Question

To what height should a cylindrical vessel be filled with a homogeneous liquid to make the force with which the liquid passes on the side of the vessel equal to the force exerted be the liquid on the bottom of the vessel?

Solution

Correct option is

h = r

Consider a cylindrical vessel of radius r filled with a liquid of density  to a height h. If p0 is the atmospheric pressure, the pressure difference inside and outside the base of the cylinder,

    

  

Now as pressure at a depth y below the surface of liquid inside will be  while outside p0, so pressure difference on two sides of the curved surface at depth 0 below the surface will be 

        

So the force on a strip of curved surface of thickness dy at depth below the surface of the liquid as show in figure will be

               

  

But according to given problem FB = FC; so from eq. (i) and (ii),  

        

SIMILAR QUESTIONS

Q1

In English the phrase ‘tip of the iceberg’ is used to mean a small visible fraction of something that is mostly hidden. For a real iceberg what is this fraction if the density of sea water is 1.03 g/cc and that of ice is 0.92 g/cc?

Q2

A piece of metal floats on mercury. The coefficient of volume expansion of the metal and mercury are  respectively. If the temperature of both mercury and metal are increased by an amount , by what factor the fraction of the volume of the metal submerged in mercury changes?

Q3

A block of wood floats in water with two-thirds of its volume submerged. In oil the block floats with 0.90 of its volume submerged. Find the density of (a) wood and (b) oil, if density of water is 103 kg/m3  

Q4

Air is streaming past a horizontal aeroplane wing such that its speed is 120 m/s over the upper surface and 90 m/s at the lower surface. If the density of air is 1.3 kg/m3, find the difference in pressure between the top and bottom of the wing. If the wing is 10 m long and has an average width 2m, calculate the gross lift of the wing. 

Q5

A horizontal pipe line carries water in a streamline flow. At a point along the pipe where the cross-section area is 10 cm2, the water velocity is 1 m/s and the pressure is 2000 Pa. What is the pressure of water at another point where the cross-section area is 5 cm2?

Q6

Water from a tap emerges vertically downward with an initial speed of 1.0 ms–1. The cross-sectional area of the tap is 10–4 m2. Assume that the pressure is constant throughout the stream of water, and that the flow is steady. The cross-sectional area of the stream 0.15 m below the tap is : (g = 10 m/s2 

Q7

Water is filled in a cylindrical container upto a height of 3 m. A hole of cross-section area a is made in the wall of the container at a height of 52.5 cm from the bottom. The cross-section area of the container is A. If a/A = 0.1, then the squire of the speed of water coming out from the hole is (g = 10 m/s2)

Q8

The vertical section of a wing of a fan is shown below. Maximum upthrust is in

                                            

Q9

The vertical of a mass ball of mass M and density d1, when dropped in a container filled with glycerine becomes constant after some time. If the density of glycerine is d2, the viscous force acting on the ball will be

Q10

 

The density of air in atmosphere decreases with height and can be expressed by the relation:  

           

Where  is the density at sea-level, A is a constant and h is the height. Calculate the atmospheric pressure at sea-level. Assume g to be constant.