Find the time period of mass $M$ when displaced from its equilibrium position and then released for the system shown in figure.
Find the time period of mass $M$ when displaced from its equilibrium position and then released for the system shown in figure.
Here, for calculation of time period we have neglected acceleration of gravity. Because, it is constant at all places and doesn't affect on resultant restoring force.
Let in the equilibrium position, the spring has extended by an amount $x_{0}$. Let the mass be pulled through a distance $x$ and then released. But, string is inextensible, hence the spring alone will contribute the total extension $x+x=2 x$. So net extension in the spring will be $2 x+x_{0}$.
When the extension of spring is $x_{0},(\mathrm{M}$ is not suspended) the restoring force in spring $\mathrm{F}=2 k x_{0}, \ldots$ (1) $\left[\because \mathrm{F}=\mathrm{T}+\mathrm{T}\right.$ and $\left.\mathrm{T}=k x_{0}\right]$
When mass is suspended, the net extension in spring $2 x+x_{0}$ and restoring force in spring, $\mathrm{F}^{\prime}=2 k\left(2 x+x_{0}\right)=4 k x+2 k x_{0} \quad \ldots$ (2)
Restoring force on system,
$f^{\prime} =-\left(\mathrm{F}^{\prime}-\mathrm{F}\right)$
$=\mathrm{F}-\mathrm{F}^{\prime}$
$=2 k x_{0}-4 k x-2 k x_{0} \quad \text { [From equ. (1) and (2)] }$
$=-4 k x$
Similar Questions
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 the mass $M$ is displaced in the horizontal direction, then the frequency of oscillation of the system 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 the mass $M$ is displaced in the horizontal direction, then the frequency of oscillation of the system is
Two masses $m_1$ and $m_2$ are suspended together by a massless spring of constant $K$. When the masses are in equilibrium, $m_1$ is removed without disturbing the system. The amplitude of oscillations is
Two masses $m_1$ and $m_2$ are suspended together by a massless spring of constant $K$. When the masses are in equilibrium, $m_1$ is removed without disturbing the system. The amplitude of oscillations is
Two masses $m_1$ and $m_2$ are supended together by a massless spring of constant $k$. When the masses are in equilibrium, $m_1$ is removed without disturbing the system; the amplitude of vibration is
Two masses $m_1$ and $m_2$ are supended together by a massless spring of constant $k$. When the masses are in equilibrium, $m_1$ is removed without disturbing the system; the amplitude of vibration 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? (In ${m} / {s}^{2}$)
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? (In ${m} / {s}^{2}$)
- [JEE MAIN 2021]
A block with mass $M$ is connected by a massless spring with stiffiess 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 perticle with mass $m$ is placed on the mass $M$ ?
($A$) The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$, whereas in the second case it remains unchanged
($B$) The final time period of oscillation in both the cases is same
($C$) The total energy decreases in both the cases
($D$) The instantaneous speed at $x_0$ of the combined masses decreases in both the cases
A block with mass $M$ is connected by a massless spring with stiffiess 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 perticle with mass $m$ is placed on the mass $M$ ?
($A$) The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$, whereas in the second case it remains unchanged
($B$) The final time period of oscillation in both the cases is same
($C$) The total energy decreases in both the cases
($D$) The instantaneous speed at $x_0$ of the combined masses decreases in both the cases
- [IIT 2016]