Coordinates of centre of mass

Let us consider a system of  two particles m1 and m2 and placed at A and B having position coordinates x1 and x2 respectively from the origin as shown in above figure. If  Xcm is position coordinate of the centre of mass from origin
              Then d1=Xcm-x1
                       d2=x2-Xcm
                       d=x2-x1
         According to the definition of the centre of mass of the particles, if the centre of mass is defined  at C along AB  such that 
       product of (m1,AC)= product of (m2,CB)
       product of (m1,d1)= product of (m2,d2)
                Then
                   m1(Xcm-x1)=m2(x2-Xcm)
                    Xcm(m1+m2)=m1x1+m2x2
                     Xcm=(m1x1+m2x2)/(m1+m2)
        This is the position coordinate of the centre of mass.
From the above equation, the position coordinate is analog to the weighted mean displacement, i.e., where weighting factor for each particle is the fraction of the total mass that each particle has.
               Suppose, x1=0 and x2=d, then
                     Xcm=m2d/(m1+m2). This is the result when the origin is shifted to x1
               Suppose, x2=0 and x1=d, then
                          Xcm=m1d/(m1+m2). This is the result when the origin is shifted to x2
            If the origin is shifted to centre of mass i.e., Xcm=0 then x1=-d1 and x2=d2 then  
                    0=(-m1d1+m2d2)/(m1+m2) or m1d1=m2d2
                        Therefore, (d1/d2)=(m2/m1)
                 So, we can say that the ratio of the distances of centre of  mass  from masses is inverse ratio of their masses. We can say that the centre of mass is nearer to the heavier mass. The location of the centre of mass is independent of  the reference frame used to locate it. The centre of mass depends upon the masses the particles and the position of the particles relative to one another.

Stability and its applications

Mainly, the stability of a body depends upon the base  area  and height of centre of gravity for a body. Stability of a body increases with base area and decreases with the height of centre of gravity.
                There are three types of equilbrium for a body i.e.,
Stable equilbrium:-   A body is in stable equilbrium when its position is not disturbed.
Unstable equilbrium:- A body is in unstable equilbrium , if it does not regain its original position after it is being slightly disturbed
Neutral equilbrium:- A body is in neutral equilbrium if it changes its position without change in equilbrium after being disturbed.
          The stability principle is mainly used in constructing ship. For that purpose, the base of the ship  is made as large as possible and the height of the centre of gravity is made as small as possible to maintain the stability of the ship very high. Due to this reason, even the ship tilts a side but it regains its original position so that the ship is in stable equilbrium.
          Another application of stability is person walking  on a rope. When a person is walking on a rope by holding a long pole in his hand, he changes  the orientation of the pole  such that the line of action of the total weight passes through the rope  so that he does not fall down.

Difference between the centre of mass and centre of gravity

Consider a body consisting of large number of particles each of mass 'm' and each particle is attracted by the earth towards its centre with weight of  'mg'. Like this, the weight of  various particles constitute a system of  like parallel forces. The resultant of these forces are passing through a point in the body. That point is called the centre of gravity of the body.
               Therefore the centre of gravity of a body  is the point through which the weight of the body acts.
In case of small bodies, the centre of gravity and centre of mass almost coincide. Suppose if the body is large and so extensive that the acceleration due to gravity changes from one point to another point in the space occupied by the body where centre of gravity and centre of mass are different.
               The main difference between the centre of gravity and centre of mass is that centre of gravity depends upon acceleration due to gravity on different particles of the body, on other hand, centre of mass is independent of acceleration due to gravity.
                Centre of gravity is defined as the stability of the body when supported, on the other hand, centre of mass is defined to describe the motion of the body as a whole.

