Factors affecting gravitation acceleration(g).

Complete hand written Derivation on Factors affecting 'g'

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5-September-2019

Index:

• Coulomb's Law
• Principle of Superposition

Coulomb’s Law: 

The force of interaction (attraction or repulsion) between two point charges is directly proportional to the product of their masses and inversely proportional to the square of their distance.

Let q₁, q₂ are two point charges, the distance between them is r,

Where,   

K is the proportionality constant. It depends upon nature of medium separating the charges and on the system of units.

Formula of K in SI System:

Principle of superposition:

The resultant force on any charge is equal to the vector sum of coulomb forces acting on individual charges.

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Relation between universal constant (G) and gravitational acceleration (g)

Suppose if,  

M = Mass of earth

m = mass of falling object

R = Radius of earth 

Then, gravitational acceleration force,

By the second law of newton's velocity law,

Now on comparison of eqⁿ (1) and (2), we say that eqⁿ (1) =  eqⁿ (2)

We know that the formula of density is,

On putting the value of 'M' in eqⁿ (3)

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Index:

1. Newton's law of gravitation
2. Acceleration due to gravity

Newton's law of gravitation:

Every particle in the universe attracts another particle towards them, this attraction force is called gravitation force.

Two particle 'm1' and 'm2' the distance between them is 'R' and attraction force between them is 'F'.

According to newton's law of gravitation:

• The attraction force between two particle is directly proportional to the product of their masses,

            F α m₁m₂                             (1)

• The attraction force between two particle is inversely proportional to the square root of there distance,

            F α 1/R²                                (2)

--According to (1) and (2)

{F α m₁m₂}        { F α 1/R²}

--On removing proportionality sign(α) a constant comes,

Here,

G = universal gravitational constant

value of G = 6.67×10^-11 Nm²/kg² 

Dimension = [M^-1 L³ T²]

Acceleration due to gravity:

 The rate of change of velocity of an object falling down from any height due to effect of earth gravitational force is called gravitational acceleration (g).

The value of 'g' is 9.8m/s² 

Dimension = [M⁰ L¹ T^-2]

Here are few pictures of my hand written notes for better understanding,

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Escape velocity and Escape energy

Escape velocity
When we project a body upward then after reaching a certain height, the body return back due to gravitational attraction(g) of the earth. If we go increasing the initial velocity of he body, then it returns back after reaching greater and greater height. Ultimately(finally), at a certain velocity of projection the body will go out the gravitation field of the earth and will never return to the earth. This initial velocity of the body is called the 'escape velocity'.

                                                                                                                                                                                              in simple words- The minimum velocity with which a body should be projected from the surface of the earth so that it escapes the earth gravitational field called the escape velocity of the body.

Escape energy
The kinetic energy given to the body to project it with the escape velocity is called 'escape energy'.

Derivation:

Potential energy
The initial gravitational potential energy between two objects at some separations is:

where:

PE_i is the initial gravitation potential energy.   

R_i is the initial separation between the centers of the object in Km.

M is the mass of attracting object(earth) in Kg. 

m is the mass of escaping object in Kg.

NOTE: Since the work required to move an object from the zero reference point of Potential energy to some point in space is negative, the potential energy at that point is also considered negative.

Kinetic energy

The initial kinetic energy of an object projected at some velocity away from the earth or other astronomical body is:

where:

KE_i is initial kinetic energy.

m is the mass of object in Kg.

v_e is the escape velocity.

NOTE: The velocity given to the body is the escape velocity, therefore we considered v_e (escape velocity) in place of v.

Total energy or Total initial energy:

The total initial energy or the total initial energy is the sum of the potential energy and kinetic energy at the release point.

Final energy:

gravitational fields hypothetically extend to infinity. Thus, if the initial velocity is great enough, the object will travel to an infinite separation and thus, escape the gravitational force.

Potential energy at infinity:

PE∞ = gravitational potential energy at infinity.

R∞ = infinite separation between the centers of the objects. 

we know that R∞ = ∞(infinity), thus PE∞ = 0 

Kinetic energy at infinity:    

KE∞ = the Final kinetic energy..

v∞ = final velocity at infinite distance.

At infinity, the velocity of the object is zero (0); 

i.e. v∞ = 0, Thus KE∞ = 0.

Total final energy:

we know that the kinetic energy is moving upward and the potential energy is acting downward, the total energy at the initial position is

  

Now, by the law of conservation of energy

TE_i= TE ∞

KE_i+ PE_i = 0

substitute values: 

Take a square root of each expression to get,

we know that gravitation acceleration (g) is, 

Now by substitute the value of gR_i² in eq. 1

where, 

value of g = 9.81    m/s², or 9.807 m/s²

R_i= R= radius of earth 6378.1 km

To find the escape velocity from the earth surface:

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redupoint

Resistance

                          All About Electric Resistance

At this page, we have a tendency to fully outline resistance & every important data regarding resistance

INDEX

• resistance or resistance is a constant term
• factors upon which resistance depends
• resistivity
• temperature coefficient of resistance 
• combination of resistance (SCR&PCR)                                          

RESISTANCE

Electric Resistance is a property of a resistor to oppose the flow of electric current through it.
OR
The flow of free electrons is called current, and resistance is the obstruction in the path of flow of electron.

