Revision Test

A* Batch
[Physics]
[Fluid]

1. A ball of density d is dropped onto a horizontal solid surface. It bounces elastically from the surface and
returns to its original position in a time t1. Next, the ball is released and it falls through the same height before
striking the surface of a liquid of density dL.
(a) If d < dL, obtain an expression (in terms of d, t 1 and dL) for the time t2 the ball takes to come back to the
position from which it was released.
(b) Is the motion of the ball simple harmonic?
(c) If d = dL, how does the speed of the ball depend on its depth inside the liquid ?
Neglect all frictional and other dissipative forces. Assume the depth of the liquid to be large.

2. Two identical cylindrical vessels with their bases at the same level each contain a liquid of density  as shown
in figure. The height of the liquid in one vessel is h2 and other vessels h1, the area of either base is A. Find the
work done by gravity in equalizing the levels when the two vessels are connected.

h1
h2

Figure (1)

3. A cyllindrical wooden stick of length L, and radius R and density  has a small metal piece of mass m
(of negligible volume) attached to its one end. Find the minimum value for the mass m (in terms of given
parameters) that would make the stick float vertically in equilibrium in a liquid of density  (>).

4. A container of large uniform cross−sectional area A resting on a horizontal surface, holds two immiscible,
H
non−viscous and incompressible liquids of densities d and 2d, each of height as shown in figure. The lower
2
density liquid is open to the atmosphere having pressure P 0.
 H A
(a) A homogeneous solid cylinder of length L  L  cross−sectional area is immersed such that it floats
 2  5
L
with its axis vertical at the liquid−liquid interface with the length in the denser liquid. Determine:
4

Motion Education | 394-Rajeev Gandhi Nagar | : 1800-212-1799 | url : www.motion.ac.in | Page # 1

 (i) The density D of the solid and (ii) The total pressure at the bottom of the container.
(b) The cylinder is removed and the original arrangement is restored. A tiny hole of area s (s << A) is punched on
  H 
the vertical side of the container at a height h  h     . Determine :
  2 
(i) The initial speed of efflux of the liquid at the hole
(ii) The horizontal distance x travelled by the liquid initially and
(iii) The height hm at which the hole should be punched so that the liquid travels the maximum distance x m
initially. Also calculate xm. [Neglect air resistance in these calculations]

5. A container of cross-section area ‘S’ and height ‘h’ is filled with mercury up to the brim. Then the container is
sealed airtight and a hole of small cross section area ' S/n ' (where ‘n’ is a positive constant) is punched in its
bottom. Find out the time interval upto which the mercury will come out from the bottom hole.
[Take the atmospheric pressure to be equal to h0 height of mercury column: h > h0]

6. A Pitot tube is shown in figure. Wind blows in the direction shown. Air at inlet A is brought to rest, whereas
its speed just outside of opening B is unchanged. The U tube contains mercury of density m. Find the speed
of wind with respect to Pitot tube. Neglect the height difference between A and B and take the density of air
as a.

7. One end of a long iron chain of linear mass density  is fixed to a sphere of mass
m and specific density 1/3 while the other end is free. The sphere along with the
chain is immersed in a deep lake. If specific density of iron is 7, the height h h
above the bed of the lake at which the sphere will float in equilibrium is (Assume
that the part of the chain lying on the bottom of the lake exerts negligible force on
the upper part of the chain.) :

8. Two very large open tanks A & F both contain the same liquid. A horizontal pipe BCD, having a small
constriction at C, leads out of the bottom of tank A, and a vertical pipe E containing air opens into the
constriction at C and dips into the liquid in tank F. Assume streamline flow and no viscosity. If the cross
section area at C is one-half that at D, and if D is at distance h1 below the level of the liquid in A, to what
height h2 will liquid rise in pipe E? Express your answer in terms of h1. [Neglect changes in atmospheric
pressure with elevation. In the containers there is atmosphere above the water surface and D is also open to
atmosphere.]
P

Motion Education | 394-Rajeev Gandhi Nagar | : 1800-212-1799 | url : www.motion.ac.in | Page # 2

9. A side wall of a wide open tank is provided with a narrowing tube (as shown in figure) throught which water
flows out. The cross-sectional area of the tube decrease from S = 3.0 cm2 to s = 1.0 cm2. The water level in the
tank is h = 4.6 m higher than that in the tube. Neglecting the viscosity of the water, find the horizontal
component of the force tending to pull the tube out of the tank.

10. The horizontal bottom of a wide vessel with an ideal fluid has a round orifice of radius R 1 over which a round
closed cylinder is mounted, whose radius R 2 > R1. The clearance between the cylinder and the bottom of the
vessel is very small, the fluid density is . Find the static pressure of the fluid in the clearance as a function of
the distance r from the axis of the orifice (and the cylinder), if the height of the fluid is equal to h.

R1
h
R2

11. Water flows out of a big tank along a tube bent at right angles, the inside radius of the tube is equal to
r = 0.50 cm. The length of the horizontal section of the tube is equal to  = 22 cm.The water flow rate is
Q = 0.50 litres per second. Find the moment of reaction forces of flowing water, acting on the tube’s walls,
relative to the point O.



