Adegree is angstroms it is 10^-10 where as AU is astronomical unit distance between sun and earth 15 crore km

The industrial revolution starts in England and Western Europe in mid-18th Century. Its main contributions are inventions to perform work in effective and efficient manner. Some of them are,

1. Blast Furnace

It is the closed type of furnace which is capable of generating high temperatures. It is used to convert Iron into steel which is most widely used element nowadays.

2. Cotton Jenney

It is said to be a most valuable invention in the 18thcentury because it reduces time to collect cotton from cotton flowers. It is almost 300 times efficient than a normal hand.

3. Steam engine

Steam engine invented by James Watt. It revolutionizes the transport system which ultimately leads to connectivity between countries. So, different peoples, cultures and their ideas mingle to form revolutionary products and inventions to the modern world.

Fig. Steam Engine James Watt_1784

There are many other inventions that are outcomes of the industrial revolution which pave the path for modern technology.

The flow speed of a fluid can be measured using a device such as a Venturi meter or an orifice plate, which can be placed into a pipeline to reduce the diameter of the flow. For a horizontal device, the continuty plate shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed. Subsequently Bernoulli's principle then shows that there must be a decrease in the pressure in the reduced diameter region. This phenomenon is known as the Venturi effect.

Bernoulli’s Principle

• For a streamline fluid flow, the sum of the pressure (P), the kinetic energy per unit volume (ρv²/2) and the potential energy per unit volume (ρgh) remain constant.
• Mathematically:- P+ ρv²/2 + ρgh = constant
  • where P= pressure ,
  • E./ Volume=1/2mv²/V = 1/2v²(m/V) = 1/2ρv²
  • E./Volume = mgh/V = (m/V)gh = ρgh

Derive: Bernoulli’s equation

Assumptions:

1. Fluid flow through a pipe of varying width.
2. Pipe is located at changing heights.
3. Fluid is incompressible.
4. Flow is laminar.
5. No energy is lost due to friction:applicable only to non-viscous fluids.

• Mathematically: -
• Consider the fluid initially lying between B and D. In an infinitesimal timeinterval Δt, this fluid would have moved.
  • Suppose v₁= speed at B and v₂= speedat D, initial distance moved by fluid from to C=v₁Δt.
  • In the same interval Δtfluid distance moved by D to E = v₂Δt.
  • P₁= Pressureat A_1, P₂=Pressure at A₂.
  • Work done on the fluid atleft end (BC) W₁ = P₁A₁(v₁Δt).
  • Work done by the fluid at the other end (DE)W₂ = P₂A₂(v₂Δt)
• Net work done on the fluid is W₁ – W₂ = (P₁A₁v₁Δt− P₂A₂v₂Δt)
• By the Equation of continuity Av=constant.
  • P₁A₁ v₁Δt - P₂A₂v₂Δt where A₁v₁Δt =P₁ΔV and A₂v₂Δt = P₂ΔV.
• Therefore Work done = (P₁− P₂) ΔVequation (a)
  • Part of this work goes in changing Kinetic energy, ΔK = (½)m (v₂² – v₁²) and part in gravitational potential energy,ΔU =mg (h₂ − h₁).
• The total change in energy ΔE= ΔK +ΔU = (½) m (v₂² – v₁²) + mg (h₂ − h₁). (i)
• Density of the fluid ρ =m/V or m=ρV

• Therefore in small interval of time Δt, small change in mass Δm
  • Δm=ρΔV (ii)
• Putting the value from equation (ii) to (i)

• ΔE = 1/2 ρΔV (v₂² – v₁²) + ρgΔV (h₂ − h₁)  equation(b)
• By using work-energy theorem: W = ΔE
  • From (a) and (b)
  • (P₁-P₂) ΔV =(1/2) ρΔV (v₂² – v₁²) + ρgΔV (h₂ − h₁)
  • P₁-P₂ = 1/2ρv₂² - 1/2ρv₁²+ρgh₂ -ρgh₁(By cancelling ΔV from both the sides).
• After rearranging we get,P₁ + (1/2) ρ v₁² + ρg h₁ = (1/2) ρ v₂² + ρg h₂
• P+(1/2) ρv²+ρg h = constant.
• This is the Bernoulli’s equation.