NOTE: Conductors (metals) have very little resistance while insulators (glass rubber, mica, etc.) have a large amount of resistance.

Resistance is a constant term, that present where the flow of charge takes place

Mathematically,

V ∝ I
On removing proportionality sign (as a constant resistance comes),

V = IR

Where, 
v= volt, 
I= current, 
R= resistance (as a constant term)

Resistance can be denoted by (R or r)

The SI unit of resistance is ohm ( Ω).

If the potential difference (V) is 1 volt and current (I) is 1 ampere, then the Resistance (R) is 1 ohm

Resistance can be found by the "BBROYGBVGW" method.

Very high resistance is measured in mega-ohm and very low resistance in micro-ohm.

S.No.	Prefix	Symbol	Value in ohm (℧)
1.	Micro i.e. one millionth		10-6
2.	Milli i.e. one thousandth		10-3
3.	Kilo i.e. one thousand		103
4.	Mega i.e. one million		106

Factors upon which resistance depends:

• Length: The resistance R of a conductor is directly proportional to its length i.e. R directly proportional L                                               

R ∝ L

i.e., an increase in the length of a conductor increases its resistance and vice versa.  

• Area of cross-section: The resistance R of a uniform conductor is inversely proportional to its area of cross-section (A),                           

i.e., if we increase the cross-sectional area (A) of the conductor, it's resistance decreases and vice-versa. 

• Nature of material: The resistance of a material conductor also depends on the nature of its material. In a simple word, a material having a large number of free electron, lesser is the electric resistance.

For e.g. the resistance of nichrome wire is 60 times that of copper wire and of equal length of the cross-section area.

• Changes with temperature: i.e., resistance increase with the increase in temperature  

Neglecting the effect of temperature, for the time being, the electrical resistance of a conductor

  by combining both equations (1)&(2)

Where rho (ρ) is a constant and known as electrical resistivity or specific resistance of a conductor.

Resistivity:

The electrical resistivity of the material of a conductor is defined as the resistance of unit length and the unit cross-section area of the conductor.

                                               

The SI unit of resistivity is Ohm.    
              

Electrical resistivity depends on the nature of material and relaxation time.

Effect of temperature on resistance:

The effect of temperature on resistance depends upon the type of material

• The resistance increases with increase in temperature (for metals)
• 
  
• The resistance increase rise in temperature but he increases is very small or irregular (for alloy)
• 
  
• The resistance of semiconductors (Ge & Si), insulators (.e.g., glass, paper, mica, etc) and electrolytes decrease with an increase in temperature
• 
  

Temperature coefficient of Resistance:

·         

       suppose at 0 degree Celsius resistance is R_o

·         At 't' deg. C resistance increase to R_t

·         So increment in resistance is:
                                                                                       R_t - R_o

This increment in resistance depends on

·       
            Initial resistance of conductor (R_o)

·         R_t - R_o ∝ R_o

·         (R_t - R_o) ∝ t

·         On the nature of the material

So (R_t - R_o) ∝ R_ot  
  here, ∝ sign denotes proportional

On removing proportionality sign, a constant come (∝)

(R_t - R_o) = ∝R_ot

                      here, ∝ (alpha) is temperature co-efficient of resistance

  

                        here, Rt  is the value of resistance at any to temperature  

                        now, alpha (temperature of constant of resistance) is:

Combination of Resistance

There are two types of combination of Resistance

• ·         Series combination of Resistance
• ·         Parallel combination of Resistance

• Series combination of Resistance

·       

  
          -Current through each resistor is the same

·       

        -Sum of potential difference across individual Resistor is equal to the       potential difference applied by source,                                                 

V = V₁+V₂+V₃

·    
            -Equivalent Resistance,            

         R = R₁+R₂+R₃               ... (1)

·       
             -By Ohm’s law              

V₁ = IR₁                     V₂ = IR₂                      V₃ = IR₃

 Total potential difference (V) between A&B

From equation (1) we know that, R= R1+R2+R3

Therefore, 

• Parallel combination of Resistance:

·      

 -Voltage through each resistor is the same

·        
     -Sum of electric current flowing through the individual resistor is equal to   electric current drawn by the source,                       

I=I₁+I₂+I₃ 

·          
      -Equivalent Resistor,                 

       -By Ohm’s law,      

So, the total current:

We know that, 

Therefore,                                                  

Series and Parallel combination of Resistance

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