Other SHM Systems

A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______

A small sphere oscillates simple harmonically in a watch glass whose radius of curvature is $1.6$ m. The period of Oscillation of the sphere is (Acceleration due to gravity $g = 10$ m/s$^{2}$)

A uniform disc of radius $R$ and mass $M$ is free to oscillate about the axis $A$ as shown in the figure. For small oscillations the time period is ______.

(g is acceleration due to gravity)

A wooden cube (density $d$) of side $L$ floats on a liquid of density $\rho$ with its upper and lower surfaces horizontal. If the cube is pushed slightly down and released, it performs S.H.M. of period $T$. The value of $T$ is ($g$ = acceleration due to gravity)

A rectangular block of mass 'm' and cross-sectional area 'A' floats on a liquid of density '$\varrho$'. It is given a small vertical displacement from equilibrium, it starts oscillating with frequency (g=acceleration due to gravity)

A cylindrical cork of uniform density floats in a liquid of density $\rho_1$. If the cork is depressed slightly and released, it oscillates harmonically with time period $T$. If the same cork floats in another liquid of density $\rho_2$, then the similar oscillation has time period $2T$. The value of $\rho_2/\rho_1$ is:

A small sphere oscillates simple harmonically in a watch glass whose radius of curvature is 1.6 m. The period of oscillation of the sphere is (acceleration due to gravity $g = 10\text{ m/s}^2$)

A light hollow cube of side length 10 cm and mass 10 g, is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $y\pi \times 10^{-2}\text{ s}$, where the value of $y$ is (Acceleration due to gravity, $g = 10\text{ m/s}^2$, density of water $= 10^{3}\text{ kg/m}^3$)

A small spherical ball of radius ' $r$ ' is rolling on a curved surface which is frictionless and has a radius of curvature ' R '. Its motion is simple harmonic. Then its time period of oscillation is proportional to ( $g =$ acceleration due to gravity)

A small sphere oscillates simple harmonically in a watch glass whose radius of curvature is 1.6 m. The period of oscillation of the sphere in second is (acceleration due to gravity, $g = 10\text{ m/s}^2$)

A piece of wood has length, breadth and height, ' $a$ ', ' $b$ ' and ' $c$ ' respectively. Its relative density, is ' $d$ '. It is floating in water such that the side ' $a$ ' is vertical. It is pushed down a little and released. The time period of S.H.M. executed by it is ($g =$ acceleration due to gravity)

A tube of uniform bore of cross-sectional area ' $A$ ' has been set up vertically with open end facing up. Now ' $M$ ' gram of a liquid of density ' $d$ ' is poured into it. The column of liquid in this tube will oscillate with a period ' T ', which is equal to [ $g =$ acceleration due to gravity]

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance (R/2) from the earth's centre, where 'R' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period:

A cylindrical plastic bottle of negligible mass is filled with 310 ml of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $\omega$. If the radius of the bottle is 2.5 cm then $\omega$ is close to – (density of water = 103 kg/m3).

Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length $\ell$ and mass m. The rod is pivoted at its centre 'O' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is:

A rod of mass 'M' and length '2L' is suspended at its middle by a wire. It exhibits torsional oscillations; If two masses each of 'm' are attached at distance 'L/2' from its centre on both sides, it reduces the oscillation frequency by 20%. The value of ratio m/M is close to:

A cylindrical block of wood (density = 650 kg m${}^{-3}$), of base area 30 cm2 and height 54 cm, floats in a liquid of density 900 kg m${}^{-3}$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly):

A pendulum made of a uniform wire of cross sectional area $A$ has time period $T.$ When an additional mass $M$ is added to its bob, the time period changes to $T_M.$ If the Young's modulus of the material of the wire is $Y$ then $\frac{1}{Y}$ is equal to:

($g =$ gravitational acceleration)

A uniform cylinder of length L and mass M having cross-sectional area A is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density $\sigma$ at equilibrium position. When the cylinder is given a downward push and released, it starts oscillating vertically with a small amplitude. The time period T of the oscillations of the cylinder will be:

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M.$ The piston and the cylinder have equal cross sectional area $A$. When the piston is in equilibrium, the volume of the gas is $V_0$ and its pressure is $P_0.$ The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency

A rectangular block of mass m and area of cross-section A floats in a liquid of density $\rho$. If it is given a small vertical displacement from equilibrium it undergoes with a time period T, then