Speed of Efflux

Q.

Two identical cylindrical vessels are kept on the ground and each contain the same liquid of density $\mathrm{d}$. The area of the base of both vessels is S but the height of liquid in one vessel is ${\mathrm{x}}_1$ and in the other ${\mathrm{x}}_2$. When both cylinders are connected through a pipe of negligible volume very close to the bottom, the liquid flows from one vessel to the other until it comes to equilibrium at a new height. The change in energy of the system in the process is:

1. $\mathrm{gdS}{\left({\mathrm{x}}_2+{\mathrm{x}}_1\right)}^2$

2. $\mathrm{gdS}{\left({{\mathrm{x}}^2}_2+{\mathrm{x}}_1\right)}^2$

3. $\frac{1}{4}\mathrm{gdS}{\left[{\mathrm{x}}_2-{\mathrm{x}}_1\right]}^2$

4. $\frac{3}{4}\mathrm{gdS}{\left[{\mathrm{x}}_2-{\mathrm{x}}_1\right]}^2$

Q.

The pressure at any point inside a liquid is directly proportional to density of the liquid.

1. True
2. False

Q.

Derive an expression for the pressure exerted by the liquid having density d, at a depth h.

Q. A tank is filled with water upto a height H. Water is allowed to come out of a hole P in one of the walls at a depth h below the surface of water as shown in figure. Express the horizontal distance X in terms of H and h

1. $X=\sqrt{h(H-h)}$
2. $X=\sqrt{\frac{h}{2}(H-h)}$
3. $X=2\sqrt{h(H-h)}$
4. $X=4\sqrt{(H-h)}$

Q.

The table shows a familiar pattern from geometry. Determine what the variables $\mathrm{x}$ and $\mathrm{y}$ represent. Then write a function rule that relates $\mathrm{y}$ to $\mathrm{x}$.

Is the function a linear function?

Q. A cylindrical vessel is filled with a liquid up to a height $H$. A small hole is made in the vessel at a distance $y$ below the liquid surface as shown in figure. The liquid emerging from the hole strikes the ground at distance $X$.

1. $X$ remains same if the hole is at depth $y$ or $H-y$.
2. $X$ is maximum for $y=\frac{H}{2}$.
3. both (a) and (b) are correct.
4. both (a) and (b) are wrong.

Q. A very small hole is made at the bottom of a tank filled with water (density $=1000{\text{kg/m}}^3$). If the total pressure at the bottom of the tank is $3\text{atm}$ $(1\text{atm}={10}^5{\text{N/m}}^2)$, then the velocity of efflux is:
Take $g=10{\text{m/s}}^2$.

1. $20\text{m/s}$
2. $\sqrt{300}\text{m/s}$
3. $\sqrt{600}\text{m/s}$
4. $\sqrt{500}\text{m/s}$

Q. If the velocity head of a stream of water is equal to $10\text{cm}$, then its speed of flow is approximately

1. $1.2\text{m/s}$
2. $1.4\text{m/s}$
3. $1.7\text{m/s}$
4. $1.8\text{m/s}$

Q. A large tank is filled with water $(\text{density}={10}^3{\text{kg/m}}^3).$ A small hole is made at a depth of $10\text{m}$ below water surface. The range of water coming out of the hole is $R$ on the ground. What extra pressure must be applied on the water surface so that the range become $2R$ ?
$(p_0=1\text{atm}={10}^5\text{Pa}$ and $g=10{\text{m/s}}^2)$

1. $1\text{atm}$
2. $2\text{atm}$
3. $4\text{atm}$
4. $3\text{atm}$

Q.

Write the mathematical relationship describing the pressure exerted by a liquid at a point inside it.