Centre of mass and its introduction

There are two motions for a rigid body i.e., rotatory motion and translatory motion.  When a wheel moves on a horizontal plane, it rotates moves on a straight line path. If a particle on the rim of the wheel is considered, it will have a complicated path. But the particle at the axis of rotation is considered, then it has straight line motion. Suppose  a  ball is moved in spinning manner, then any particle on the surface having complicated path of motion. But the particle at the centre of the ball moves in a parabolic path.
            From the above examples, we come to the conclusion that when a rigid body has both rotatory and translatory motions, the different particles on the body moves in different directions. But the particle at a specific point is moved along a particular direction and the body moves as if the whole external force  is acting on it and represents the motion of the entire body.
           The location of centre mass is determined as a fixed point of a system of particles or a rigid body within the boundaries of the system, where the entire mass of the system or body is supposed to be concentrated.
            For a uniform rod, the centre of mass is the middle point of the rod. For a rectangular plate, the centre of mass is intersecting point of thediagonals of the rectangular plate. For a circular plate, the centre of mass is at the centre of the circular plate. For a ring, the centre of mass is not within the body, but within the boundaries of the system
               So far we discussed about the motion of a rigid body, but in general, we have to deal with the system of bodies consisting of  large number of particles which have mutual interactions such that they may change the positions with respect to each other in a complicated way during the motion. Such system is called Non-rigid body system. Planets moving around the sun and each planet having satellites moving around them is an example of Non-rigid system..

Kirchoff's first and second laws

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emf of the cell

In order to maintain continuous flow of current through a conductor AB of resistance R, we should always keep A at a positive(higher) potential and B at a negative(lower) potential. We connect A to positive terminal P of the cell and B to negative terminal Q of the cell. Through chemical reaction, the cell always maintains P at a constant positive potential and Q at a constant negative potential.

In external circuit, current(+ve charge) flows from P to Q via the conductor AB. But, inside the cell the same positive charge moves from lower potential to higher potential. To do this, the cell must be able to do work on the charge. The energy to do this work is derived from the chemical process inside the cell.

The influence that makes charge move from lower potential to higher potential is called the electromotive force and is denoted by E.

The emf of a cell is defined as the work done in carrying a unit positive charge through the complete circuit including the charge flow inside the cell.

The emf is measured in the units of Joule/Coloumb or Volt. Thus emf has the same units as the units of potential difference.

The resistance to flow of current inside the electrolyte solution of the cell is called internal resistance of the cell. The emf and internal resistance of the cell will be fairly constant only when small current is drawn from the cell.

Thermistor and its applications

Thermistor is widely used in measuring the rate of energy flow in micro wave beams. The beams fall on a thermistor and heats it. A relatively small rise in temperature results in a very large change in resistance because, for a thermistor alpha is very high. By measuring the change in resistance, we can accurately measure micro wave power.

In a radio circuit, there will be several heater elements in series. A sudden change in the current through the circuit will damage the device. To prevent such a sudden surge of current, we place a thermistor with a high negative temperature coefficient of resistance, in series in the circuit. Initially, the thermistor is at cold state and hence has a very high resistance. This prevents the current to moderate level. Later, as the thermistor gets heated, its resistance decreases and it allows normal flow of current through the heater elements there by preventing surge. A thermistor can also be used as a thermostat.

Thermistor

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Temperature Dependence of Resistivity

The resistivity of the metallic conductors is found to increase with temperature. Over a limited range of temperature, the resistivity of metallic conductors is found to increase linearly (i.e., approximately) with temperature.


Thus, the temperature coefficient of resistivity may be defined as fractional change in resistivity per unit rise in temperature.

Above graph shows that variation of resistivity of a ohmic conductor with temperature(in kelvin)
Some important details about temperature dependence of resistivity as follows
a) For certain metals at lower temperature, the temperature dependence of resistivity is quite non-linear. The graph will be curve.
b)Metallic alloys like Nichrome have high resistivity. Therefore, it is used widely in electric heaters.
c)Resistivity of manganin and constantan is nearly independent of temperature. So, manganin is widely used in making resistance boxes, standard resistances and wires used in metre bridge and potentiometer.
d) The resistivity of carbon decreases with increase of temperature and has a negative temperature coefficient of resistance. Semiconductors like germanium and silicon, also behave in the same way.

conductance and conductivity

Conductance is defined as the reciprocal of resistance of a conductor, i.e., the ratio of the current i and voltage V.