COMPREHENSION BASED QUESTIONS

Comprehension # 1
A wooden cylinder of diameter 4r, height H and density /3 is kept on a hole of diameter 2r of a tank, filled
with liquid of density  as shown in the figure.
12. If level of the liquid starts decreasing slowly when the level of liquid is at a height h1 above the cylinder the
block starts moving up. At what value of h1, will the block rise :

4H 5H 5H
(A) (B) (C) (D) Remains same
9 9 3

Motion Education | 394-Rajeev Gandhi Nagar | : 1800-212-1799 | url : www.motion.ac.in | Page # 3

13. The block in the above question is maintained at the position by external means and the level of liquid is
lowered. The height h2 when this external force reduces to zero is

4H 5H 2H
(A) (B) (C) Remains same (D)
9 9 3

14. If height h2 of water level is further decreased then,

(A) cylinder will not move up and remains at its original position.
(B) for h2 = H/3, cylinder again starts moving up
(C) for h2 = H/4, cylinder again starts moving up
(D) for h2 = H/5 cylinder again starts moving up

Comprehension # 2
A fixed thermally conducting cylinder has a radius R and height L0. The cylinder is open at its bottom and has
a small hole at its top. A piston of mass M is held at a distance L from the top surface, as shown in the figure.
The atmospheric pressure is P0.
2R

L0

Piston

15. The piston is now pulled out slowly and held at a distance 2L from the top. The pressure in the cylinder
between its top and the piston will then be
P0 P0 Mg P0 Mg
(A) P0 (B) (C) + (D) −
2 2 R 2 2 R 2

16. While the piston is at a distance 2L from the top, the hole at the top is sealed. The piston is then released, to a
position where it can stay in equilibrium. In this condition, the distance of the piston from the top is
 
2P0 R 2  P0 R 2 − Mg   P0 R 2 + Mg   P0 R 2 
(A)   (2L) (B)   (2L) (C)   (2L) (D)   (2L)
 R P0
2
+ Mg   R 2
P0   R 2
P0   R 2
P0 − Mg 

17. The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below
the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the
cylinder as shown in the figure. The density of the water is . In equilibrium, the height H of the water column
in the cylinder satisfies

L0

(A)  g (L0 – H)2 + P0 (L0 – H) + L0P0 = 0 (B)  g (L0 – H)2 – P0 (L0 – H) – L0P0 = 0
(C)  g (L0 – H)2 + P0 (L0 – H) – L0P0 = 0 (D)  g (L0 – H)2 – P0 (L0 – H) + L0P0 = 0
Motion Education | 394-Rajeev Gandhi Nagar | : 1800-212-1799 | url : www.motion.ac.in | Page # 4
18. A cylindrical vessel of height 500 mm has an orifice (small hole) at its bottom. The orifice is initially closed
and water is filled in it up to height H. Now the top is completely sealed with a cap and the orifice at the
bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady
with height of water column being 200 mm. Find the fall in height (in mm) of water level due to opening of
the orifice.
[Take atmospheric pressure = 1.0 × 105 N/m2 , density of water = 1000 kg/m3 and g = 10 m/s2. Neglect any
effect of surface tension]

19. Two solid spheres A and B of equal volumes but of different densities d A and dB are connected by a string.
They are fully immersed in a fluid of density dF. They get arranged into an equilibrium state as shown in the
figure with a tension in the string. The arrangement is possible only if

(A) dA < dF (B) dB > dF (C) dA > dF (D) dA + dB = 2dF

20. A solid sphere of radius R and density  is attached to one end of a mass-less spring of force constant k. The other
end of the spring is connected to another solid sphere of radius R and density 3. The complete arrangement is
placed in a liquid of density 2 and is allowed to reach equilibrium. The correct statement(s) is (are)
4 R 3  g
(A) the net elongation of the spring is
3k
8R3 g
(B) the net elongation of the spring is
3k
(C) the light sphere is partially submerged.
(D) the light sphere is completely submerged.

Comprehension # 3
A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross
section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston
pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the
spray gun shown, the radii of the piston and the nozzle are 20 mm and 1mm respectively. The upper end of the
container is open to the atmosphere.

21. If the piston is pushed at a speed of 5mms–1 , the air comes out of the nozzle with a speed of
(A) 0.1ms–1 (B) 1ms–1 (C) 2ms–1 (D) 8ms–1

22. If the density of air is a and that of the liquid , then for a given piston speed the rate (volume per unit time)
at which the liquid is sprayed will be proportional to
a 
(A) (B) a (C) (D) 
 a

Motion Education | 394-Rajeev Gandhi Nagar | : 1800-212-1799 | url : www.motion.ac.in | Page # 5

23. A closed tube in the form of an equilateral triangle of side  = 3m contains equal volumes of three liquids
which do not mix and is placed vertically with its lowest side horizontal. Find 'x' (in meter) in the figure if the
densities of the liquids are in A.P.

24. A wooden block with a coin placed on its top, floats in water as shown in figure. The distance  and h are
shown here. After some time the coin falls into the water. Then :

(A)  decreases and h increase (B)  increases and h decreases

(C) both  and h increases (D) both  and h decrease

Motion Education | 394-Rajeev Gandhi Nagar | : 1800-212-1799 | url : www.motion.ac.in | Page # 6