass $M$
Hence $F equals increment P. A equals fraction numerator P A x over denominator h end fraction$
Comparing it with $open vertical bar F close vertical bar equals k x rightwards double arrow k equals M omega to the power of 2 end exponent equals fraction numerator P A over denominator h end fraction$
$rightwards double arrow omega equals square root of fraction numerator P A over denominator M h end fraction end root rightwards double arrow T equals 2 pi square root of fraction numerator M h over denominator P A end fraction end root$
Short trick: by checking the options dimensionally. Option [a] is correct

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