16 Pressures:

The thrust acting on a body per unit area is called pressure 

i.e.    

Pressure is a scalar quantity because it is transmitted equally in all directions when force is applied, which shows that a definite direction is not associated with pressure.

• Unit of Pressure:

In SI, unit of pressure is  or Pascal  

i.e.    

If one Newton force is exerted on one metre square area then pressure applied by the body is called one Pascal. And in  unit of pressure is         

 Dimensional formula 

17.     Practical Applications of Pressure:

   A Sharpe Knife Cuts Than a Blunt One:

For the same total force, the effective force per unit area is more for the Sharpe edge then the blunt edge. Hence shape knife cut better then a flaunt one.

   Railway Tracks are Laid on a Large Sized Wooden or Iron Sleeper:

Because sleeper reduce the pressure which prevent the tracks to yield of the ground under the weight of the train

   It is difficult for a Man to Walk on sand while a camel walks easily on sand:

In spite of the fact that a camel is much heavier than a man, camel can easily walk on sand or because the area of the camel feet’s is much larger than a man, due to large area pressure is a leg.

   Pins and Nails are made to have Pointed Ends:

When a force is applied over head of a pin or a nail, it transmits a large pressure on the surface and easily penetrates the surface. So pins and nails are made to have pointed ends.

18.    Density and Relative Density:

• Density:-  The mass per unit volume of a substance is called density. It is denoted by ρ.

If M is the mass of a substance of volume V, 

Then density of substance is given by

 Unit: In SI unit of ρ is    . 

Dimensional formula 

As pressure is increased, volume decreases and hence density will increase.

• Relative Density:

 It is defined as the ratio of density of substance to the density of water at.

i.e. 

     Relative density has no unit, it is a pure number.

     The density of water 4℃ is maximum i.e. 1000

Illustration 13: Find the density and specific gravity of gasoline if 51 g occupies 75 cm³? 

  Sol:  

Question 14: The mass of a liter of milk is 1.032 kg. The butterfat that it contains has a density of 865 kg/m3 when pure, and it constitutes 4 percent of the milk by volume. What is the density of the skimmed milk? 

Sol: Find the mass of butterfat present in the milk. Subtract this from total mass to get mass of fat-free milk. The density of fat-free milk is equal to its mass divided by its volume.

Volume of fat in 1000 cm³ of milk = 4% × 1000 cm³ = 40 cm³                                                                                  

Mass of 40 cm³ fat

= 

  Density of skimmed milk

19   Pascal’s Law:

According to this law, the pressure applied to an enclosed liquid is transmitted to every portion of the liquid and the wall of the contained vessel.

Experimental Verification of Pascal’s Law:

Suppose a vessel of three openings A, B, and C of are A, 2A and A/2 respectively. Again suppose a force F is applied to piston A then to maintain B and C, the force 2F and F/2 should be applied at B and C. this shows that pressure P is transmitted equally in all directions because    

20 Applications of Pascal’s Law (Hydraulic Machines):

The device which works on Pascal’s law is called Hydraulic Machines.

e.g.: Hydraulic lift, Hydraulic Breaks, Hydraulic Press, and Dentist’s Chair etc. 

(i) Hydraulic Lift:

It consist of a hydraulic fluid filled container whose both end are fitted with two cylinder of area  and. The piston of small area  exerts a force directly on the fluid,

so that pressure  is transmitted to the whole container.

So pressure obtained at  will be same as that of at .

⟹   

⟹ 

As   ⟹ 

Thus force on larger piston must be large as compared to smaller piston. This can be used to lift a heavy body.

(ii)  Hydraulic Breaks:

 Hydraulic Breaks are used in automobile to retard the motion.

Construction:-

It consists of a master cylinder filled with a piston. The piston P is attached to breaks pedal with the help of lever system. The master cylinder is connected to wheel cylinder through a tube and wheel cylinder is consist of two piston’s  these pistons press against the break shoe  

Working:-

 When brake pedal is pressed, piston P is pushed into the master cylinder the pressure so produced is transmitted equally to piston  and moves in outward direction and forces the brake shoes to expand outwardly and wheel stops. Hence break is applied.

