Spring Systems

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 ______.

A spring of force constant $15\text{ N/m}$ is cut into two pieces. If the ratio of their length is $1:3$, then the force constant of smaller piece is $\rule{3em}{0.4pt}\text{ N/m}$.

A spring of force constant $180$ N/m is loaded with a mass $0.20$ kg. The amplitude of oscillations is $4$ cm. When mass comes to equillibrium position, its velocity in m/s is

A body of mass $64$ g is made to oscillate turn by turn on two different springs A and B. Spring A and B has force constant $4$ N/m and $16$ N/m respectively. If $T_1$ and $T_2$ are the period of oscillations of springs A and B respectively, then $(T_1+T_2)/(T_1-T_2)$ will be

A spring force constant $180$ N/m is loaded with a mass $0.2$ kg. The amplitude of oscillations is $4$ cm. When mass comes to equilibrium position, its velocity is

Two particles A and B of equal masses are suspended from two massless springs of spring constants $K_A$ and $K_B$ respectively. The maximum velocities of the particles during oscillations are equal. The ratio of the amplitude of 'B' to that of 'A' is

Two springs of force constants '$2k$' and '$k$' are connected to a mass '$m$' as shown. Mass is displaced slightly to one side and released. The frequency of oscillation of the two springs-mass system is

A spring has length $l$ and force constant $k$. It is cut into 2 springs of length $l_1$ and $l_2$ such that $l_1 = nl_2$ ($n$=an integer). The force constant of the spring of length $l_2$ is $\left(k \propto \frac{1}{l}\right)$

Three masses 100g, 300g and 500g are suspended at the end of a spring as shown in figure and are in equilibrium. When the 500 g mass is removed, the system oscillates with a period of 3 second. When the 300 g mass is also removed, it will oscillate with a period of

A light spring is suspended with mass '$m_1$' at its lower end and its upper end is fixed to a rigid support. The mass is pulled down a short distance and then released. The period of oscillation is T second. When a mass ' ' is added to '$m_1$' and the system is made to oscillate the period is found to be $\frac{3}{2}$ T. The ratio $m_2\ \left(\frac{m_1}{m_2}\right)$ is

A spring executes S.H.M. with mass 10 kg attached to it. The force constant of spring is 10 N/m. If at any instant its velocity is 40 cm/s, the displacement at that instant is

(Amplitude of S.H.M. is 0.5 m)

A mass is suspended from a vertical spring which is executing S.H.M. of frequency $10$ Hz. The spring is unstretched at the highest point of oscillation. Maximum speed of the mass is ($g = 10$ m/s$^{2}$)

A spring produces extension $'x'$ by applying a force $'F'$ N. A body of mass $'m'$ suspended from spring oscillates vertically with period $'T'$. The mass of the suspended body is (Neglect mass of spring)

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 maximum energy in the spring are ______ Hz and ______ J, respectively.

(Take $g = 10\text{ m/s}^2$)

$S_1$ and $S_2$ are two identical springs. The frequency of oscillation of mass m is F. If one spring is removed, the frequency will become

A body has time period $T_1$ under the action of one force and $T_2$ under the action of another force. The square of the time period when both the forces are acting in the same direction is

A simple harmonic oscillator having four identical springs (as shown in figure) has a time period $T_1$. Another similar simple harmonic oscillator of the same mass $m$ but the four identical springs with spring constants $2k$ has a time period $T_2$, then $\frac{T_1}{T_2}$ is

All the springs in fig (a), (b) and (c) are identical, each one having force constant K. Mass m is attached to each system. If $T_a, T_b$ and $T_c$ are the periodic time of oscillations of the three systems in fig (a), (b) and (c) respectively, then

Two massless springs of spring constants $K_1$ and $K_2$ are connected one after the other forming a single chain, suspended vertically and a certain mass is attached to the free end. If $x_1$ and $x_2$ are their respective extensions and 'F' is their stretching force, the total extension produced is

