Two identical cylindrical vessels are kept on | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The initial potential energy of the system is given by the sum of the potential energies of the liquid in both vessels:$U_{i} = (dSx_{1})g \cdot \

Vedclass · Accepted Answer

$\frac{1}{4} gdS(x_{2}-x_{1})^{2}$

Vedclass · Answer

(C) The initial potential energy of the system is given by the sum of the potential energies of the liquid in both vessels:$U_{i} = (dSx_{1})g \cdot \frac{x_{1}}{2} + (dSx_{2})g \cdot \frac{x_{2}}{2} = \frac{dSg}{2}(x_{1}^{2} + x_{2}^{2})$By the principle of conservation of volume,the total volume of liquid remains constant:$Sx_{1} + Sx_{2} = S(x_{f} + x_{f}) = 2Sx_{f}$$x_{f} = \frac{x_{1} + x_{2}}{2}$The final potential energy of the system is:$U_{f} = 2 	imes (dSx_{f})g \cdot \frac{x_{f}}{2} = dSgx_{f}^{2} = dSg \left( \frac{x_{1} + x_{2}}{2} ight)^{2}$The change in potential energy is $\Delta U = U_{f} - U_{i}$:$\Delta U = dSg \left[ \left( \frac{x_{1} + x_{2}}{2} ight)^{2} - \frac{x_{1}^{2} + x_{2}^{2}}{2} ight]$$\Delta U = dSg \left[ \frac{x_{1}^{2} + x_{2}^{2} + 2x_{1}x_{2}}{4} - \frac{2x_{1}^{2} + 2x_{2}^{2}}{4} ight]$$\Delta U = dSg \left[ \frac{2x_{1}x_{2} - x_{1}^{2} - x_{2}^{2}}{4} ight] = -\frac{dSg}{4}(x_{1} - x_{2})^{2}$The magnitude of the change in energy is $\frac{1}{4} gdS(x_{2} - x_{1})^{2}$.

Similar Questions

$A$ pump motor is used to deliver water at a certain rate from a given pipe. To obtain twice as much water from the same pipe in the same time, the power of the motor has to be increased to: (in $times$)

$A$ capillary tube $(A)$ is dipped in water. Another identical tube $(B)$ is dipped in a soap-water solution. Which of the following shows the relative nature of the liquid columns in the two tubes?

Two cylindrical vessels of equal cross-sectional area of $2 \ m^2$ contain water up to heights of $10 \ m$ and $6 \ m$,respectively. If the vessels are connected at their bottom,then the work done by the force of gravity is (Density of water is $10^3 \ kg/m^3$ and $g = 10 \ m/s^2$):

Two blocks of ice when pressed together join to form one block because

Vedclass Test Series

Exam Paper Generator

Online Exam Module