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
A body of mass $m $ is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass $m$ is slightly pulled down and released , it oscillates with a time period of $3\,s$ . When the mass $m$ is increased by $1\,kg$ , the time period of oscillations becomes $5\,s$ . The value of $m$ in $kg$ is
A body of mass $m $ is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass $m$ is slightly pulled down and released , it oscillates with a time period of $3\,s$ . When the mass $m$ is increased by $1\,kg$ , the time period of oscillations becomes $5\,s$ . The value of $m$ in $kg$ is
- [NEET 2016]
Two oscillating systems; a simple pendulum and a vertical spring-mass-system have same time period of motion on the surface of the Earth. If both are taken to the moon, then-
Two oscillating systems; a simple pendulum and a vertical spring-mass-system have same time period of motion on the surface of the Earth. If both are taken to the moon, then-
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
A spring having with a spring constant $1200\; N m ^{-1}$ is mounted on a hortzontal table as shown in Figure A mass of $3 \;kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $2.0 \;cm$ and released. Determine
$(i)$ the frequency of oscillations,
$(ii)$ maximum acceleration of the mass, and
$(iii)$ the maximum speed of the mass.
A spring having with a spring constant $1200\; N m ^{-1}$ is mounted on a hortzontal table as shown in Figure A mass of $3 \;kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $2.0 \;cm$ and released. Determine
$(i)$ the frequency of oscillations,
$(ii)$ maximum acceleration of the mass, and
$(iii)$ the maximum speed of the mass.
Three masses $700g, 500g$ and $400g$ are suspended at the end of a spring a shown and are in equilibrium. When the $700g$ mass is removed, the system oscillates with a period of $3\,seconds$, when the $500g$ mass is also removed, it will oscillate with a period of .... $s$
Three masses $700g, 500g$ and $400g$ are suspended at the end of a spring a shown and are in equilibrium. When the $700g$ mass is removed, the system oscillates with a period of $3\,seconds$, when the $500g$ mass is also removed, it will oscillate with a period of .... $s$