21. Expression for Pressure Exerted by a Liquid Column:

Suppose a vessel filled with a liquid of density  and imagine an area  at depth. Now the force on the area 

A=Weight of the liquid contained in the imaginary liquid Column

I.e. F=mass of the liquid in the column of depth

So the pressure of the liquid at a depth  is 

⟹   

Thus pressure exerted by liquid is directly proportional to 

                            Height of the liquid column.

                            Density of the liquid.

     The blood pressure of human being I grater at the feet than at the brain.

22. Variation of Pressure with Depth: or Effect of Gravity on Fluid Pressure:

Suppose a vessel filled with a liquid, as the liquid is in equilibrium, consider a imaginary cylinder in the vessel between point  and  of area  and height. Again suppose that  is pressure at A in vertical downward direction and  is pressure at B in upward direction. 

So net force acting in vertical downward direction is 

And net force acting in vertically upward direction is 

Now at equilibrium   

Or    

Or     

Or 

If we shift  at top and  at bottom

Then   and 

⟹     

Or  

     Liquid pressure is same at same depth.

     Pressure inside liquid vessel depends upon depth.

     The pressure  is called gauze pressure.

     Pressure does not depend upon the area of cross section.

     If we neglected effect of gravity then 

     The pressure exerted by liquid does not depend upon the shape, size of the containing vessel; this property is called hydraulic paradox.

23. Atmospheric Pressure:

The pressure exerted by the atmosphere of earth on a body is called atmospheric pressure. The atmospheric pressure at sea level is 

Torricelli’s experiment of atmospheric Pressure:

Torricelli found that when a tube of  long filled with mercury is dipped into a mercury vessel then the mercury remains in the tube at a height of  above the surface of the mercury vessel. The remaining upper empty level is a pure vacuum.

So the pressure between two ends B and A is       

⟹  

Here    

So atmospheric pressure

Thus atmospheric Pressure

Height of the atmosphere:

As   

As we know Atmospheric Pressure

⟹    ⟹   

But in actual practice both  decreases with height so atmosphere extends with decreasing pressure even beyond 100 Km.

Question 15: Atmospheric pressure is about. How large a force does the atmosphere exert on a 2 cm² area on the top of your head?  

Sol: Force 

24.  Different Units of Pressure:

                            In SI

                         In

25.  PRESSURE MEASURING DEVICES

1 Manometer:-

A manometer is a tube open at both ends and bent into the shape of a “U” and is partially filled with mercury. When one end of the tube is subjected to an unknown pressure p, the mercury level drops on that side of the tube and rises on the other so that the difference in mercury level is h as shown in the figure.

When we move down in a fluid, pressure increases with depth and when we move up the pressure decreases with height. When we move horizontally in a fluid, pressure remains constant. 

Therefore,  

where  is atmospheric pressure, and  is the density of the fluid inside the vessel. 

2 The Mercury Barometer:-

It is a straight glass tube (closed at one end) completely filled with mercury and inserted into a dish which is also filled with mercury as shown in the figure. Atmospheric pressure supports the column of mercury in the tube to a height h. 

The pressure between the closed end of the tube and the column of mercury is zero, p = 0. 

Therefore, pressure at points A and B are equal and thus   

.

 Hence   , 

26  Buoyancy:

When a body is immersed in a fluid, then fluid exerts pressure on all the faces of the body, this upward force acting on the body immersed in the liquid is called buoyant force and the phenomenon is called Buoyancy.

   Archimedes Principle:

According to Archimedes Principle, when a body is immersed wholly or partially in a fluid then it loses its weight equal to weight of the liquid displaced by the body. 

Suppose a body of mass  and height  and area of cross section  is immersed fully in a liquid of density. 

Then net down ward force acting on body at upper face.          

And upward force acting on lower face of the body is

Or upward force 

Net upward force  

Or  

Or  

Or  

Or  

Hence upward thrust acting on the body is equal to mass of the liquid displaced by the body.