A horizontal spring executes S.H.M. with amplitude $A_1$ when mass $m$ is attached to it. When it passes through mean position another mass $m$ is placed on it. Both masses move together with amplitude $A_2$. The ratio $\frac{A_1}{A_2}$ is

A mass $m_1$ performs S.H.M. with amplitude A which is connected to horizontal spring. While mass $m_1$ passing through mean position, another mass $m_2$ ($m_2 < m_1$) is placed on it so that both masses move together with amplitude $A_1$. The ratio of $A/A_1$ is

As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses $1$ kg and $0.2$ kg with a separation more than spring natural length and are released. Assuming the horizontal surface to be frictionless, the angular frequency (in SI unit) of the system is:

If the period of a oscillation of mass ' m ' suspended from a spring is 2 s, then the period of suspended mass ' 4 m ' with the same spring will be

Two identical point masses $P$ and $Q$, suspended from two separate massless springs of spring constants $k_1$ and $k_2$, respectively, oscillate vertically. If their maximum speeds are the same, the ratio ( $A_Q/A_P$) of the amplitude $A_Q$ of mass $Q$ to the amplitude $A_P$ of mass $P$ is

Two bodies A and B of equal mass are suspended from two massless springs of spring constant k1 and k2, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is

A spring executes S.H.M. with mass 1 kg attached to it. The force constant of the spring is $4\text{ N/m}$. If at any instant its velocity is $20\text{ cm/s}$, the displacement at that instant is (Amplitude of S.H.M. is 0.4 m)

A mass $m$ is suspended from a spring of negligible mass. The spring is pulled a little and then released, so that mass executes S.H.M. of time period $T$. If the mass is increased by $m_0$, the periodic time becomes $\frac{5T}{4}$. The ratio $\frac{m_0}{M}$

A mass $x$ gram is suspended from a light spring. It is pulled in downward direction and released so that mass performs S.H.M. of period T. If mass is increased by Y gram, the period becomes $\frac{4T}{3}$. The ratio of $Y/x$ is

In an oscillating spring mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly out of the box vertically so that the average frequency $\omega(t)$ and average amplitude $A(t)$ of the system change with time $t$. Which one of the following options schematically depicts these changes correctly?

A block of mass 2 kg is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is 2 m and spring constant is 200 N/m. The block is pushed such that the length of the spring becomes 1 m and then released. At distance x m (x < 2) from the wall, the speed of the block will be

A vertical spring oscillates with period 6 second with mass $m$ is suspended from it. When the mass is at rest, the spring is stretched through a distance of (Take, acceleration due to gravity, $g = \pi^2 = 10\text{ m/s}^2$)

A mass suspended from a vertical spring performs S.H.M. of period 0.1 second. The spring is unstretched at the highest point of suspension. Maximum speed of the mass is (Gravitational acceleration $g = 10\text{ m/s}^2$)

All the springs in fig. (a), (b) and (c) are identical, each having force constant K each. Mass m is attached to each system. If $T_a, T_b$ and $T_c$ are the time periods of oscillations of the three systems in fig. (a), (b) and (c) respectively, then

As shown in the figure, $S_1$ and $S_2$ are identical springs with spring constant K each. The oscillation frequency of the mass ' $m$ ' is ' $f$ '. If the spring $S_2$ is removed, the oscillation frequency will become

A mass ' $M$ ' attached to a horizontal spring executes S.H.M. of 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 $\left(\frac{A_1}{A_2}\right)$ is

Two blocks of masses $m$ and $M, (M > m)$, are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released, then ( $\mu =$ coefficient of friction between the two blocks) A. The time period of small oscillation of the two blocks is $T = 2\pi\sqrt{\frac{(m+M)}{k}}$

