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

  1. T equals 2 pi square root of open parentheses fraction numerator M h over denominator P A end fraction close parentheses end root    
  2. T equals 2 pi square root of open parentheses fraction numerator M A over denominator P h end fraction close parentheses end root    
  3. T equals 2 pi square root of open parentheses fraction numerator M over denominator P A h end fraction close parentheses end root    
  4. T equals 2 pi square root of M P h A end root    

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 capital delta P is increase in pressure and capital delta V is decrease 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 capital delta P right parenthesis left parenthesis V minus capital delta V right parenthesis
    rightwards double arrow P V equals P V plus capital delta P V minus P capital delta V minus capital delta P capital delta V
    rightwards double arrow capital delta P. V minus P. capital delta V equals 0 (neglecting capital delta P. capital delta V right parenthesis
    capital delta P left parenthesis A h right parenthesis equals P left parenthesis A x right parenthesis rightwards double arrow capital delta P equals fraction numerator P. x over denominator h end fraction
    This excess pressure is responsible for providing the restoring force (F) to the piston of mass M.
    Hence F equals capital delta P. A equals fraction numerator P A x over denominator h end fraction
    Comparing it with vertical line F vertical line 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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