Now the apparent weight of the body

27.  Law of Floatation:

The law of floatation states that a body will float in a liquid if the weight of the liquid displaced by the immersed part of the body is equal to or greater than the weight of the body,

i.e.:-

(i) When weight of the body is greater than upward thrust then body will sink.

(ii) When weight of the body is equal to upward thrust then body will remain in equilibrium with fluid.

(iii)     If weight of the body is smaller than upward thrust then body will float on the surface of the liquid.

       Relation between Density of Solid and Density of Liquid:

Suppose a solid of density  and volume  is placed in a liquid of density  and volume  now for equilibrium condition.

Or 

Or 

Or               

⟹    

If     then  i.e. body will sink.

  If  then  i.e. body will float

   Hence a body having density smaller then density of fluid then body will float in liquid.

Some Examples of Floating Bodies:

     Ice float on water because density of ice is smaller than density of water.

     A person can swim in sea water easily then in river water because density of sea water is larger.

     A ship is made of steel but its interior is given a hollow concave shape which displaces more water than its weight hence can float.

     A piece of iron sink in water but a ship made of iron float on water because ship displace much larger water than piece of iron, so it happens.

SURFACE TENSION

28. Intermolecular Force 

 The force of attraction or repulsion acting between the molecules is known as intermolecular force..  The intermolecular forces of attraction may be classified into two types

Cohesive force :

• The force of attraction between molecules of same substance is called the force of cohesion. This force is lesser in liquids and least in gases.
• Ex. (i) Two drops of a liquid collapse into one when brought in mutual contact.
• (ii) It is difficult to separate two sticky plates of glass welded with water.
• (iii) It is difficult to break a drop of mercury into small droplets because of large cohesive force between the mercury molecules.

Adhesive force : 

• The force of attraction between the molecules of the different substances is  called the force of adhesion
• Ex. (i) Adhesive force enables us to write on the blackboard with a chalk.
• (ii) A piece of paper sticks to another due to large force of adhesion between the paper and gum molecules.
• (iii) Water wets the glass surface due to force of adhesion.

• Cohesive or adhesive forces are inversely proportional to the eighth power of distance between the molecules.   

29.  Surface Tension:- 

The property of a liquid due to which its free surface tries to have minimum surface area and behaves like a stretched elastic membrane is called surface tension.  A small liquid drop has spherical shape, because the sphere has minimum surface area. 

Surface tension of a liquid is measured by the force acting per unit length on the free surface of liquid, the direction of this force being perpendicular to the line and tangential to the free surface of liquid. So if F is the force acting on one side of length L, 

then  

• It depends only on the nature of liquid and is independent of the area of surface or length of line considered. 
• It is a scalar as it has a unique direction which is not to be specified. 
• Dimension: [MT ^{– 2}].         (Similar to force constant) 
• Units: N/m (S.I.) and Dyne/cm [C.G.S.] 
• It is a molecular phenomenon and its root cause is the electromagnetic forces. 

30.  Examples of Surface Tension:-

(1) When mercury is split on a clean glass plate, it forms globules. Tiny globules are spherical on the account of surface tension because force of gravity is negligible. The bigger globules get flattened from the middle but have round shape near the edges.

(2) When a greased iron needle is placed gently on the surface of water at rest, so that it does not prick the water surface, the needle floats on the surface of water despite it being heavier because the weight of needle is balanced by the vertical components of the forces of surface tension. If the water surface is pricked by one end of the needle, the needle sinks down.

(3) When a molten metal is poured into water from a suitable height, the falling stream of metal breaks up and the detached portion of the liquid in small quantity acquire the spherical shape.

(4) Oil drop spreads on cold water. Where as it may remains as a drop on hot water. This is due to the fact that the surface tension of oil is less than that of cold water and is more than that of hot water.

(5) Hair of shaving brush/painting brush when dipped in water spread out but as soon as it is taken out, its hair sticks together.

(6) Rain drops are spherical in shape because each drop tends to acquire minimum surface area due to surface tension, and for a given volume, the surface area of sphere is minimum.

