The bob of a simple pendulum executes simple harmonic motion in water with a period $t$, while the period of oscillation of the bob is $t _ { 0 }$ in air. Neglecting frictional force of water and given that the density of the bob is $\left( \frac { 4 } { 3 } \right) \times 1000 \mathrm {~kg} / \mathrm { m } ^ { 3 }$. What relationship between $t$ and $t _ { 0 }$ is true?
(1) $t = t _ { 0 }$
(2) $t = t _ { 0 } / 2$
(3) $t = 2 t _ { 0 }$
(4) $t = 4 t _ { 0 }$

A particle at the end of a spring executes simple harmonic motion with a period $t _ { 1 }$, while the corresponding period for another spring is $t _ { 2 }$. If the period of oscillation with the two springs in series is $T$, then
(1) $T = t _ { 1 } + t _ { 2 }$
(2) $T^2 = t_1^2 + t_2^2$
(3) $\mathrm { T } ^ { - 1 } = \mathrm { t } _ { 1 } ^ { - 1 } + \mathrm { t } _ { 2 } ^ { - 1 }$
(4) $\mathrm { T } ^ { - 2 } = \mathrm { t } _ { 1 } ^ { -2 } + \mathrm { t } _ { 2 } ^ { -2 }$

The total energy of a particle executing simple harmonic motion is
(1) $\propto x$
(2) $\propto x ^ { 2 }$
(3) independent of $x$
(4) $\propto x ^ { 1 / 2 }$

A particle of mass $m$ is attached to a spring (of spring constant $k$) and has a natural angular frequency $\omega _ { 0 }$. An external force $F ( t )$ proportional to $\cos \omega t \left( \omega \neq \omega _ { 0 } \right)$ is applied to the oscillator. The time displacement of the oscillator will be proportional to
(1) $\frac { \mathrm { m } } { \omega _ { 0 } ^ { 2 } - \omega ^ { 2 } }$
(2) $\frac { 1 } { m \left( \omega _ { 0 } ^ { 2 } - \omega ^ { 2 } \right) }$
(3) $\frac { 1 } { m \left( \omega _ { 0 } ^ { 2 } + \omega ^ { 2 } \right) }$
(4) $\frac { m } { \omega _ { 0 } ^ { 2 } + \omega ^ { 2 } }$

The function $\sin^2(\omega t)$ represents
(1) a periodic, but not simple harmonic motion with a period $2\pi/\omega$
(2) a periodic, but not simple harmonic motion with a period $\pi/\omega$
(3) a simple harmonic motion with a period $2\pi/\omega$
(4) a simple harmonic motion with a period $\pi/\omega$

Two simple harmonic motions are represented by the equation $\mathrm{y}_1 = 0.1\sin\left(100\pi t + \frac{\pi}{3}\right)$ and $y_2 = 0.1\cos\pi t$. The phase difference of the velocity of particle 1 w.r.t. the velocity of the particle 2 is
(1) $-\pi/6$
(2) $\pi/3$
(3) $-\pi/3$
(4) $\pi/6$

If a simple harmonic motion is represented by $\frac{d^2x}{dt^2} + \alpha x = 0$, its time period is
(1) $\frac{2\pi}{\alpha}$
(2) $\frac{2\pi}{\sqrt{\alpha}}$
(3) $2\pi\alpha$
(4) $2\pi\sqrt{\alpha}$

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $\omega' =$
(1) $\frac{\omega m}{(m + 2M)}$
(2) $\frac{\omega(m + 2M)}{m}$
(3) $\frac{\omega(m - 2M)}{(m + 2M)}$
(4) $\frac{\omega m}{(m + M)}$

Four point masses, each of value $m$, are placed at the corners of a square $ABCD$ of side $\ell$. The moment of inertia through $A$ and parallel to $BD$ is
(1) $m\ell^2$
(2) $2m\ell^2$
(3) $3m\ell^2$
(4) $3m\ell^2$

