Physics Formulas to Memorize For Upcoming Exams!

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Surface Tension

 

We observe many things in our day-to-day life.Surface tension Phenomenon is one among them. Often we confuse the Phenomena of Surface tension with Buoyancy. Both the phenomena are entirely different to each other in the sense, in Buoyancy a portion of the body gets dipped in the liquid whereas in Surface tension the body will be remaining on the layer of water without getting wet.
Let us observe these leaves on the surface of water. We could see them moving in the water without getting wet.

For these leaves to be on the layer, there should be some force acted by the upward layer of water which keeps the leaf on the surface. This is nothing but the Surface tension.
Let us study more about the Surface tension in this section. 

What is Surface Tension?

The Surface tension is defined as:
The dragging force observed in the given liquid per unit length. It is given by the formula:

T = FL

where F = Force per unit length and
L = Length over which the force acts.
The Surface tension is expressed in Newton per meter.

What causes Surface Tension?

 

Surface tension is a physical property of water. Here the cohesive force keeps the water intact. Each molecule in the beaker is pulled in every direction equally by adjacent molecules.

Let us observe the following diagram:

The dragging force acts between the each molecules by the other molecule. Therefore the resulting net force is zero.
At the surface of water in a beaker, the water molecule does not have another water molecule around the sides of them. Therefore creation of internal pressure. Water molecules at the surface are pulled inwards. Top layer of liquid surface of beaker are compressed to minimum area.

Surface Tension formula

 

The Surface tension is expressed by the formula:

T = FL

Where F = Force per unit length and
L = Length over which the force acts.

To Calculate the tension we use the formula:

T = 12 ρ grh.

where h = h + r3.
Here
r = radius of the capillary tube at the liquid meniscus
h = height of the liquid in the capillary tube above the free surface of liquid in beaker.
ρ = Density of water (ρ = 1 × 10³ kg/m³ for water).

Capillarity

 

Capillarity is the phenomena observed in the capillary tube to draw the fluid upward against the force of gravity.

The Capillarity is given by the formula:

h = 2 T cosθρgr

Where h = height of liquid
σ = Surface tension
θ = Contact angle
ρ = Density of the liquid
g = acceleration due to gravity
r = radius of tube.

Examples of Surface Tension

Below are given some Illustration for Surface tension:

1. When a sewing needle or U-pin is gently placed on water surface , it floats. The water surface below the needle gets depressed slightly .The force of surface tension acts tangentially. The vertical component of the force of surface tension balances the weight of the needle or U-pin.
2. Impurities present in a liquid affects its surface tension. A highly soluble substance like salt increases the surface tension whereas sparingly soluble substances like soap decreases the surface tension.
3. When a small paper boat is placed in a glass jar containing water. Paper boat floats on the surface of water. When a drop of soap detergent is added to water. Soap molecules spreads on water surface which creates a reaction force which moves the small paper boat forward.

How to find Surface Tension

Tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object.

Two masses M₁ and M₂ are attached to a string which passes over a pulley attached to the edge of a horizontal table. The mass M₁ lies on the frictionless surface of the table. Let the tension in the string be T and the acceleration of the system be a, then

T = M₁a

M₂g - T = M₂a

∴ a = M2gM1+M2

The Tension is given by:

T = M₁a = M1M2gM1+M2.

Below are given some problems Which are based on surface tension:

Solved Examples

Question 1: Three blocks masses 2kg, 3kg and 5kg are connected to each other with light strings and are then placed on a smooth frictionless surface. See fig. below. Let the system be pulled with a force F from the side of lighter mass so that it moves with an acceleration of 1ms^-2. T₁ and T₂ denote the tensions in the strings. Calculate the value of T₁ and T₂.
Solution:

Using the formula Tension T = Ma,
Where M = Mass and
           a = acceleration

For T₁ total effective mass is mass of block B₂ + mass of block B₁ and acceleration applied is of 1ms^-2.
therefore Tension T₁ = (3+5)kg × 1ms^-2 = 8 N
Similarly tension T₂ = 5kg × 1ms^-2 = 5N.

Question 2: A mass M is suspended by a rope from a rigid support at P as shown in the figure. Another rope is tied at the end Q, And it is pulled horizontally with a force F. If the rope PQ makes angle A with the vertical, then find the tension in the string PQ?
Solution:
In the triangle PLQ, QP =T, PL= mg, and LQ= F, (all are vectors ). The point Q is in equilibrium under the action of T, Mg and F.
Here, T = PQ = LQsinA
                   = FsinA.
The Tension is given by T = FsinA.

Bernoulli's Principle

 

In the 18^th century Daniel Bernoulli noticed that fluids flow faster when forced through constrictions. We usually come across these kinds of phenomenon when we observe and compare rapids and meandering rivers.