31.   Factors Affecting Surface Tension:- 

1.      Temperature:

The surface tension of liquid decreases with rise of temperature

 The surface tension of liquid is zero at its boiling point and it vanishes at critical temperature. At critical temperature, intermolecular forces for liquid and gases become equal and liquid can expand without any restriction. For small temperature differences, the variation in surface tension with temperature is linear and is given by the relation   www.vartmaaninstitute.com

Where  are the surface tensions at   and  respectively and  is the temperature coefficient of surface tension. 

Examples: (i) Hot soup tastes better than the cold soup.  

          (ii) Machinery parts get jammed in winter. 

2.      Impurities:

 The presence of impurities either on the liquid surface or dissolved in it, considerably affect the surface tension, depending upon the degree of contamination. A highly soluble substance like sodium chloride when dissolved in water increases the surface tension of water. But the sparingly soluble substances like phenol when dissolved in water, decreases the surface tension of water. 

32.  Applications of Surface Tension 

    The oil and grease spots on clothes cannot be removed by pure water. On the other hand, when detergents (like soap) are added in water, the surface tension of water decreases. The   force of adhesion between soap solution and oil or grease on the clothes increases. Thus, oil, grease and dirt particles get mixed with soap solution easily. Hence clothes are washed easily. 

    The antiseptics have very low value of surface tension. The low value of surface tension prevents the formation of drops that may otherwise Block the entrance to skin or a wound. Due to low surface tension, the antiseptics spread properly over wound. 

    Surface tension of all lubricating oils and paints is kept low so that they spread over a large area.

    Oil spreads over the surface of water because the surface tension of oil is less than the surface tension of cold water.  

    A rough sea can be calmed by pouring oil on its surface. 

    In soldering, addition of ‘flux’ reduces the surface tension of molten tin, hence, it spreads.

33.Molecular Theory of Surface Tension:- 

• The maximum distance up to which the force of attraction between two molecules is appreciable is called molecular range. 
• A sphere with a molecule as centre and radius equal to molecular range is called the sphere of influence. 

The liquid enclosed between free surface (PQ) of the liquid and an imaginary plane (RS) at a distance r (equal to molecular range) from the free surface of the liquid form a liquid film. 

To understand the concept of tension acting on the free surface of a liquid, let us consider four liquid molecules like A, B, C and D. Their sphere of influence is shown in the figure. 

(1) Molecule A is well within the liquid, so it is attracted equally in all directions. Hence the net force on this molecule is zero and it moves freely inside the liquid. 

(2) Molecule B is little below the free surface of the liquid and it is also attracted equally in all directions. Hence the resultant force acts on it is also zero. 

(3) Molecule C is just below the upper surface of the liquid film and the part of its sphere of influence is outside the free liquid surface. So the number of molecules in the upper half (attracting the molecules upward) is less than the number of molecule in the lower half (attracting the molecule downward). Thus the molecule C experiences a net downward force. 

(4) Molecule D is just on the free surface of the liquid. The upper half of the sphere of influence has no liquid molecule. Hence the molecule D experiences a maximum downward force. 

Thus all molecules lying on surface film experiences a net downward force. Therefore, free surface of the liquid behaves like a stretched membrane. 

34. Surface Energy: 

 The potential energy of surface molecules per unit area of the surface is called surface energy. 

  If a rectangular wire frame ABCD, with a sliding wire LM dipped in soap solution, a film is formed over the frame. Due to the surface tension, the film will have a tendency to shrink and thereby, the sliding wire LM will be pulled in inward direction. 

However, the sliding wire can be held in this position under a force F, which is equal and opposite to the force acting on the sliding wire LM all along its length due to surface tension in the soap film.  

If T is the force due to surface tension per unit length,

 Then    

Here   is length of the sliding wire LM. The length of the sliding wire has been taken as   for the reason that the film has got two free surfaces. 

 Suppose that the sliding wire LM is moved through a small distance x, so as to take the position  M’ L’. In this process, area of the film increases by  (on the two sides) and to do so, the work done is given by 

Or Total increase in area of the film]

 From the above expression

Or

I.e. surface tension may be defined as the amount of work done in increasing the area of the liquid surface by unity against the force of surface tension at constant temperature.