Starting from the origin, a body oscillates simple harmonically with a period of 2 s. After what time will its kinetic energy be $75\%$ of the total energy?
(1) $\frac{1}{12}$ s
(2) $\frac{1}{6}$ s
(3) $\frac{1}{4}$ s
(4) $\frac{1}{3}$ s

The maximum velocity of a particle, executing simple harmonic motion with an amplitude 7 mm, is $4.4$ m/s. The period of oscillation is
(1) 100 s
(2) 0.01 s
(3) 10 s
(4) 0.1 s

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency $\omega$. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time
(1) at the highest position of the platform
(2) at the mean position of the platform
(3) for an amplitude of $\frac{g}{\omega^2}$
(4) for an amplitude of $\frac{g^2}{\omega^2}$

A mass $M$, attached to a horizontal spring, executes S.H.M. with amplitude $A_{1}$. When the mass $M$ passes through its mean position then a smaller mass m is placed over it and both of them move together with amplitude $A_{2}$. The ratio of $\left(\frac{A_{1}}{A_{2}}\right)$ is:
(1) $\frac{M+m}{M}$
(2) $\left(\frac{M}{M+m}\right)^{1/2}$
(3) $\left(\frac{M+m}{M}\right)^{1/2}$
(4) $\frac{M}{M+m}$

A solid sphere is rolling on a surface as shown in figure, with a translational velocity $v \mathrm{~ms}^{-1}$. If it is to climb the inclined surface continuing to roll without slipping, then minimum velocity for this to happen is
(1) $\sqrt{2gh}$
(2) $\sqrt{\frac{7}{5}gh}$
(3) $\sqrt{\frac{7}{2}gh}$
(4) $\sqrt{\frac{10}{7}gh}$

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) $2\pi \left[ \frac { M } { ( k + A\sigma g ) } \right] ^ { 1/2 }$

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 particle is moving in a circular path of radius $a$, with a constant velocity $v$ as shown in the figure. The centre of circle is marked by 'C'. The angular momentum from the origin O can be written as:
(1) $va(1+\cos 2\theta)$
(2) $va(1+\cos\theta)$
(3) $va\cos 2\theta$
(4) $va$

A pendulum with the time period of 1 s is losing energy due to damping. At a certain time, its energy is 45 J. If after completing 15 oscillations its energy has become 15 J, then its damping constant (in $\mathrm { s } ^ { - 1 }$) will be
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 15 } \ln 3$
(3) $\frac { 1 } { 30 } \ln 3$
(4) 2

A cylindrical block of wood (density $= 650 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$), of base area $30 \mathrm {~cm} ^ { 2 }$ and height 54 cm, floats in a liquid of density $900 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$. 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):
(1) 52 cm
(2) 26 cm
(3) 39 cm
(4) 65 cm

An oscillator of mass $M$ is at rest in its equilibrium position in a potential, $V = \frac { 1 } { 2 } k ( x - X ) ^ { 2 }$. A particle of mass $m$ comes from the right with speed $u$ and collides completely inelastic with $M$ and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is: ( $M = 10 , m = 5 , u = 1 , k = 1$ )
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { \sqrt { 3 } }$
(3) $\sqrt { \frac { 3 } { 5 } }$
(4) $\frac { 1 } { 2 }$

A particle executes simple harmonic motion and it is located at $x = a , b$ and $c$ at time $t _ { 0 } , 2 t _ { 0 }$ and $3 t _ { 0 }$ respectively. The frequency of the oscillation is:
(1) $\frac { 1 } { 2 \pi t _ { 0 } } \cos ^ { - 1 } \left( \frac { a + c } { 2 b } \right)$
(2) $\frac { 1 } { 2 \pi t _ { 0 } } \cos ^ { - 1 } \left( \frac { a + 2 b } { 3 c } \right)$
(3) $\frac { 1 } { 2 \pi t _ { 0 } } \cos ^ { - 1 } \left( \frac { a + b } { 2 c } \right)$
(4) $\frac { 1 } { 2 \pi t _ { 0 } } \cos ^ { - 1 } \left( \frac { 2 a + 3 c } { b } \right)$

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) $K _ { 2 } = K$