What are Fluids ? Why can?t fluids retain their shape ? What is Viscosity ?

What is Pressure ? Derivation of pressure due to a liquid at depth h inside it.

Water is filled to height h in a vessel as shown in the figure.

Find the pressure exerted by water on the botton face of vessel.

Water is poured upto same height in the three vessels having the same area of base, as shown in the figure.

Is the force exerted by water on the base of vessels equal ?

Is the weight of water in all the vessels equal ?

Atmospheric Pressure and Absolute Pressure.

Consider a rectangular tank of size (l b w) filled with a liquid of density r to a height H as shown in fig. Find the force at the base and on the walls of the tank.

What is the effective point of application of force on the side wall?

Find the force exerted by the liquid on the side walls of the vessel in the situations shown in the figure.

Density of liquid is = r, Width of vessel is = w

Water is contained in a vessel as shown in the figure. Compute the horizontal and vertical components of force due to hydrostatic pressure on the section AB, which is a quarter of a cylinder of radius r. Given that

r = 5m and that the gate is 4 m wide.

Pascals Law

In a hydraulic press, the cross sectional area of the two cylinders is A1 and A2 respectively. A force F1 is applied to smaller cylinder.

a) What is the pressure produced in the cylinders?

b) What is the thrust exerted on the larger plunger?

c) How much work is done by the operator, if the smaller plunger moves down a distance d1?

Pressure inside a fluid uniformly accelerating in horizontal direction.

Figure shows an L-shaped tube filled with a liquid to a height h. What should be the horizontal acceleration a of the tube so that the pressure at the point B becomes atmospheric.

Pressure inside a fluid uniformly accelerating in vertical or slanted direction.

A trolley containing a liquid slides down a smooth inclined of angle a with the horizontal. Find the angle of inclination q of the free surface with the horizontal.

Equation of Surface of a Rotating Fluid.

A cylindrical vessel of radius R and height H is filled up to 4H/5 with a liquid of specific gravity . The vessel is rotated about its axis.

a) Determine the speed of rotation when the liquid just starts spilling.

b) Determine the height of lowest point of surface of liquid at the above speed.

c) Find the speed of rotation when the base is just visible.

Buoyancy and Archimedes? Principle.

A wooden object floats in water kept in a beaker. The object is near a side of the beaker. Let P1, P2, P3 be the pressures at the three points A, B and C of the bottom as shown in figure.

(a) P1 = P2 = P3

(b) P1 < P2 < P3

(c) P1 > P2 > P3

(d) P1 = P2 ? P3

A piece of wood floats in water kept in a beaker. If the beaker moves with a vertical acceleration a, the wood will

(a) sink deeper in the liquid if a is upward

(b) sink deeper in the liquid if a is downward, with a < g

(c) come out more from the liquid if a is downward, with a < g

(d) remain in the same position relative to the water

Relative placement of Center of Gravity and Center of Buoyancy determines the Equilibrium of a Floating body. What is Meta-Center ?

A wooden stick of length L, radius R and density r 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 s ( > r ).

Understanding Streamlined and Turbulent flow. Equation of Continuity.

Derivation and Understanding of Bernoulli?s Equation. Is Bernoulli?s Equation fluid version of Work-Kinetic Energy theorem ?

Some applications of the Bernoulli?s Equation to some special cases.

In a streamline flow,

(a) the speed of the particle always remains same

(b) the velocity of particle always remains same

(c) the kinetic energies of all the particles arriving at a given point are the same

(d) the momentum of all the particles arriving at a given point are the same.

Consider a uniform cylindrical tube completely filled with water. Water enters the tube through end A with speed v1 and leaves through end B with speed v2. In case I the tube is horizontal, in case II it is vertical with the end A upward and in case III it is vertical with the end B upward.

We have v1 = v2 for

(a) case I

(b) case II

(c) case III

(d) all cases.

