Spring-mass systems

A spring stretches by 2 mm when it is loaded with a mass of 200 g. From equilibrium position the mass is further pulled down by 2 mm and released. The frequency associated with the system and maxmimum energy in the spring are ________ Hz and ________ J, respectively.

(Take $g=10$ m/s²)

The frequency of oscillation of a mass $m$ suspended by a spring is $v_1$. If the length of the spring is cut to half, the same mass oscillates with frequency $v_2$. The value of $v_2/v_1$ is ________.

The center of a disk of radius $r$ and mass $m$ is attached to a spring of spring constant $k$, inside a ring of radius $R > r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $T = \dfrac{2\pi}{\omega}$. The correct expression for $\omega$ is ($g$ is the acceleration due to gravity):

A block of mass $5\,\mathrm{kg}$ moves along the $x$-direction subject to the force $F = (-20x + 10)\,\mathrm{N}$, with the value of $x$ in metre. At time $t = 0\,\mathrm{s}$, it is at rest at position $x = 1\,\mathrm{m}$. The position and momentum of the block at $t = (\pi/4)\,\mathrm{s}$ are

If the collision occurs at time $t_0 = 0$, the value of $v_{\text{cm}}/(a\omega)$ will be _____.

If the collision occurs at time $t_0 = \pi/(2\omega)$, then the value of $4b^2/a^2$ will be _____.

On a frictionless horizontal plane, a bob of mass $m = 0.1$ kg is attached to a spring with natural length $l_0 = 0.1$ m. The spring constant is $k_1 = 0.009\ \mathrm{N\,m^{-1}}$ when the length of the spring $l > l_0$ and is $k_2 = 0.016\ \mathrm{N\,m^{-1}}$ when $l < l_0$. Initially the bob is released from $l = 0.15$ m. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T = (n\pi)$ s, then the integer closest to $n$ is _____.

Two massless springs with spring constants 2 k and 9 k, carry 50 g and 100 g masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be :

In figure (A), mass '2 m' is fixed on mass 'm' which is attached to two springs of spring constant k. In figure (B), mass 'm' is attached to two springs of spring constant 'k' and '2k'. If mass 'm' in (A) and in (B) are displaced by distance '$x$' horizontally and then released, then time period $T_1$ and $T_2$ corresponding to (A) and (B) respectively follow the relation.

As per the given figure, two blocks each of mass 250 g are connected to a spring of spring constant 2 Nm$^{-1}$. If both are given velocity $v$ in opposite directions, then maximum elongation of the spring 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 0.9 kg, attached to a horizontal spring, executes SHM with an amplitude $A_1$. When this mass passes through its mean position, then a smaller mass of 124 g is placed over it and both masses move together with amplitude $A_2$. If the ratio $\dfrac{A_1}{A_2}$ is $\dfrac\alpha{\alpha-1}$, then the value of $\alpha$ will be ______.

In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillation will be :

One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin (0,0). A point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\vec p$ pointing towards the charge $q$ is fixed at the origin, the spring gets stretched to a length $l$ and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $\Delta l \ll l$ from its equilibrium position and released, it is found to oscillate at frequency $\dfrac1\delta\sqrt{\dfrac km}$. The value of $\delta$ is ______.

A block of mass $2M$ is attached to a massless spring with spring-constant $k$. This block is connected to two other blocks of masses $M$ and $2M$ using two massless pulleys and strings. The accelerations of the blocks are $a_1$, $a_2$ and $a_3$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $x_0$. Which of the following option(s) is/are correct?

[$g$ is the acceleration due to gravity. Neglect friction]

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0\ \mathrm{N\,m^{-1}}$ and the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0 kg moving with a speed of $2.0\ \mathrm{m\,s^{-1}}$ collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is ______.

A 1 kg block attached to a spring vibrates with a frequency of 1 Hz on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an 8 kg block placed on the same table. So, the frequency of vibration of the 8 kg block is :

A block with mass $M$ is connected by a massless spring with stiffness constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_0$. Consider two cases: (i) when the block is at $x_0$; and (ii) when the block is at $x=x_0+A$. In both the cases, a particle with mass $m\,(<M)$ is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass $m$ is placed on the mass $M$?

Two bodies of masses 1 kg and 4 kg are connected to a vertical spring, as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency 25 rad/s, and amplitude 1.6 cm while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is (take $g=10$ ms$^{-2}$).

A solid sphere of radius $R$ and density $\rho$ is attached to one end of a mass-less spring of force constant $k$. The other end of the spring is connected to another solid sphere of radius $R$ and density $3\rho$. The complete arrangement is placed in a liquid of density $2\rho$ and is allowed to reach equilibrium. The correct statement(s) is (are)