Answer
Verified
440k+ views
Hint: Here, we first know what Bernoulli's principle states, and then further we can prove the theorem by deriving the expression which relates the pressure and potential energy of the fluid. At last we will also note down the limitations of this theorem.
Formula used:
$\eqalign{
& K.{E_{gained}} = \dfrac{1}{2}\rho ({v_2}^2 - {v_1}^2) \cr
& P.{E_{gained}} = \rho g({h_2} - {h_1}) \cr} $
Complete answer:
Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.
To prove Bernoulli's theorem, consider a fluid of negligible viscosity moving with laminar flow, as shown in Figure.
Let the velocity, pressure and area of the fluid column be ${p_1}$, ${v_1}$ and ${A_1}$ at Q and ${p_2}$, ${v_2}$ and ${A_2}$ at R. Let the volume bounded by Q and R move to S and T where QS =${L_1}$, and RT = ${L_2}$.
If the fluid is incompressible:
The work done by the pressure difference per unit volume = gain in kinetic energy per unit volume + gain in potential energy per unit volume. Now:
${A_1}{L_1} = {A_2}{L_2}$
Work done is given by:
$\eqalign{& W = F \times d = p \times volume \cr
& \Rightarrow {W_{net}} = {p_1} - {p_2} \cr} $
$\eqalign{& \Rightarrow K.E = \dfrac{1}{2}m{v^2} = \dfrac{1}{2}V\rho {v^2} = \dfrac{1}{2}\rho {v^2}(\because V = 1) \cr
& \Rightarrow K.{E_{gained}} = \dfrac{1}{2}\rho ({v_2}^2 - {v_1}^2) \cr} $
$\eqalign{& {P_1} + \dfrac{1}{2}\rho {v_1}^2 + \rho g{h_1} = {P_2} + \dfrac{1}{2}\rho {v_2}^2 + \rho g{h_2} \cr
& \therefore P + \dfrac{1}{2}\rho {v^2} + \rho gh = const. \cr} $
For a horizontal tube
$\eqalign{& \because {h_1} = {h_2} \cr
& \therefore P + \dfrac{1}{2}\rho {v^2} = const. \cr} $
Therefore, this proves Bernoulli's theorem. Here we can see that if there is an increase in velocity there must be a decrease in pressure and vice versa.
Additional information:
Bernoulli's principle is named after Daniel Bernoulli who published this in his book Hydrodynamica in 1738.
Pressure applied on an object is given by the force exerted on the object per unit area. The S.I unit of pressure is Pascal.
Work done on an object is defined as the force applied on the object for a certain displacement. Further, if we define a volume, it is the quantity of three-dimensional space enclosed by a closed surface.
Note:
We should remember that no fluid is totally incompressible whereas in practice the general qualitative assumptions still hold for real fluids. One should also notice that in Bernoulli's theorem, it is given that the velocity of every particle of liquid across any cross-section is uniform which is not correct, because the velocity of the particles is different in different layers.
Formula used:
$\eqalign{
& K.{E_{gained}} = \dfrac{1}{2}\rho ({v_2}^2 - {v_1}^2) \cr
& P.{E_{gained}} = \rho g({h_2} - {h_1}) \cr} $
Complete answer:
Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.
To prove Bernoulli's theorem, consider a fluid of negligible viscosity moving with laminar flow, as shown in Figure.
Let the velocity, pressure and area of the fluid column be ${p_1}$, ${v_1}$ and ${A_1}$ at Q and ${p_2}$, ${v_2}$ and ${A_2}$ at R. Let the volume bounded by Q and R move to S and T where QS =${L_1}$, and RT = ${L_2}$.
If the fluid is incompressible:
The work done by the pressure difference per unit volume = gain in kinetic energy per unit volume + gain in potential energy per unit volume. Now:
${A_1}{L_1} = {A_2}{L_2}$
Work done is given by:
$\eqalign{& W = F \times d = p \times volume \cr
& \Rightarrow {W_{net}} = {p_1} - {p_2} \cr} $
$\eqalign{& \Rightarrow K.E = \dfrac{1}{2}m{v^2} = \dfrac{1}{2}V\rho {v^2} = \dfrac{1}{2}\rho {v^2}(\because V = 1) \cr
& \Rightarrow K.{E_{gained}} = \dfrac{1}{2}\rho ({v_2}^2 - {v_1}^2) \cr} $
$\eqalign{& {P_1} + \dfrac{1}{2}\rho {v_1}^2 + \rho g{h_1} = {P_2} + \dfrac{1}{2}\rho {v_2}^2 + \rho g{h_2} \cr
& \therefore P + \dfrac{1}{2}\rho {v^2} + \rho gh = const. \cr} $
For a horizontal tube
$\eqalign{& \because {h_1} = {h_2} \cr
& \therefore P + \dfrac{1}{2}\rho {v^2} = const. \cr} $
Therefore, this proves Bernoulli's theorem. Here we can see that if there is an increase in velocity there must be a decrease in pressure and vice versa.
Additional information:
Bernoulli's principle is named after Daniel Bernoulli who published this in his book Hydrodynamica in 1738.
Pressure applied on an object is given by the force exerted on the object per unit area. The S.I unit of pressure is Pascal.
Work done on an object is defined as the force applied on the object for a certain displacement. Further, if we define a volume, it is the quantity of three-dimensional space enclosed by a closed surface.
Note:
We should remember that no fluid is totally incompressible whereas in practice the general qualitative assumptions still hold for real fluids. One should also notice that in Bernoulli's theorem, it is given that the velocity of every particle of liquid across any cross-section is uniform which is not correct, because the velocity of the particles is different in different layers.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE
Try out challenging quizzes on this topic
made by experts!
made by experts!
Take me there!