Water flows smoothly through the pipe shown in the figure, descending in the process. Rank the four numbered sections of pipe according to

a) the volume flow rate Rv through them,

b) the flow speed v through them, and

c) the water pressure P at them, greatest first.

Explanation of Surface Tension and Surface Energy.

Water is filled up to a height h in a beaker of radius R as shown in figure. The density of water is r, the surface tension of water is T and the atmospheric pressure is Po. Consider a vertical section ABCD of the water column through a diameter of the beaker. The force on water on one side of this section by water on the other side of this section has magnitude

Explanation of Viscosity.

An open U-tube contains two liquids of different densities. If the density of heavier liquid is , find the density of lighter liquid in terms of the heights. h1 and h2

A U-tube of uniform cross section contains mercury (density r) in both of its arms.

Liquids of different densities are poured into each arm of the tube until the upper surfaces of both the liquids are in the same horizontal level.

If the density of the liquids is h1 and h2 times the density of mercury, find the ratio of heights of the two liquids.

A circular tube of uniform cross section is filled with two liquids of densities r1 and r2 such that each liquid occupies a quarter of volume of the tube.

If the line joining the interface of liquids makes an angle q with vertical, find the value of q.

A solid hemisphere of radius R is made to just sink in a liquid of density r. Find

(a) the vertical thrust on the curved surface,

(b) the side thrust on the hemisphere,

(c) the vertical thrust on the flat surface,

(d) the total hydrostatic force acting on the hemisphere.

The vessel shown in figure has two section of areas of cross section A1, and A2.

A liquid of density r fills both the section up to a height h in each. Neglect air pressure. Mark the correct options.

A metallic block weighs 100 g in air, and weighs only 93.6 g when immersed in water. It is known that some copper is mixed with the gold. Find the amount of copper added. (density of gold is 19.3 g/cm3 and that of copper is 8.9 g/cm3)

A piece of ice is floating in water. What will happen to the level of water when all ice melts? What will happen if the vessel is filled not with water but with liquid

a) denser than water

b) lighter than water

A cubical block of iron 5 cm on each side is floating on mercury in a vessel

a) What is the height of the block above mercury level?

b) Water is poured into the vessel so that it just covers the iron block? What is the height of the water column?

Density of mercury = 13.6 gm/cm3, density of iron = 7.2 gm/cm3

A block of wood is floating in water in a closed vessel as shown in the figure. The vessel is connected to an air pump. When more air is pushed into the vessel, the block of wood floats with ( neglect compressibility of water )

a) larger part in the water

b) smaller part in the water

c) same part in the water

d) at some instant it will sink

A uniform cylinder of density r and cross-sectional area A floats in equilibrium in two non-mixing liquids of densities r1 and r2 as shown in the figure. The length of the part of the cylinder in air is h and the lengths of the part of cylinder immersed in the liquid are h1 and h2 as shown in the figure.

A boat floating in a water tank is carrying a large stone. If the stone is unloaded into water, what will happen to the water level?

A rod of length 6m has mass 12 kg. If it is hinged at one end at a distance of 3m below the water surface. (Specific gravity of the material of the rod is 0.5). Find

a) the length of rod under water

b) angle made by rod with the vertical

c) What weight must attached to the other end of the rod so that 5m of the rod is submerged?

d) Find the magnitude and direction of the force exerted by the hinge on the rod.

A tension in a string holding a solid block below the surface of a liquid as in figure is T when the system is at rest.

Then what will be the tension in the string if the system has upward acceleration a ?

A U-tube contains two liquids of densities r1 and r2. The tube is now given an acceleration a in the horizontal direction and the height difference in the sections is as shown. What is the ratio r1:r2 ?

A no uniform cylinder of mass m, length l and radius r is having its center of mass at a distance l/4 from the center and lying on the axis of the cylinder. The cylinder is kept in a liquid of uniform density r. The moment of inertia of the rod about the center of mass is I. The angular acceleration of the point A relative to point B just after the rod is released from the position as shown in figure.

