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:
(1) Smaller than $2\pi \left[ \frac { M } { ( k + A\sigma g ) } \right] ^ { 1/2 }$
(2) $2\pi \sqrt { \frac { M } { k } }$
(3) Larger than $2\pi \left[ \frac { M } { ( k + A\sigma g ) } \right] ^ { 1/2 }$
(4) $2\pi \left[ \frac { M } { ( k + A\sigma g ) } \right] ^ { 1/2 }$

Two simple pendulums of length 1 m and 4 m respectively are both given small displacement in the same direction at the same instant. They will be again in phase after the shorter pendulum has completed number of oscillations equal to:
(1) 2
(2) 7
(3) 5
(4) 3

A mass $m = 1.0\mathrm{~kg}$ is put on a flat pan attached to a vertical spring fixed on the ground. The mass of the spring and the pan is negligible. When pressed slightly and released, the mass executes simple harmonic motion. The spring constant is $500\mathrm{~N/m}$. What is the amplitude A of the motion, so that the mass $m$ tends to get detached from the pan? (Take $g = 10\mathrm{~m/s^2}$). The spring is stiff enough so that it does not get distorted during the motion.
(1) $\mathrm{A} > 2.0\mathrm{~cm}$
(2) $\mathrm{A} = 2.0\mathrm{~cm}$
(3) $\mathrm{A} < 2.0\mathrm{~cm}$
(4) $\mathrm{A} = 1.5\mathrm{~cm}$

A block of mass $m$, lying on a smooth horizontal surface, is attached to a spring (of negligible mass) of spring constant $k$. The other end of the spring is fixed, as shown in the figure. The block is initially at rest in its equilibrium position. If now the block is pulled with a constant force $F$, the maximum speed of the block is:
(1) $\frac { F } { \sqrt { m k } }$
(2) $\frac { 2 F } { \sqrt { m k } }$
(3) $\frac { \pi F } { \sqrt { m k } }$
(4) $\frac { F } { \pi \sqrt { m k } }$

Two masses $m$ and $\frac { m } { 2 }$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system (see figure). Because of torsional constant $k$, the restoring torque is $\tau = k \theta$ for angular displacement $\theta$. If the rod is rotated by $\theta _ { 0 }$ and released, the tension in it when it passes through its mean position will be:
(1) $k \theta _ { 0 } ^ { 2 }$
(2) $\frac { 3 k \theta _ { 0 } { } ^ { 2 } } { l _ { 0 } }$
(3) $\frac { 2 k \theta _ { 0 } { } ^ { 2 } } { l }$
(4) $\frac { k \theta _ { 0 } { } ^ { 2 } } { l }$

Two light identical springs of spring constant $k$ are attached horizontally at the two ends of a uniform horizontal rod $AB$ of length $l$ and mass $m$. The rod is pivoted at its center '$O$' 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:
(1) $\frac { 1 } { 2 \pi } \sqrt { \frac { 3 k } { m } }$
(2) $\frac { 1 } { 2 \pi } \sqrt { \frac { k } { m } }$
(3) $\frac { 1 } { 2 \pi } \sqrt { \frac { 6 \mathrm { k } } { m } }$
(4) $\frac { 1 } { 2 \pi } \sqrt { \frac { 2 k } { m } }$

A particle undergoing simple harmonic motion has time dependent displacement given by $x ( t ) = \mathrm { A } \sin \frac { \pi t } { 90 }$. The ratio of kinetic to potential energy of this particle at $t = 210 s$ will be
(1) $\frac { 1 } { 9 }$
(2) 1
(3) 2
(4) $\frac { 1 } { 3 }$

A particle executes simple harmonic motion with an amplitude of $5 cm$. When the particle is at $4 cm$ from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is:
(1) $\frac { 8 \pi } { 3 }$
(2) $\frac { 3 } { 8 } \pi$
(3) $\frac { 4 \pi } { 3 }$
(4) $\frac { 7 } { 3 } \pi$

A pendulum is executing simple harmonic motion and its maximum kinetic energy is $\mathrm { K } _ { 1 }$. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is $\mathrm { K } _ { 2 }$
(1) $K _ { 2 } = 2 K _ { 1 }$
(2) $\mathrm { K } _ { 2 } = \frac { \mathrm { K } _ { 1 } } { 2 }$
(3) $K _ { 2 } = \frac { K _ { 1 } } { 4 }$
(4) $K _ { 2 } = K$

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 $= 10 ^ { 3 } \mathrm {~kg} / \mathrm { m } ^ { 3 }$)
(1) $5.00 \mathrm { rad } \mathrm { sec } ^ { - 1 }$
(2) $2.50 \mathrm { rad } \mathrm { sec } ^ { - 1 }$
(3) $7.9 \mathrm { rad } \mathrm { sec } ^ { - 1 }$
(4) $3.75 \mathrm { rad } \mathrm { sec } ^ { - 1 }$

A simple pendulum of length 1 m is oscillating with an angular frequency $10 \mathrm { rad } / \mathrm { s }$. The support of the pendulum starts oscillating up and down with a small angular frequency of $1 \mathrm { rad } / \mathrm { s }$ and an amplitude of $10 ^ { - 2 } \mathrm {~m}$. The relative change in the angular frequency of the pendulum is best given by:
(1) $10 ^ { - 3 } \mathrm { rad } / \mathrm { s }$
(2) $1 \mathrm { rad } / \mathrm { s }$
(3) $10 ^ { - 1 } \mathrm { rad } / \mathrm { s }$
(4) $10 ^ { - 5 } \mathrm { rad } / \mathrm { s }$

A body is performing simple harmonic with an amplitude of 10 cm. The velocity of the body was tripled by air Jet when it is at 5 cm from its mean position. The new amplitude of vibration is $\sqrt { x } \mathrm {~cm}$. The value of $x$ is $\_\_\_\_$ .

As per given figures, two springs of spring constants $K$ and $2K$ are connected to mass $m$. If the period of oscillation in figure (a) is 3 s, then the period of oscillation in figure (b) will be $\sqrt { x }$ s. The value of $x$ is $\_\_\_\_$.

A mass $m$ is attached to two springs as shown in figure. The spring constants of two springs are $K_1$ and $K_2$. For the frictionless surface, the time period of oscillation of mass $m$ is
(1) $2\pi\sqrt{\frac{m}{K_1 + K_2}}$
(2) $\frac{1}{2\pi}\sqrt{\frac{K_1 - K_2}{m}}$
(3) $2\pi\sqrt{\frac{m}{K_1 - K_2}}$
(4) $\frac{1}{2\pi}\sqrt{\frac{K_1 + K_2}{m}}$

A simple harmonic oscillator has an amplitude $A$ and time period $6 \pi$ second. Assuming the oscillation starts from its mean position, the time required by it to travel from $x = A$ to $x = \frac { \sqrt { 3 } } { 2 } A$ will be $\frac { \pi } { x } \mathrm {~s}$, where $x =$ $\_\_\_\_$ .

The time period of simple harmonic motion of mass $M$ in the given figure is $\pi\sqrt{\dfrac{\alpha M}{5K}}$, where the value of $\alpha$ is $\_\_\_\_$.

A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s . The maximum velocity of the particle is $\_\_\_\_$ $\mathrm { cm } / \mathrm { s }$.

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $4 \mathrm {~m} , 2 \mathrm {~ms} ^ { - 1 }$ and $16 \mathrm {~ms} ^ { - 2 }$ at a certain instant. The amplitude of the motion is $\sqrt { x } , \mathrm {~m}$ where $x$ is $\_\_\_\_$