B. The acceleration of the blocks is $a = -\frac{kx}{M+m}$ ( $x =$ displacement of the blocks from the mean position)

C. The magnitude of the frictional force on the upper block is $\frac{m\mu|x|}{M+m}$

D. The maximum amplitude of the upper block, if it does not slip, is $\frac{\mu(M+m)g}{k}$

E. Maximum frictional force can be $\mu(M+m)g$.

Choose the correct answer from the options given below:

A spring has a certain mass suspended from it and its period of vertical oscillations is $T_1$. The spring is now cut into two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillations is now $T_2$. The ratio of $T_2/T_1$ is

A mass $m$ is suspended from a spring of negligible mass and the system oscillates with a frequency $f_1$. The frequency of oscillations if a mass $9m$ is suspended from the same spring is $f_2$. The value of $\frac{f_1}{f_2}$ is ________.

Two bodies A and B of equal mass are suspended from two separate massless springs of spring constants $K_1$ and $K_2$ respectively. The two bodies oscillate vertically such that their maximum velocities are equal. The ratio of the amplitude of $B$ to that of $A$ is

A mass ' $m$ ' attached to a spring oscillates with a period of 3 second. If the mass is increased by 0.6 kg, the period increases by 3 second. The initial mass ' $m$ ' is equal to

Three masses $500\text{ g}, 300\text{ g}$ and 100 g are suspended at the end of spring as shown in figure and are in equilibrium. When the 500 g mass is removed, the system oscillates with a period of 3 second. When the 300 g mass is also removed it will oscillate with a period of

Frequency of a particle performing S.H.M. is 10 Hz. The particle is suspended from a vertical spring. At the highest point of its oscillation the spring is unstretched. Maximum speed of the particle is $(g = 10\text{ m/s}^2)$

All the springs in fig. (a), (b) and (c) are identical, each having force constant K. Mass attached to each system is ' $m$ '. If $T_a, T_b$ and $T_c$ are the time periods of oscillations of the three systems respectively, then

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

A spring has a certain mass suspended from it and its period for vertical oscillations is '$T_1$'. The spring is now cut in to two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillations is now '$T_2$ '. The ratio $T_1/T_2$ is

For a simple harmonic motion in a mass spring system shown, the surface is frictionless. When the mass of the block is $1\text{ kg}$, the angular frequency is $\omega_1$. When the mass block is $2\text{ kg}$ the angular frequency is $\omega_2$. The ratio $\omega_2/\omega_1$ is

Four massless springs whose force constants are $2K,\ 2K,\ K$ and $2K$ respectively are attached to a mass $M$ kept on a frictionless plane as shown in figure. If mass $M$ is displaced in horizontal direction then frequency of oscillating system 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:

A block of mass 2 kg is attached with two identical springs of spring constant 20 N/m each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is $\frac{\pi}{\sqrt{x}}$ in SI unit. The value of $x$ is ____________.

A light spring is suspended with mass $m_1$ at its lower end and its upper end fixed to a rigid support. The mass is pulled down a short distance and then released. The period of oscillation is $T$ second. When a mass $m_2$ is added to $m_1$ and the system is made to oscillate, the period is found to be $\frac{3}{2}T$. The ratio $m_1 : m_2$ is

The upper end of the spring is fixed and a mass '$m$' is attached to its lower end. When mass is slightly pulled down and released, it oscillates with time period 3 second. If mass '$m$' is increased by $1\text{ kg}$, the time period becomes 5 second. The value of '$m$' is (mass of spring is negligible)

A mass m attached to free end of a spring executes SHM with a period of 1s. If the mass is increased by 3 kg the period of the oscillation increases by one second, the value of mass m is ___________ kg.