Unit 

Question 16:  Calculate the force required to take away a flat circular plate of radius 4 cm from the surface of water, surface tension of water being 75 dyne cm^-1.    

Sol: Force = Surface tension × length of the surface

Length of the surface = circumference of the circular plate

Required force = T × L = 72 × 8 = 1810 dyne.

35. EXCESS PRESSURE

The pressure inside a liquid drop or a soap bubble must be in excess of the pressure outside the bubble drop because without such pressure difference, a drop or a bubble cannot be in stable equilibrium. Due to the surface tension, the drop or bubble has got the tendency to contract and disappear altogether. To balance this, there must be excess of pressure inside the bubble.

Excess Pressure Inside a Liquid Drop:-

Consider a liquid drop of radius R. The molecules lying on the surface of liquid drop, due to surface tension will experience resultant force acting inward to the surface.
Let  S = Surface tension of liquid drop P = excess pressure inside the drop Due to excess of pressure, let there be increase in the radius of the drop by a small quantity in figure
Then work done by the excess pressure.

Or   

Increase in surface area of the drop  

 =

So    increase in surface energy

    =   

As the increase in surface energy is at the cost of work done by the excess pressure, therefore from (I) and (ii)    

Or     

36. Inside a Soap Bubble:-

The molecules lying on the surface of soap bubble, due to surface tension will experience resultant force acting inward to the surface.
Let S = Surface tension of soap bubble P = excess pressure inside the soap bubble.  Due to excess of pressure, let there be increase in the radius of the soap bubble by a small quantity in figure.
Then work done by the excess pressure.

Or          

Soap bubble has two surfaces so effective increase in surface area of the soap bubble   

     =   2 [

=

So increase in surface energy

As the increase in surface energy is at the cost of work done by the excess pressure,

 Therefore from (I) and (ii)     

  ⟹

Question 17: A 0.02 cm liquid column balances the excess pressure inside a soap bubble of radius 7.5 mm. Determine the density of the liquid. Surface tension of soap solution = 0.03 Nm^–1

 Sol: Pressure inside the soap bubble is larger than that outside it by amount 4T/R, where T is surface tension and R is its radius.

 Gauge pressure =  ρgh 

The excess pressure inside a soap bubble is

The pressure due to 0.02 cm of the liquid column is 

Thus, 

Question 18: How much work will be done in increasing the diameter of a soap bubble from 2 cm to 5 cm? Surface tension of soap solution is 3.0 × 10^-2 N/m. 

 Sol: Work done will be equal to the increase in the surface potential energy, which is surface tension multiplied by increase in area of surface of liquid.

Soap bubble has two surfaces. 

Hence,  W = T ∆A

Here, 

37. Air Bubble Inside Liquid:-

 Consider a air bubble of radius R inside a liquid. Let S = Surface tension of air bubble  P = excess pressure inside the air bubble.  Due to excess of pressure, let there be increase in the radius of the drop by a small quantity in figure. Then work done by the excess pressure     

As     

Or     

=   

Increase in surface area of the air bubble

=

So increase in surface energy

As the increase in surface energy is at the cost of work done by the excess pressure, 

Therefore from (I) and (ii) 

⟹        

Question 19: What should be the pressure inside a small air bubble of 0.1 mm radius situated just below the water surface? Surface tension of water = 7.2×10^–2 N/m and atmospheric pressure = 1.013 ×10⁵ N/m².  

 Sol: Pressure inside the air bubble is larger than that outside it by amount, where T is surface tension and R is its radius. Surface tension of water

  N/m; 

Radius of air bubble

  m

The excess pressure inside the air bubble is given by,

 Pressure inside the air bubble,

 Substituting the values, we have,

Question 20:  Calculate the energy released when 1000 small water drops each of same radius m coalesce to form one large drop. The surface tension of water is N/m.

Sol: Energy released will be equal to the loss in surface potential energy.

Let r be the radius of smaller drops and R of bigger one. Equating the initial and final volumes, we have

 Further, the water drops have only one free surface.

Therefore

Here, negative sign implies that surface area is decreasing. Hence, energy is released in the process.

38. Angle of contact:- 

Angle which the tangent to the liquid surface makes at the points of contact with the solid surface inside liquid is called angle of contact.

• It is particular for a given pair of liquid and solid. Thus the angle of contact changes with the pair of solid and liquid. 
• It does not depend upon the inclination of the solid in the liquid. 
• On increasing the temperature, angle of contact decreases. 
• Soluble impurities increase the angle of contact. 
• A partially soluble impurity decreases the angle of contact.

39.  Meniscus Of the Liquid:- 

When a capillary tube is dipped in a liquid, the liquid surface becomes curved near the point of contact. This curved surface is due to the resultant of two forces i.e. the force of cohesion and the force of adhesion. The curved surface of the liquid is called meniscus of the liquid.  If liquid molecule A is in contact with solid (i.e. Wall of capillary tube) then forces acting on molecule A are   

   Force of adhesion  (acts outwards at right angle to the wall of the tube). 

   Force of cohesion  (acts at an angle to the vertical).

Concave Meniscus:-

If force of adhesion between the container and liquid is larger than force of cohesion between liquid particles then concave meniscus is formed.

Here   . 

Or in form of pressure we may say that if atmospheric pressure is greater than liquid pressure than concave meniscus is formed.

 e.g.:- Example: Water in glass capillary tube.

Convex  Meniscus:- 

If force of adhesion between the container and liquid is smaller than force of cohesion between liquid particles then concave meniscus is formed.

Here   . 

In form of pressure we may say that if atmospheric pressure is smaller than liquid pressure than convex meniscus is formed.

 e.g.:- Example: Mercury in glass capillary tube.

Plane Meniscus:-

 If force of adhesion between the container and liquid is equal to force of cohesion between liquid particles then plane meniscus is formed.

Here   . 

In form of pressure we may say that if atmospheric pressure is smaller than liquid pressure than plane meniscus is formed.

 e.g.:- Example: Pure water in silver coated capillary tube.

40. Capillarity:- 

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

The root cause of capillarity is the difference in pressures on two sides of (concave and convex) curved surface of liquid. 

• Examples of capillarity:-

   Ink rises in the fine pores of blotting paper leaving the paper dry. 

   A towel soaks water. 

   Oil rises in the long narrow spaces between the threads of a wick. 

   Wood swells in rainy season due to rise of moisture from air in the pores. 

   Plugging of fields is essential for preserving moisture in the soil. 

   Sand is drier soil than clay. This is because holes between the sand particles are not so fine as compared to that of clay, to draw up water by capillary action.

41. Ascent Formula For Rise Of Liquid In A Capillary:-

When one end of capillary tube of radius  is immersed into a liquid of density  which wets the sides of the capillary tube (water and capillary tube of glass), the shape of the liquid meniscus in the tube becomes concave upwards.

            R = radius of curvature of liquid meniscus.

            T = surface tension of liquid.

             P = atmospheric pressure at point A,

 Pressure at point   

Pressure at points C and D just above and below the plane surface of liquid in the vessel is also P (atmospheric pressure).

The points B and D are in the same horizontal plane in the liquid but the pressure at these points is different. In order to maintain the equilibrium the liquid level rises in the capillary tube up to height h.

 Pressure due to liquid column = pressure difference due to surface tension

Or   

             [ as ]

1.                  The capillary rise depends on the nature of liquid and solid both i.e. on T, ,   and R.

(ii)          For a given liquid and solid at a given place  [As T, ,  and g are constant]

 i.e.  Lesser the radius of capillary greater will be the rise and vice-versa. This is called Jurin’s law.

(iii) In case of capillary of insufficient length  

 i.e. L < h, the liquid will neither overflow from the upper end nor will it tickle along the vertical sides of the tube. The liquid after reaching the upper end will increase the radius of its meniscus without changing nature such that: 

Illustration 21: A capillary tube of radius 0.20 mm is dipped vertically in water. Find the height of the water column raised in the tube. Surface tension of water = 0.075 N m^–1 and density of water = 1000 kg m^–3. Take g = 10 m s^–2.  

Sol: Use the formula for height of the liquid in the capillary. 

We have,  

1