Water is emerging slowly and smoothly from a tap. Find the radius of cross-section of water as a function of depth h fallen from the tap.

m is gently placed on the middle of the surface is depressed by a distance y. The surface tension of liquid is given by

Water is filled in a vessel to a height h. A small orifice is made at the bottom of vessel. Find the speed of efflux with which water comes out from the orifice.

(Area of cross-section of orifice is negligible as compared to the area of cross-section of the vessel)

A vessel with a small orifice in its bottom is field with water and kerosene. Density of water is r1 and density of kerosene is r2 (r1 > r2). Find the velocity of water flow if the height of the water layer is h1 and that of the kerosene layer is h2.

Neglect viscosity.

A Venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two sections of the pipe; The cross sectional area A of the entrance and exit of the meter matches the pipe?s cross-sectional area. Between the entrance the exit, the fluid flows from the pipe with speed v and then through a narrow region of cross-sectional area a with speed V. A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid?s speed is accompanied by a pressure difference between the wider and narrower region, which causes a height difference h of the liquid in the two arms of the nanometer.

Find the speed of flow?

A pitot tube is mounted along the axis of a gas pipeline having cross-sectional area A. If the densities of the liquid and the gas are and respectively and the difference in the height of liquid columns in the two arms of the pitots tube is Dh, find the speed of gas flowing across the section of the pipe.

A tube bent at right angle is lowered into a water stream, as shown in figure. The velocity of the steam relative to the tube is v The closed upper end of the tube situated at a height h0 from the water surface has a small orifice. Find the height h upto which the water jet will spurt.

Figure shows a Siphon, which is a device for removing liquid from a container. The tube must initially be filled, but once this has been done, liquid will flow through the tube until the liquid surface in the container is in level with the lower end of the tube.

Find the work that has to be done to squeeze all water from a horizontally placed cylinder of volume V through an orifice of cross-sectional area a during the time t by means of constant force acting on the piston. The cross-sectional area of the orifice is considerably less than the piston area and there are no resistive forces.

Two liquids are filled upto heights h1 and h2 behind a wall of width w.

Find out

a) forces in part AB and BC

b) point of application of total force

(neglect atmospheric pressure)

Length of a horizontal arm of a tube is L and ends of both the vertical arms are open to atmospheric pressure Po. A liquid of density is poured in the tube such that liquid just fills the horizontal part of the tube as shown in figure. Now one end of the open end is sealed and the tube is then rotated about a vertical axis passing through the other vertical arm with angular speed w. If the liquid rises to a height h in the sealed arm, find the pressure in the sealed tube during rotation.

A wide cylindrical vessel of height H is filled with water and is placed on the ground. Find at what height h from the bottom of the vessel a small hole should be made in the vessel so that the water coming out of this hole strikes the ground at the maximum distance from the vessel. What is this maximum distance?

A vessel filled with water is free to slide on a frictionless surface. A small hole is made at a depth h from the surface. What is force required to be applied on the vessel to keep it stationary immediately after the water starts leaking.

The tube shown is of uniform cross-section. Liquid flows through it at a constant speed in the direction shown by the arrows. The liquid exerts on the tube

(a) a net force to the right

(b) a net force to the left

(c) a clockwise torque

(d) an anticlockwise torque

Water is flowing out of a tank through a tube bent at right angle. The radius of the tube is r and the length of its horizontal section is l The rate of water flow is Q. What is the moment of reaction forces of flowing water acting on the tubes wall, relative to the point O?

The side wall of a wide vertical cylindrical vessel of height h has a narrow vertical slit running all the way down to the bottom of the vessel. The width of the slit is = w. With the slit closed, the vessel is filled with water. What is the resultant force of reaction of the water flowing out of the vessel immediately after the slit is opened?

A cylindrical vessel of height H and base area A is filled with water. The vessel has a small oriface of area a in the bottom. Find the time in which the vessel will be empty.

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