Specific resistance or resistivity

In order to understand the behaviour of the materials, it is necessary to study certain properties like (a) Resistivity (b) Conductivity and (c) conductance.

The resistance (R) of a conductor depends on its length, area of cross section.

In the above equation if l=1m and A=1sq.m, then p(rou)=R. There is the difference between the resistance and resistivity i.e., resistance is the bulky property of a material and resistivity is specific property of the material.Since, specific resistance is proportionality constant, so that it is independent of length and area of cross section for that material.

For conductors, the specific resistivity like follows


From the above figure we found that for mercury, the resistivity is high.
For semi conductors, insulators resistivities like follows

From the above the above tables, we see that resistivity of insulators is about 10power22 times the metallic conductors. Since metals like silver, copper and aluminium have lowest resistivities, so that there are used in manufacture electric cables, connecting wires etc.

graphical properties of non-ohmic resistances

This figure shows that the relation between the current i and v in case of vaccum tubes i.e., an electronic device
This figure shows the V-i characterstics of a P-N junction diode rectifier which shows that non-linear relation between the voltage and the current in positive direction and in the reverse direction. This figure shows increase in the current is not possible beyond zeroth value of the voltage, so that either in positive value of V or negative value of V, for large value of the voltage, there is increase in the current.
This figure shows that the V-i characterstics of thermistor. The detail study of thermistor we can see in electronics subject.

Ohmic and Non-ohmic resistances(devices)

Resistances that obey the Ohm's law are called ohmic resistances. A metallic conductor at a constant temperature is called Ohmic resistance. For such ohmic resistances, the V-I characterstic curve will be a straight line passing through the origin.
In ohmic resistance, current is reversed in the direction when the potential difference is reversed, but the magnitude of the current remain the same.

There are many resistances that do not obey the Ohm's law. These are called Non-ohmic resistances. For these resistances, the V-I characterstic curve will not be linear. The function of modern electronic devices that do not obey the Ohm's law.

Ohm's law

Generally, for a metallic conductor has a constant resistance R when other physical conditions remain same. For a metallic conductor the current passing through conductor will be directly proportional to the potential difference V applied across its ends. George Simon Ohm gave the relationship between V and i, which is called ohm's law.
Ohm's law is just a empirical relationship. It is not a fundamental physical principle and does not specify any general property of matter. For electrolytes, another essential condition required for application of Ohm's law is that the physical state must remain the same. Ohm's law is valid only for metallic conductors in which V/i has a constant value irrespective of the magnitudes of V and i

Resistance and units

The resistance is defined as the ratio of the potential difference 'V' across it to the current 'i' which flows through the conductor.

Resistance (R)=V/i

When we apply a potential difference 'V' between the ends of a conductor, an electric field 'E' will be set up inside the conductor and as a result a current 'i' flows through the conductor. This causes root notion for the resistance.
Resistance is a characteristic of the conductor as a whole. Resistance depends in general on nature of material, its dimensions(length,area of cross section), its temperature.

In some type of conductors the resistance 'R' increases when 'V' is increased. In another type of conductors the resistance 'R' decreases when 'V' is decreased. In some type of conductors 'R' depends on the direction of current flows through it.

current and units

Current is defined as the rate of flow of charge through any cross section of a conductor. Electric current is defined as the net charge passing through any cross section per unit time.
Suppose the net charge 'q' passes through any cross section of the conductor in time 't', then the current 'i' is given by

i=q/t

Where 'q' is in coloumbs and 't' is in seconds and then 'i' is in coloumb/sec or ampere. Like the mass, current is macroscopic quantity in SI system and is dimensionally denoted as I or A