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
IIT 1981, Diffcult
Download our app for free and get started
(a) Let the piston be displaced through distance $x$ towards left, then volume decreases, pressure increases. If $\Delta P$ is increase in pressure and $\Delta V$ is decrease in volume, then considering the process to take place gradually (i.e. isothermal) ${P_1}{V_1} = {P_2}{V_2}$
$ \Rightarrow PV = (P + \Delta P)(V - \Delta V)$
$ \Rightarrow PV = PV + \Delta PV - P\Delta V - \Delta P\Delta V$
$ \Rightarrow \Delta P.V - P.\Delta V = 0$ (neglecting $\Delta P.\Delta V)$
$\Delta P(Ah) = P(Ax)$
$ \Rightarrow \Delta P = \frac{{P.x}}{h}$
This excess pressure is responsible for providing the restoring force $(F)$ to the piston of mass $M$.
Hence $F = \Delta P.A = \frac{{PAx}}{h}$
Comparing it with $|F| = kx$
$\Rightarrow k = M{\omega ^2} = \frac{{PA}}{h}$
$ \Rightarrow \omega = \sqrt {\frac{{PA}}{{Mh}}} $
$\Rightarrow T = 2\pi \sqrt {\frac{{Mh}}{{PA}}} $
Short trick : by checking the options dimensionally. Option (a) is correct.
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A particle is placed at the lowest point of a smooth wire frame in the shape of a parabola, lying in the vertical $xy-$ plane having equation $x^2 = 5y$ $(x, y$ are in meter). After slight displacement, the particle is set free. Find angular frequency of oscillation.....$rad/s$ (in $rad/sec$ ) (take $g = 10\ m/s^2$ )View Solution
- 2When a mass $M$ is attached to the spring of force constant $k$, then the spring stretches by $l$. If the mass oscillates with amplitude $l$, what will be maximum potential energy stored in the springView Solution
- 3A block of mass m, attached to a spring of spring constant $k$, oscillates on a smooth horizontal table. The other end of the spring is fixed to a wall. The block has a speed $v$ when the spring is at its natural length. Before coming to an instantaneous rest, if the block moves a distance $x$ from the mean position, thenView Solution
- 4A body of mass $5\; kg$ hangs from a spring and oscillates with a time period of $2\pi $ seconds. If the ball is removed, the length of the spring will decrease byView Solution
- 5The periodic time of a simple pendulum of length $1\, m $ and amplitude $2 \,cm $ is $5\, seconds$. If the amplitude is made $4\, cm$, its periodic time in seconds will beView Solution
- 6A body is executing simple harmonic motion with an angular frequency $2\,rad/s$. The velocity of the body at $20\, mm$ displacement, when the amplitude of motion is $60\, mm$, is ...... $mm/s$View Solution
- 7For a particle executing $S.H.M.$ the displacement $x$ is given by $x = A\cos \omega t$. Identify the graph which represents the variation of potential energy $(P.E.)$ as a function of time $t$ and displacement $x$View Solution
- 8A particle of mass $m$ is attached to three identical springs $A, B$ and $C$ each of force constant $ k$ a shown in figure. If the particle of mass $m$ is pushed slightly against the spring $A$ and released then the time period of oscillations isView Solution
- 9The scale of a spring balance reading from $0$ to $10 \,kg$ is $0.25\, m$ long. A body suspended from the balance oscillates vertically with a period of $\pi /10$ second. The mass suspended is ..... $kg$ (neglect the mass of the spring)View Solution
- 10A $S.H.M.$ has amplitude $‘a’$ and time period $T$. The maximum velocity will beView Solution