A mass $M$ is suspended from a light spring. An additional mass $M_1$ added extends the spring further by a distance $x$. Now, the combined mass will oscillate on the spring with period $T =$

A uniform circular disc of mass $12\text{ kg}$ is held by two identical springs. When the disc is slightly pressed down and released, it executes S.H.M. of period 2 second. The force constant of each spring is (nearly) (Take $\pi^2 = 10$)

Under the influence of force $F_1$ the body oscillates with a period $T_1$ and due to another force $F_2$ body oscillates with period $T_2$. If both forces acts simultaneously, then the resultant period is (consider displacement is same in all three cases)

In the figure given below, a block of mass $M = 490\text{ g}$ placed on a frictionless table is connected with two springs having same spring constant $\left(K = 2\text{ N m}^{-1}\right)$. If the block is horizontally displaced through '$X$' $\text{m}$ then the number of complete oscillations it will make in $14\pi$ seconds will be _____________.

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:

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 $3\text{ s}$, then the period of oscillation in figure (b) will be $\sqrt{x}\text{ s}$. The value of $x$ is ___________.

In figure $(A)$, mass '$2m$' 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.

A mass $0.9\text{ kg}$, 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 $124\text{ g}$ is placed over it and both masses move together with amplitude $A_2$. If the ratio $\frac{A_1}{A_2}$ is $\frac{\alpha}{\alpha-1}$, then the value of $\alpha$ will be ___________.

A spring is stretched by 5 cm by a force 10 N. The time period of the oscillations when a mass of 2 kg is suspended by it is -

Two particles A and B of equal masses are suspended from two massless springs of spring constants K1 and K2 respectively. If the maximum velocities during oscillations are equal, the ratio of the amplitude of A and B is

Two identical springs of constant '$K$' are connected in series and parallel in shown in figure. A mass '$M$' is suspended from them. The ratio of their frequencies is series to parallel combination will be

If two similar springs each of spring constant K1 are joined in series, the new spring constant and time period would be changed by a factor:

Two identical springs of spring constant '2k' are attached to a block of mass m and to fixed support (see figure). When the mass is displaced from equilibrium position on either side, it executes simple harmonic motion. The time period of oscillations of this system is:

Consider two identical springs each of spring constant k and negligible mass compared to the mass M as shown. Fig. 1 shows one of them and Fig. 2 shows their series combination. The ratios of time period of oscillation of the two SHM is Tb/Ta = $\sqrt{x}$, where value of x is ___________. (Round off to the Nearest Integer)

A mass '$m_1$ ' is suspended from a spring of negligible mass. A spring is pulled slightly in downward direction and released, mass performs S.H.M. of period '$T_1$ '. If the mass is increased by '$m_2$', the time period becomes '$T_2$'. The ratio $\frac{m_2}{m_1}$ is

A mass of 5 kg is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length 4 m has the same period of oscillation as the spring system. What is the value of acceleration due to gravity on the planet where these experiments are performed?

The motion of a mass on a spring, with spring constant K is as shown in figure.

The equation of motion is given by

x(t) = A sin$\omega$t + B cos$\omega$t with $\omega = \sqrt{\frac{K}{m}}$

Suppose that at time t = 0, the position of mass is x(0) and velocity v(0), then its displacement can also be represented as x(t) = C cos($\omega$t $-\phi$), where C and $\phi$ are:

A particle is suspended from a vertical spring which is executing S.H.M. of frequency $5\text{ Hz}$. The spring is unstretched at the highest point of oscillation. Maximum speed of the particle is $(g = 10\text{ m/s}^2)$

A body executes SHM under the action of force '$F_1$' with time period '$T_1$ '. If the force is changed to '$F_2$', it executes SHM with period '$T_2$'. If both the forces '$F_1$ ' and '$F_2$' act simultaneously in the same direction on the body, its time period is

A particle connected to the end of a spring executes S.H.M. with period '$T_1$'. While the corresponding period for another spring is '$T_2$'. If the period of oscillation with two springs in series is 'T', then

A body performs S.H.M. under the action of force '$F_1$ ' with period '$T_1$' second. If the force is changed to '$F_2$' it performs S.H.M. with period '$T_2$ ' second. If both forces '$F_1$' and '$F_2$' act simultaneously in the same direction on the body, the period in second will be

In the given figure, a body of mass M is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant k, the frequency of oscillation of given body is: