Strain

The change produced in the dimensions of a body under a system of forces or couples in equilibrium, is called strain, and is measured by the change per unit length [linear or longitudinal strain], per unit volume [volume strain], or the angular deformation [shear strain or simply shear] according to the change that takes place in length, volume or shape of the body.

Therefore, we have three kinds of strain: (i) Longitudinal strain (ii) Volume strain and (iii) Shearing strain

(i)  Longitudinal strain: The change in length per unit length is called longitudinal strain. That is if L is the original length and l is the change in length, then

 Longitudinal strain = Change in length / original length = l / L

(ii) Volume strain: The change in volume per unit volume is called volume strain. That is if V is the original volume and v is the change in volume, then

Volume strain = Change in Volume / original volume = v / V 

 

(iii) Shearing strain: A change in shape without change in volume is known as shearing strain. It is measured by the angle of shear measured in radians.

Elasticity

Elasticity: It is the property by which a body offers resistance to external forces tending to change its volume or shape or both and it will regain its original form when the deforming force is removed.

Perfectly Rigid body: A body is said to be perfectly rigid when it is impossible to alter its shape by the application of force. But no body is perfectly rigid.

Perfectly Elastic Body: When a body is acted upon by a force or a system of forces, it undergoes changes in its shape or size. If the body recovers its original dimension completely after withdraw of the applied force, then the body is said to be perfectly elastic but no body is perfectly elastic.

Stress:

When a force is applied on a body, internal forces are generated. These internal forces react in a direction opposite to the forces applied and tend to bring the body to its original state; the restoring force per unit area of the body is known as stress.

Stress = Force/Area

If the force is inclined to the surface, then its component, perpendicular to the surface, measured per unit area, is called normal stress and the component acting along the surface, per unit area, is called tangential or shearing stress.

(i) Normal Stress: Restoring force per unit area perpendicular to the surface as called normal stress.

(ii)        Tangential Stress: Restoring force per unit area parallel to the surface is called tangential stress

Problem

1. One kilogram of ice at -15 ^oC is heated until the whole of it evaporates. How much heat is required? Latent heat of fusion of ice = 80 cal/gm and that of steam = 540 cal/gm.

2. Calculate the energy released when (i) 20 g water at 100 °C and (ii) 20 g of steam at 100 °C are each spilt on the hand. Take the specific heat capacity of water to be 4200 J kg ^–1 K ^–1 and the specific latent heat of vaporization of water to be 2.2 MJ kg^–1. Assume that the temperature of the skin is 33 °C.

3. It takes 15 minutes for an electric kettle to heat a certain quantity of water from 0 ^oC to the boiling point 100 ^oC. It requires 80 minutes to turn all the water at 100 ^oC into steam. Determine the latent heat of steam.

4. A refrigerator converts 60 g of water at 25 ^oC into ice at -30 ^oC in one hour. Determine the quantity of heat removed per minute. (Specific heat of ice = 0.5)

5. (i) How much heat is needed to take ice of mass m = 720 g at -10 ^oC to a liquid state at 15 ^oC? (ii) If we supply the ice with a total heat only 210 kJ, what then is the final state of the water?

6. In a cold storage ice melts at the rate of 3 kg per hour when the external temperature is 28 ^oC. Find the minimum power output of the motor used to run the refrigerator just to prevent melting of ice.

7. In an industrial process 10 kg of water per hour is to be heated from 20 ⁰ C to 80 ⁰ C.To do this, steam at 150 ⁰ C is passed from a boiler into a copper coil immersed in water. The steam condenses in the coil and is returned to the boiler as water at 90⁰ C. How many kg of steam is required per hour? Specific heat of steam= 1cal /gm, Latent heat of steam =540 cal/gm

8. An aluminium container of mass 100 gm contains 200 gm of ice at -20⁰C.Heat is added to the system at the rate of 100 calories per seconds. What is the temperature of the system after 4 minutes? Specific heat of aluminium = 0.2 cal g^-1(=⁰C)^-1

9. A lead bullet at 100 ⁰ C strikes a steel plate and melts. What was its minimum speed? Specific heat of lead =0.03, latent heat = 5cal/gm and melting point = 327 ⁰ C. The heat produced is shared equally between the plate and the bullet.

10. An ice cube at 0⁰C is dropped into ground and melts to water at 0⁰C.If all the kinetic energy of the ice went into heating it, from what height did it fall?

11. When a falling hailstone is at a height of 2.0 km its mass is 2.50 g. What is its potential energy? Assuming that all of this potential energy is converted to latent heat during the fall, calculate the mass of the hailstone on reaching the ground. Take the specific latent heat of fusion of ice to be 3.36 × 10⁵ J kg^–1 and the acceleration due to gravity to be 9.81 ms^–2.

Heat Calculation

Heating / Cooling Curves

The diagram on the left shows the uptake of heat by 1 kg of water, as it passes from ice at -50 ºC to steam at temperatures above 100 ºC, affects the temperature of the sample.

A: Rise in temperature as ice absorbs heat.
B: Absorption of latent heat of fusion.
C: Rise in temperature as liquid water absorbs heat.
D: Water boils and absorbs latent heat of vaporization.
E: Steam absorbs heat and thus increases its temperature.

The above is an example of a heating curve. One could reverse the process, and obtain a cooling curve. The flat portions of such curves indicate the phase changes

Latent Heat of Vaporization

Evaporation is the change of state from liquid to vapor.  In the process of evaporation, the molecule absorbs energy. This energy is latent heat. 

How did you make the water evaporate?  Probably you added heat.  You might have set it out in the sun, or possibly put it over a fire.   To make water evaporate, you put energy into it. The individual molecules in the water absorb that energy, and get so energetic that they break the hydrogen bonds connecting them to other water molecules.  They become molecules of water vapor

l (vaporization) = 540 cal/gm or 2260 KJ/Kg 

The definition of the specific latent heat of vaporization is

'The specific latent heat of vaporization is the amount of heat required to convert unit mass of a liquid into the vapour without a change in temperature."

 

For water at its normal boiling point of 100 ºC, the latent specific latent heat of vaporization is 2260 kJ.kg^-1. This means that to convert 1 kg of water at 100 ºC to 1 kg of steam at 100 ºC, 2260 kJ of heat must be absorbed by the water. Conversely, when 1 kg of steam at 100 ºC condenses to give 1 kg of water at 100 ºC, 2260 kJ of heat will be released to the surroundings

Latent Heat of Fusion:

Fusion is the change of state from solid to liquid.  In the process of fusion, the molecule absorbs energy. This energy is latent heat. 

When a solid substance changes from the solid phase to the liquid phase, energy must be supplied in order to overcome the molecular attractions between the constituent particles of the solid. This energy must be supplied externally, normally as heat, and does not bring about a change in temperature.

The units of heat of fusion are usually expressed as joules per mole (the SI units) or calories per gram or Btu per pound-mole.

l (fusion) = 80 cal/gm or 334 KJ/Kg  

The specific latent heat of fusion is defined as

"The specific latent heat of fusion of a substance is the amount of heat required to convert unit mass of the solid into the liquid without a change in temperature."

 

The specific latent heat of fusion of ice at 0 ºC, for example, is 334 kJ.kg^-1. This means that to convert 1 kg of ice at 0 ºC to 1 kg of water at 0 ºC, 334 kJ of heat must be absorbed by the ice. Conversely, when 1 kg of water at 0 ºC freezes to give 1 kg of ice at 0 ºC, 334 kJ of heat will be released to the surroundings

Latent Heat:

Latent heat describes the amount of energy in the form of heat that is required for a material to undergo a change of phase.

A solid consists of molecules that are tightly bound to each other by forces acting between them. Energy must be supplied in the form of heat to overcome these forces. As the bonding forces weaken, groups of molecules break free, and as more heat is supplied the solid changes into a liquid, in which small groups of molecules constantly break apart, and reform and groups slide freely past each other. While this is happening, the temperature of the substance remains unchanged. All of the heat energy is used in loosening the bonds between molecules.

If still more energy is applied, it has the effect of making the groups of molecules move faster. When they move faster they strike harder against any object with which they come into contact. It is the speed of motion of molecules that a thermometer measures as temperature. Once a solid has melted to become a liquid, the application of additional heat raises the temperature of the liquid.

When a liquid is cooled, its molecules lose energy and move more slowly, and the temperature of the liquid falls. When their energy falls to a certain level the molecules start bonding together. This requires less energy than moving about, and so heat energy is released as the liquid solidifies. The temperature of the substance remains unchanged, but the surrounding medium is warmed by the release of energy.

For a phase change, the heat liberated or absorbed is given by 

Q= ml ------------------------- (1) 

where l is the latent heat of fusion (latent heat of melting) or latent heat of vaporization.