Bernoulli reasoned that when a particular energy increases (kinetic) in a system the rest of the forms of energy must decrease.

To understand this principle the best example would be observing the lifting of an airplane. An airplane wing is shaped in such a way that the air passing over it must travel further and faster than the air passing underneath. This leads to a lower air pressure build up on the wing than the air below. This produces the lift force which keeps the airplane afloat.

 

Bernoulli's Principle Definition

 

This principle basically gives us a relation between velocity, pressure and height of the flowing non viscous fluid in a horizontal flow.

According to it, the speed and pressure of the flowing fluid are inversely proportional to each other, that is, if the velocity increases, it will lead to an automatic decrease in the pressure of the fluid.

or

The theorem states that for the streamline flow of an ideal liquid, the total energy (sum of pressure energy, potential energy and kinetic energy) per unit mass remains constant at every cross-section, throughout the flow.

Hence according to the principle, for a horizontal flow if the velocity decreases then the pressure exerted by the fluid will decrease.

We have many forms of Bernoulli equation according to the flow of the liquid. According to it if we add all the energy components of the flowing fluid along the streamline then we will get a constant value for it throughout the flow of line. Even the sum of the potential and kinetic energy follows a constant value. We can also say that Bernoulli principle is the result of Newton's second law of motion.

Bernoulli's Principle Equation

This theorem is a consequence of the principle of conservation of energy, applied to ideal liquids in motion. As per the theorem statement that is for the streamline flow of an ideal liquid, the total energy (sum of pressure energy, potential energy and kinetic energy) per unit mass remains constant at every cross-section, throughout the flow.

Consider a tube AB of varying cross-section and at different heights. Let an ideal liquid (an ideal liquid is incompressible and non-viscous) flow through it in a streamline. Since the liquid is flowing from A to B, p1>p2.
Now A1v1r = A2v2r = m (according to the equation of continuity)
Here A1>A2 so v1<v2
The force on the liquid at A=ρ1A1 and the force on the liquid at B=ρ2A2
Now, the work done per second on the liquid at

section A = r1A1v1

= ρ1v1

Where v1 is velocity and v₁ is volume of liquid per sec.
(here, Work donesec= Force × DistanceTime = force×velocity)
Now, the work done per second on the liquid at

section B = ρ2A2v2

= ρ2v2

since v₁ = v₂ = v (equation of continuity)

Net work done per second on the liquid by the pressure energy in moving from A to B = ρ1v−ρ2v
The net work done per second, in turn, increases the P.E. per second and also increases the K.E. per sec, from A to B. This is in accordance with the law of conservation of energy.

p1v−p2v = (mgh2−mgh1) + (12mv22−12mv21)

or p1v+mgh1+12mv21 = p2v+mgh2+12mv22

or p1vm + gh1+12v21= p2vm + gh2+12v22

or p1ρ + gh1+12v21 = p2ρ + gh2+12v22

pρ + gh + 12v2 = constant

Pressure energy per unit mass (p/e) + potential energy per unit mass (gh) + kinetic energy per unit mass is constant for Streamline flow of an ideal liquid.
The Bernoulli equation is different for adiabatic as well as isothermal processes.

Bernoulli Effect

• We have already studied the Bernoulli's law. The use of this law has been seen as Bernoulli Effect. If we take the example of the air plane then the air flowing under the wing is faster than the air flowing above the wing. Hence the air pressure is higher under the wing than over the wing. This results in air plane lift. And this is also called Bernoulli Effect.
• The Bernoulli Effect is same as the Bernoulli law or statement or whatever we call it.

Bernoulli's Principle Examples

1. The most common example of this principle is the air plane lift. The wings of the plane are designed in such a manner that the air flowing under the wings moves at a speed which is greater than that of the upper part. Hence it results into a difference in pressure such that the pressure exerted is more upwards than in downwards direction. This results in a lift.
2. It is used to make Venturi meter. It is a device that is used to measure the rate of flow of a fluid through a pipe. It has three parts: a short converging pipe, a throat and a diverging part as well.
3. It is used in the manufacturing of the orifice meter which has the same function as above but is available at a cheaper rate.
4. It is used in pilot tube. It is a device used for measuring the velocity of flow at any point in a pipe or a channel. It is based on the principle that if the velocity of flow at a point becomes zero the pressure there is increased due to the conversion of the kinetic energy into pressure energy.
5. The carburetor available in engine has a venture which operates on the principle of Bernoulli.
6. If we want to calculate the maximum drain rate through the hole in the tank then we can use the above principle of Bernoulli.
7. It is also used in open channel hydraulics.
8. Used in the Bernoulli grip in order to create the non contact adhesive force between surface and gripper.
9. The spoiler of the race car is shaped in a way to obtain the maximum speed during the race.
10. Race cars also use this principle to keep the wheel pressed to the racing track.