Q. The height of water level in a tank is $H=100\text{cm}$. The water stream is coming out of a hole at the depth of $20\text{cm}$ from top surface of water. Find out the time taken by the water coming out of the hole to reach the ground and the horizontal range of water. (Take $g=1000{\text{cm/s}}^2$)

1. $0.4\text{sec},80\text{cm}$
2. $0.4\text{sec},90\text{cm}$
3. $0.2\text{sec},80\text{cm}$
4. $0.2\text{sec},90\text{cm}$

Q. A cylinder of height 20 m is completely filled with water. The velocity of efflux of water (in m/s) through a small hole on the side wall of the cylinder near its bottom is

1. 10
2. 20
3. 25.5
4. 5

Q. A water tank stands on the roof of a building as shown in the figure. The value of height of water above the hole $\left(h\right)$ for which the horizontal distance covered by the water $\left(x\right)$ is maximum -

1. $0.5\text{m}$
2. $0.8\text{m}$
3. $1\text{m}$
4. $0.2\text{m}$

Q. Consider a horizontally oriented syringe containing water whose nozzle is located at a height of $20\text{m}$ above the ground. The diameter of the plunger is $10\text{mm}$ and the diameter of the nozzle is $5\text{mm}$. The plunger is pushed with a constant speed of $1\text{m/s}$. Horizontal range of the water stream on the ground (in meters) is

Q. Water and mercury are filled in two identical cylindrical vessels up to the same height. Both vessels have a tiny hole in the wall at the bottom. The velocity of water and mercury coming out of the holes are $v_1$ and $v_2$ respectively. Then:
[Density of mercury $=13.6\text{g/cc}$]

1. $v_1=v_2$
2. $v_1=13.6v_2$
3. $v_1=\frac{v_2}{13.6}$
4. $v_1=\sqrt{13.6}{v}_2$

Q. A cylindrical vessel contains a liquid of density $\rho$ upto a height $h$. The liquid is closed by a piston of mass $m$ and area of cross-section $A$. There is a small hole at the bottom of the vessel. The speed $v$ with which the liquid comes out of the hole is

1. $\sqrt{2gh}$
2. $\sqrt{2(gh+\frac{mg}{\rho A})}$
3. $\sqrt{2(gh+\frac{mg}{A})}$
4. $\sqrt{2gh+\frac{mg}{A}}$

Q. A large tank is filled with water to a height $H$. A small hole is made at the base of the tank. It takes $T_1$ time to decrease the height of water to $\frac{H}{\eta }(\eta >1),$ and it takes $T_2$ time to take out the rest of the water. If $T_1=T_2,$ then the value of $\eta$ is

1. $2$
2. $3$
3. $4$
4. $5$

Q. A tap is attached to a water tank as shown in figure. Water level above the tap is maintained at $h=0.2\text{m}$. The cross-sectional area of the tap is $\sqrt{2}\times {10}^{-4}{\text{m}}^2$. Assuming constant pressure throughout the stream of water, find the velocity & cross-sectional area of the stream $0.2\text{m}$ below the opening of the tap :-
$[\text{Assume}g=10{\text{m/s}}^2]$

1. $2\sqrt{2}\text{m/s};{10}^{-4}{\text{m}}^2$
2. $1\text{m/s};{10}^{-4}{\text{m}}^2$
3. $2\text{m/s};{10}^{-4}{\text{m}}^2$
4. $2\text{m/s};{10}^{-5}{\text{m}}^2$

Q. Water and mercury are filled in two identical cylindrical vessels up to the same height. Both vessels have a tiny hole in the wall at the bottom. The velocity of water and mercury coming out of the holes are $v_1$ and $v_2$ respectively. Then:
[Density of mercury $=13.6\text{g/cc}$]

1. $v_1=v_2$
2. $v_1=13.6v_2$
3. $v_1=\frac{v_2}{13.6}$
4. $v_1=\sqrt{13.6}{v}_2$

Q. Water is allowed to come out of a hole $P$ in one of the walls at a depth $h_1$ below the surface of water. Express the horizontal distance covered by water jet in terms of $h_1$ and $h_2$.