How much heat does it take to get water to change state?   If the water is at a temperature of 100 degrees C (that is, the boiling point, or 212 degrees F) it takes an additional 540 calories of heat to convert one gram of water from the liquid state to the vapor state.  When the vapor converts to the liquid state, 540 calories of energy will be released per gram of water. If you are converting solid water (ice) to liquid water at 0 degrees C, it will require about 80 calories of heat to melt one gram of ice, and the 80 calories will be released when the liquid water is frozen to the solid state.

ENTROPY AND THE SECOND LAW

The results of Example 18-10 about the flow of heat from a higher to a lower temperature, or the mixing of  substances at different temperatures, are characteristic of all natural [that is, irreversible] processes. When we include the entropy changes of all the systems taking part in the process, the increases in entropy are always greater than the decreases. In the special case of reversible process, the increases and decreases are equal. Hence we can state the general principle: When all systems taking part in a process are included, the entropy either remains constant or increases. In other words, no process is possible in which the total entropy decreases, when all systems taking part in the process are included.  This is an alternative statement of the second law of thermodynamics in terms of entropy. Thus it is equivalent to the “engine” and “refrigerator” statements discussed earlier.

The increase of entropy in every natural, irreversible process measures the increase of disorder or randomness in the universe associated with that process. Consider again the example of mixing hot and cold water. We might have used the hot and cold water as the high- and low-temperature reservoirs of a heat engine. While removing heat from the hot water and giving heat to the cold water, we could have obtained some mechanical work. But once the hot and cold water have been mixed and have come to a uniform temperature, this opportunity to convert heat to mechanical work is lost irretrievably. The lukewarm water will never unmix itself and separate into hotter and colder portions. No decrease in energy occurs when the hot and cold water are mixed. What has been lost is not an energy, but opportunity, the opportunity to convert part of the heat from the hot water into mechanical work. Hence when entropy increases, energy becomes less available, and the universe becomes more random or ”run down”. 

The statement of the second law is in terms of entropy:

“The entropy of a closed system never decreases or equivalently: The change in entropy of the universe is always greater than or equal to zero”:

Equivalent Statements Of The Second Law Of Thermodynamics

The Kelvin statement:

For example, it is easy to convert mechanical work completely into thermal energy, but it is impossible to remove thermal energy from a system and convert it completely into mechanical work with no other changes. This experimental fact is one statement of the second law of thermodynamics.

“It is impossible to remove thermal energy from a system at a single temperature and convert it to mechanical work without changing the system or surroundings in some other way”.

Second Law of Thermodynamics: Kelvin Statement

The Clausius statement:

If we place an ice cube on a hot day, the ice cube will melt. From the point of view of energy, what has happened is that some of the heat from the surrounding air enters the ice cube, raising its temperature, and eventually melting it. The surrounding air subsequently cools somewhat. However, nothing from energy conversion, or Newton’s law in general, would prevent heat from leaving the ice cube, making the ice cube colder and the surrounding air warmer. Why then doesn’t this later phenomena occur?

A common example of the conversion of mechanical energy into thermal energy is movement with friction. For example, when a block slides along a rough table, the initial mechanical [kinetic] energy of the block is converted into thermal energy as the block and the table are heated. The reverse process never occurs- a block and table that are warm will never spontaneously cool by converting their thermal energy into kinetic energy that sends the block sliding across the table. Thus there is a lack of symmetry in the roles played by heat and work. This lack of symmetry is related to the fact that some processes are irreversible. It may be mentioned that thermodynamic processes that occur in nature are all irreversible.

Let us take the case of heat conduction which is an irreversible process. If we place a hot body in contact with a cold body, heat will flow from the hot body to the cold body until they are at the same temperature. The reverse does not occur; heat does not flow from one to the other making one colder and the other warmer.

The answer lies in the Clausius statement of the 2nd law of thermodynamics, which can be written as:

“A process whose only final result is to transfer thermal energy from a cooler object to a hotter one is impossible”.

Second Law of Thermodynamics: Clausius Statement

In other words, “Heat never flows spontaneously from low temperature to high temperature”.

The Refrigerator statement:

Refrigerator:

It is impossible to make heat flow from a body at a lower temperature to a body at a higher temperature without doing external work on the working substance. Energy will not flow spontaneously from a low temperature object to a higher temperature object. This precludes a perfect refrigerator. 

“No cyclic process can transfer heat from a colder place to a hotter place with no input of mechanical work”.

