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An air chamber of volume V has a neck ar...

An air chamber of volume V has a neck area of cross section A into which a ball of mass m just fits and can move up and down without any friction, figure. Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time periodof oscillations assuming pressure volume variatinos of air to be isothermal.

Text Solution

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To solve the problem step by step, we will analyze the motion of the ball in the air chamber and derive the expression for the time period of oscillation when the ball is pressed down and released. ### Step 1: Understand the System We have an air chamber with a volume \( V \) and a neck with a cross-sectional area \( A \). A ball of mass \( m \) fits into the neck and can move up and down without friction. When the ball is pressed down a distance \( Y \) and released, we need to show that it executes Simple Harmonic Motion (SHM). ### Step 2: Change in Volume When the ball is pressed down by a distance \( Y \), the volume of air in the chamber decreases. The change in volume \( \Delta V \) can be expressed as: \[ ...
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(i)One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion. Find the time period of oscillation if length of mercury column in the tube is (ii)An air chamber of volume V has a neck area of cross section a into which a ball of mass m just fits and can more up and down without any friction. Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal.

An air chamber of volume v has a neck area of cross section a into which a ball of mass m just fits and can move up and down without any friction. Show that when the ball is pressed down a little and released for the time priod of oscillation, assuming pressure-volume variations of the air to be isothermal.

Knowledge Check

  • An air chamber of volume V has a neck of cross-sectional area a into which a light ball of mass m just fits and can move up and down without friction. Consider diameter of the ball to be equal to that of the neck of the chamber. The ball is then pressed down a little and released. Let the bulk modulus of air is B. The time period of the oscillation of the ball is:

    A
    `T = 2pi sqrt((Ba^(2))/(mV))`
    B
    `T = 2pisqrt((BV)/(ma^(2)))`
    C
    `T = 2pi sqrt((mB)/(Va^(2)))`
    D
    `T = 2pi sqrt((mV)/(Ba^(2)))`

  • An air chamber of volume V has a long of cross setional area A.A ball of mass m is fixted smpthlly in the track The ball modulus of air is B ball is pressed down slightly and released, the time period of the oscillation is

    A
    `2pi sqrt((mV)/(2BA^(2)))`
    B
    `pi sqrt((2mV)/(BA^(2)))`
    C
    `2pi sqrt((mV)/(BA^(2)))`
    D
    `(pi)/(2) sqrt((m)/(BA^(2)))`

  • Figure shows initial state of an ideal gas trapped in a container with conducting walls and a piston (mass m ) which can move without any friction. The container is placed on point supports and its wall are conducting. Assuming that atmospheric pressure is P_(0) and the mass of the ideal gas is negligible as compared to the mass of the piston and the mass of container. Take the cross section area of piston to be A. The piston is slowly lifted by an external agent and held in its position. Let M be the maximum mass of container so that it may "lift off" while pulling the piston upwards and P_(i) be the pressure of ideal gas in initial state. Pick the correct choice:

    A
    `P_(i)=P_(0)+(mg)/A`
    B
    `P_(i)=(P_(0))/2+(mg)/(2A)`
    C
    `M=m/2`
    D
    `M=(P_(0)A)/(2g)-m/2`

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