1. $\sqrt{h_1{h}_2}$
2. $2\sqrt{h_1{h}_2}$
3. $\sqrt{\frac{h_1}{h_2}}$
4. $2\sqrt{\frac{h_1}{h_2}}$

Q. A cylindrical vessel is filled with water upto a height of $4\text{m}$. The cross-sectional area of the orifice at the bottom is $\frac{1}{100}$ that of the vessel. Then:

1. 1.5 minutes is the required time to empty the tank.
2. Initial velocity of discharge is $8.85\text{m/s}$.
3. Velocity of discharge will decrease with time.
4. Same amount of water can be thrown out from the orifice in $45\text{seconds}$ if the level of water is maintained at $4\text{m}$.

Q.

Water and mercury are filled in two cylindrical vessels up to same height. Both vessels have a hole in the wall near the bottom. The velocity of water and mercury coming out of the holes are v₁ and v₂ respectively.

1. v₁ = v₂.

2. v₁ = 13.6 v₂.

3. v₁ = v₂/13.6.

4. 

Q. A tank is filled upto a height $H$. The horizontal range of water coming out of the hole which is at depth $\frac{H}{4}$ from the surface level of water is

1. $\frac{2H}{\sqrt{3}}$
2. $\frac{\sqrt{3}H}{2}$
3. $\sqrt{3}H$
4. $\frac{3H}{4}$

Q. A large open tank has two holes in the wall. One is a square hole of side $L$ at a depth y from the top and the other is a circular hole of radius $R$ at a depth $4y$ from the top. When the tank is completely filled with water, if the quantity of water flowing out per second from both the holes is same, then $R$ is equal to

1. $\frac{L}{\sqrt{2\pi }}$
2. $2\pi L$
3. $L$
4. $\frac{L}{2\pi }$

Q. A wide vessel of uniform cross-section with a small hole at the bottom is filled with a $40\text{cm}$ thick layer of water and $30\text{cm}$ thick layer of kerosene. The relative density of kerosene is $0.8$. The initial velocity of flow of the water streaming out of the hole is -

1. $\frac{2}{\sqrt{5}}{\text{ms}}^{-1}$
2. $\frac{4}{\sqrt{5}}{\text{ms}}^{-1}$
3. $\frac{6}{\sqrt{5}}{\text{ms}}^{-1}$
4. $\frac{8}{\sqrt{5}}{\text{ms}}^{-1}$

Q. Water is filled in a tank upto $4\text{m}$ height. The base of the tank is at height $1\text{m}$ above the ground. At what height should the hole be made from the ground, so that the water can be sprayed upto maximum horizontal distance on ground?

1. $2\text{m}$
2. $2.5\text{m}$
3. $1.5\text{m}$
4. $3\text{m}$

Q. The velocity of efflux of a liquid for a given height of water column in an open tank, through a small orifice at the bottom of the tank does not depend on:

1. density of liquid
2. height of the liquid column above the orifice
3. acceleration due to gravity
4. none of the above

Q. A water tank standing on the floor has two small holes punched in the vertical wall, one above the other. The holes are $2.4\text{cm}$ and $7.6\text{cm}$ above the floor. If the jets of water from the holes hit the floor at the same point, then the height of water in the tank is

1. $10\text{cm}$
2. $5\text{cm}$
3. $20\text{cm}$
4. $4.8\text{cm}$

Q. A tank is filled upto height $h$ with a liquid and is placed on a platform of height $h$ from the ground. To get maximum range $x_m$, a small hole is punched at a distance $y$ from the free surface of the liquid. Then find the value of maximum range ($x_m$).

1. $x_m=h$
2. $x_m=2h$
3. $x_m=4h$
4. $x_m=\frac{h}{2}$

Q. A cylindrical tank of height $H$ is completely filled with water. On its vertical side, there are two tiny holes, one above the middle at a height of $h_1$ and other below the middle at a depth of $h_2$. If the jets of water from the holes meet at the same point on the ground, then the ratio $\frac{h_1}{h_2}$ is