Second Law of Thermodynamics: Refrigerator Statement

 

Reducing Friction

A common way to reduce friction is by using a lubricant, such as oil, that is placed between the two surfaces, often dramatically lessening the coefficient of friction. The science of friction and lubrication is called tribology. Superlubricity, a recently-discovered effect, has been observed in graphite: it is the substantial decrease of friction between two sliding objects, approaching zero levels - a very small amount of frictional energy would be dissipated due to electronic and/or atomic vibrations.

Lubricants to overcome friction need not always be thin, turbulent fluids or powdery solids such as graphite and talc; acoustic lubrication actually uses sound as a lubricant.

Causes of friction

Friction is caused by the roughness of the materials rubbing against each other, deformations in the materials, and a molecular attraction between molecules of two surfaces.

1. Surfaces not completely smooth:

Most friction results because the surfaces of materials being rubbed together are not completely smooth. If you looked at what seems to be a smooth surface under a microscope, you would see bumps, hills and valleys that would interfere with sliding motion. Of course, the rougher the surface, the more is the friction.

If both surfaces become ultra-smooth and flat, the friction from surface roughness becomes negligible, but then friction from molecular attraction comes into play, often becoming greater than the normal friction.

2. Deformations:

Soft materials will deform when under pressure. This also increased the resistance to motion. For example, when you stand on a rug, you sink in slightly, which causes resistance when you try to drag your feet along the rug's surface. Another example is how rubber tires flatten out at the area on contact with the road.

When materials deform, you must "plow" through to move, thus creating a resistive force.

3. Molecular attraction:

There is another factor in friction, and that is stickiness caused by molecular attraction. This was mentioned above where surfaces are so smooth that the materials stick together due to molecular forces.

Soft rubber is an example of a material that can have this type of friction. This factor is usually seen in rolling friction. The stickiness will create a resistance to any motion. Although this force is the smallest, it still can be a factor when the other causes of friction are low.

The fundamental forces in nature

All the different forces observed in nature can be explained in terms of four basic interactions that occur between elementary particles:

1.       The gravitational force

2.       The electromagnetic force

3.       The strong nuclear force

4.       The weak nuclear force

The everyday forces that we observe between macroscopic objects are due to either the gravitational force or the electromagnetic force.

Forces may be placed into two broad categories, based on whether the force resulted from the contact or non-contact of the two interacting objects.

Action at a distance:

The fundamental forces of gravity and electromagnetism act between particles that are separated in space. This creates a philosophical problem referred to as action at a distance.

Contact forces:

Many forces we encounter are exerted by objects in direct contact. These forces are electromagnetic in origin and are exerted between the molecules of each object.

Normal force:

Consider a book on a table. The weight of the book pulls it downward, pressing it against the molecules in the table’s surface, which resist compression and exert a force upward on the book. Such a force, perpendicular to the surface, is called a normal force.

Frictional force:

Objects in contact can also exert forces on each other that are parallel to the surfaces in contact. The parallel component of a contact force is called a frictional force.

Static friction:

Friction is a complicated, incompletely understood phenomenon that arises due to the bonding of molecules between two surfaces that are in close contact. This bonding is the same as the molecular bonding that holds an object together. When you apply a small horizontal force to a large box resting on the floor, the box may not move because of the force of static friction,exerted by the floor on the box, balances the force you are applying. The force of static friction, which opposes the applied force, can adjust from zero to some maximum force f _{s, max} depending on how hard you push. You might expect f _{s, max} to be proportional to the area of contact between the two surfaces, but this is not the case. To a good approximation, f _{s, max} is independent of the area of contact and is simply proportional to the normal force exerted by one surface on the other:

f _{s, max} = m _s F_n

where, m_s is called the coefficient of static friction, a dimensionless quantity that depends on the nature of the surfaces in contact. If you exert a horizontal force smaller than f _{s, max} on the box, the frictional force will just balance this horizontal force. In general, we can write

f _s £ m _s F_n

 Kinetic friction:

If you push the box hard enough, it will slide across the floor. When the box is sliding, molecular bonds are continually being formed and ruptured, and small pieces of the surfaces are being broken off. The result is a force of kinetic friction,that opposes the motion. To keep the box sliding with constant velocity, you must exert a force on the box that is equal in magnitude and opposite in direction to the force of kinetic friction exerted by the floor.

The coefficient of kinetic friction m _k is defined as the ratio of magnitudes of the kinetic frictional force f _k and the normal force F_n:

f _k =m _k F_n

where m _k depends on the nature of the surfaces in contact. Experimentally, it is found that m _k is less than m _s and is approximately constant for speeds ranging from about 1 cm/s to several meters per second.

