Torricelli's theorem

 Consider a vessel which is filled with water, at the surface of water P1 pressure is applied and the velocity of water is zero (V1 = 0). Let a hole is made at a point b from which water is coming out with velocity V & at that point the pressure is P2 .

 Let ‘H’ is the height from surface to the bottom of the vessel & ‘h’ is the height from the surface to the hole. Then the remaining height is

(H – h).  

 According to Pascal's law -

     P1 = P2 = P

     V1 = 0

According to Bernoulli's theorem-

P1 + ½ ρv12 + ρgh1 = P2 + ½ ρv22 + ρgh2

P + ½ ρv12 + ρgH = P+ ½ ρv22 + ρg(H – h)

½ ρ(0)2 + ρgH =  ½ ρv2 + ρg(H – h)

       ρgH =  ½ ρv2 + ρgH - ρgh

       ½ ρv2 = ρgh

       v2 = 2gh

        v = √2gh

Hence the speed of e-flux is  v = √2gh.

 Note: a) velocity of e-flux is independent on the nature of liquid, quantity of liquid in the vessel and area of orffice.

b) Path of liquid will be parabola.

c) Greater is the distance of the hole from the free surface greater will be the velocity.

  

                                                                                     https://youtu.be/O3D2belxKBs

Density

Density (ρ) of any substance can be defined as the mass per unit volume or

                      density = mass / volume

                               ρ = m / v

Relative density

 The ratio of density of substance to the density of water at 4°C is called relative density or specific gravity.

    Relative density = density of substance / density of water at 4°C .

 

Note :  Relative density is a pure ratio. So, it has no units. 

            Density of water at 4°C in CGS is 1 g/cm3 . Therefore, numerically the relative density  and density of substance (in CGS) are equal. 

exam: Relative density of an oil is 0.6. find the absolute density in CGS and SI unit.

sol:  Density of oil (in CGS) = relative density (g/cm3) 

                     absolute density = 0.6 g/cm3

                        absolute density = 600 kg/m3

                                                         https://youtu.be/UCHIvvAvekA

Venturimeter

It is a device based on the horizontal extension of the Bernoulli’s theorem. It is a device used to measure the rate of flow of liquid through the pipe.

According to Bernoulli’s theorem –

P1 + ½ ρv12 + ρgh1 = P2 + ½ ρv22 + ρgh2

 P1 + ½ ρv12 + ρgh =  P2 + ½ ρv22 + ρgh

since h1 = h2 , so

P1 – P2 = ½ ρ ( v22 – v12 )

Divide and multiply by v12 -----

∆P = ½ ρv12 ( v22 – v12 ) / v12

∆P = ½ ρv12 (v22 /v12  - 1)

          According to equation of continuity-----

          A1 v1 = A2 v2 ,   v2 /v1 = A1 /A2

          ∆P = ½ ρv12 { (A1 /A2 )2 – 1 }

          ∆P = ½ ρv12 (A12 – A22 ) /A22

          hρHg g = ½ ρv12 (A12 – A22 ) /A22    {ρHg  = density of mercury}

          v12 = 2 hρHggA22 / ρ(A12 – A22)

                         v1 = A2 {2 hρHgg / ρ(A12 – A22)}1/2

                                           https://youtu.be/UCHIvvAvekA

Atmospheric pressure 

The pressure exerted by the earth's atmosphere is called atmospheric pressure. Atmospheric pressure changes with weather and elevation. 

Normal atmospheric pressure at sea level (an average value) is 1.013 x 105 Pa.

                        1 atm = 1.013 x 105 Pa.

Absolute pressure and Gauge pressure

The excess pressure above atmospheric pressure is usually called gauge pressure and the total pressure is called absolute pressure.

Thus,

         Gauge pressure = absolute pressure - atmospheric pressure.

                                         

                                                https://youtu.be/UCHIvvAvekA

Capillary Tube

If a tube of very narrow bore (capillary tube) is dipped in a liquid, it is formed that the liquid in the tube either ascent or descents relative to the surrounding  liquid. This phenomenon is called capillarity.

The main cause of capillarity is the difference in pressure on two sides concave or convex surface of liquid.

Consider a capillary tube of radius 'r' dipped in a liquid of surface tension 'T' and density ρ.

    cosӨ = r/R    {R=radius of meniscus}

    R = r/ cosӨ   

    

   As we know, P = 2T/R

          hρg = 2T/R    {as P=hρg}

          h = 2T / ρgR

          h = 2TcosӨ / ρgR

 So the height ascent or descent in the capillary tube will be  ----

                    h = 2TcosӨ / ρgR

                                                      https://youtu.be/xNREviwPrOg

                

 

Barometer

It is a device used to measure atmospheric pressure. In principle, any liquid can be used to fill the barometer, but mercury is the liquid generally filled because its great density makes possible any instrument of reasonable size.

         

         P₁ = P₂

          P₂ = atmospheric pressure (P₀)

         P₁ = 0 + ρgh   { ρ= density of mercury}

         P₀ = ρgh

Thus mercury barometer reads the atmospheric pressure (P₀)  directly from the height of the mercury column.

For example, if the height of mercury in the barometer is 760 mm, then the atmospheric pressure will be ----

As,     P₀ = ρgh

        P₀ = (13.6 x 10³)(9.8)(0.760)

         P₀ = 1.013 x 10⁵ N/m²

                                                                 https://youtu.be/O3D2belxKBs

Law of Floatation

Consider an object of volume V and density ρs floating in a liquid of density ρl. Let Vi be the volume of object immersed in the liquid. For equilibrium of object--

                 Weight = Upthrust

                 Vρs g = Viρl g

                 Vi / V = ρs / ρl

 This is the fraction of volume immersed in liquid.

Percentage of volume immersed in liquid----------------

  (Vi/V)100 = (ρs / ρl)100

Three possibilities are now arise----

a) If ρs < ρl, only fraction of object will be immersed in the liquid. This fraction can be calculated by the above equation.

b) If ρs = ρl, the whole of the rigid body will be immersed in the liquid. Hence , the body remains floating in the liquid wherever it is left.

c) If ρs > ρl, the body will sink.

                                                        https://youtu.be/O3D2belxKBs