A $ U$ tube of uniform bore of cross-sectional area $A$ has been set up vertically with open ends facing up. Now $m$ gm of a liquid of density $d$ is poured into it. The column of liquid in this tube will oscillate with a period $T$ such that
Diffcult
Download our app for free and get started
(d) If the level of liquid is depressed by $y\, cm$ on one side, then the level of liquid in column $P$ is $2y\, cm $ higher than $B$ as shown.
The weight of extra liquid on the side $P = 2Aydg$.
This becomes the restoring force on mass $M$.
$\therefore $ Restoring acceleration $ = \frac{{ - \,2Aydg}}{M}$
This relation satisfies the condition of $SHM$ i.e. $a \propto - y$.
Hence time period $T = 2\pi \sqrt {\frac{{{\rm{Displacement}}}}{{{\rm{|Acceleration|}}}}} $
$ = 2\pi \sqrt {\frac{y}{{\frac{{2Aydg}}{M}}}} $
The weight of extra liquid on the side $P = 2Aydg$.
This becomes the restoring force on mass $M$.
$\therefore $ Restoring acceleration $ = \frac{{ - \,2Aydg}}{M}$
This relation satisfies the condition of $SHM$ i.e. $a \propto - y$.
Hence time period $T = 2\pi \sqrt {\frac{{{\rm{Displacement}}}}{{{\rm{|Acceleration|}}}}} $
$ = 2\pi \sqrt {\frac{y}{{\frac{{2Aydg}}{M}}}} $
$ \Rightarrow T = 2\pi \sqrt {\frac{M}{{2Adg}}} $
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A particle starts from a point $P$ at a distance of $A/2$ from the mean position $O$ and travels towards left as shown in the figure. If the time period of $SHM$ , executed about $O$ is $T$ and amplitude $A$ then the equation of motion of particle isView Solution
- 2If the displacement of a particle executing $SHM $ is given by $y = 0.30\sin (220t + 0.64)$ in metre, then the frequency and maximum velocity of the particle isView Solution
- 3View SolutionIn the adjacent figure, if the incline plane is smooth and the springs are identical, then the period of oscillation of this body is
- 4Time period of a particle executing $SHM$ is $8\, sec.$ At $t = 0$ it is at the mean position. The ratio of the distance covered by the particle in the $1^{st}$ second to the $2^{nd}$ second is :View Solution
- 5An $LCR$ circuit is equivalent to a damped pendulum. In an $LCR$ circuit the capacitor is charged to $Q_0$ and then connected to the $L$ and $R$ as shown below.View Solution
If a student plots graphs of the square of maximum charge $( Q_{Max} ^2 )$ on the capacitor with time$(t)$ for two different values $L_1$ and $L_2 (L_1 > L_2)$ of $L$ then which of the following represents this graph correctly? (plots are schematic and not drawn to scale)
- 6A particle is executing simple harmonic motion with a time period $T.$ At time $t = 0$, it is at its position of equilibrium. The kinetic energy-time graph of the particle will look likeView Solution
- 7An object of mass $0.2\ kg$ executes simple harmonic along $X-$ axis with frequency of $\frac{{25}}{\pi } Hz$ . At the position $x$ = $0.04\ m$ , the object has kinetic energy of $0.5\ J$ and potential energy of $0.4\ J$ amplitude of oscillation in meter is equal toView Solution
- 8Two simple pendulums of lengths $1.44 \,m$ and $1\, m$ start swinging together. After how many vibrations will they again start swinging togetherView Solution
- 9For a simple pendulum, a graph is plotted between its kinetic energy $(KE)$ and potential energy $(PE)$ against its displacement $d$. Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)View Solution
- 10The potential energy of a particle of mass $0.1\,kg,$ moving along $x-$ axis, is given by $U = 5x(x-4)\,J$ where $x$ is in metres. It can be concluded thatView Solution