The plot of the frictional force vs. the applied force illustrates some of the features of the frictional force. Note that the frictional force equals the applied force (in magnitude) until it reaches the maximum possible value µ_sN. Then the object begins to move as the applied force exceeds the maximum frictional force. When the object is moving the frictional force is kinetic and roughly constant at the value µ_kN which is below the maximum static friction force.

Examples of kinetic friction:

¨         Sliding friction is when two objects are rubbing against each other. Putting a book flat on a desk and moving it around is an example of sliding friction.

¨         Rolling friction occurs when the two objects are moving relative to each other and one "rolls" on the other (like a car's wheels on the ground). The coefficient of rolling friction is typically denoted as μ _r.

¨         Fluid friction is the friction between a solid object as it moves through a liquid or a gas. The drag of air on an airplane or of water on a swimmer are two examples of fluid friction.

Contact Forces	Action-at-a-Distance Forces
Frictional Force	Gravitational Force
Tensional Force	Electrical Force
Normal Force	Magnetic Force
Air Resistance Force
Applied Force
Spring Force

Force & Friction

The science of mechanics is based on three natural laws relating force and motion. These were clearly stated for the first time by Sir Isaac Newton [1642 – 1727] and were published in 1686 in his Philosophiae Naturalis Principia Mathematica. Newton’s three laws relate an object’s acceleration to its mass and the forces acting on it. A modern wording of Newton’s laws follows:

I.  Newton's First Law of Motion: Every object continues to be at rest or in a state of uniform motion unless acted on by an external force.

This we recognize as essentially Galileo's concept of inertia, and this is often termed simply the "Law of Inertia".

To say that something is moving always implies a specific frame of reference. An inertial frame of reference is one in which Newton’s first law of motion holds.

II. Newton's Second Law of Motion:  Newton's second law of motion explains how an object will change velocity if it is pushed or pulled upon.

The rate of change of momentum of a body is directly proportional to the applied force acting on the body.

Firstly, this law states that if you do place a force on an object, it will accelerate, i.e., change its velocity, and it will change its velocity in the direction of the force.

It accelerates in the direction…………..

That you push it.

Secondly, this acceleration is directly proportional to the force. For example, if you are pushing on an object, causing it to accelerate, and then you push, say, three times harder, the acceleration will be three times greater.

If you push twice as hard…………..

It accelerates twice as much.

Thirdly, this acceleration is inversely proportional to the mass of the object. For example, if you are pushing equally on two objects, and one of the objects has five times more mass than the other, it will accelerate at one fifth the acceleration of the other.

If it gets twice the mass……………..

It accelerates half as much.

III. Newton's Third Law of Motion: 

The word force is used to describe the interaction between two objects. When two objects interact, they exert force on each other. Newton’s third law states that these forces are equal in magnitude and opposite in direction.

For example, if you push on a wall, it will push back on you as hard as you are pushing on it.

If you push on it…………………

It pushes on you.

The full power of Newton’s second law emerges when it is combined with the force laws that describe the interactions of objects. For example, Newton’s law for gravitation, gives the gravitational force exerted by one object on another in terms of the distance between the objects and the masses of each. This, combined with Newton’s second law, enables us to calculate the orbits of planets around the sun, the motion of the moon, and variations with altitude of g, the acceleration due to gravity

Units of Heat:

Before scientists realized that heat is transferred energy, heat was measured in terms of its ability to raise the temperature of water. Thus, the calorie (cal) was defined as the amount of heat that would raise the temperature of 1 g of water from 14.5⁰C to 15.5⁰C.

In 1948, the scientific community decided that since heat (like work) is transferred energy, the SI unit for heat should be the one we use for energy, namely, the joule. The calorie is now defined to be 4.1860 J (exactly). The “calorie” used in nutrition is really is kilocalorie.

1.                   Calorie (cal): It is the amount of heat required to increase the temperature of 1 g of water from 14.5^oC to 15.5^oC. (1 cal=4.186 J)

2.                   Kilocalorie (kcal): It is the amount of heat required to raise the temperature of 1 kg of pure water through 1^oC.

3.                   British Thermal Units (BTU): It is defined as the quantity of heat required to raise the temperature of 1 pound (lb) of pure water through 1^oF. It is also referred to as pound-degree Fahrenheit unit.

Conversion:            1 B.T.U = 251.996 Cal ( or 252 Cal)
1 calorie = 4.186 Joule ( or 4.2 joule)