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The Gravitational Constant

Gravitationis otherwise called as gravity. It includes with these four fundamentalinteractions such as strong interaction, weak interaction and electromagnetic force. An object can be combined with mass attract one another. It gives weight to objects with mass and it can cause the weight to be fall down on the ground when it is dropped. Gravitation canbe described using general theory of relativity. Let us determine the gravitational constant using Cavendish apparatus.


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Apparatus Set Up for Determining the Gravitational Constant


 By using Cavendish apparatus, the value of the gravitational constant can be determined.

              


A cross bar which consists of PQ with 2 meters long. The bar can be rotated about a vertical axis. From the ceiling of a room, a rod can be suspended. A heavy large lead spheres C and D can be suspended on the two ends of P and Q with metallic rods. A torsion head H with wooden rodRS can be suspended with fine torsion wire W made by silver-copper and it is suspended below the midpoint of PQ.


From the ends of Rs two small lead balls can be weighing down with 6.80kg namely A and B. the center of the four balls namely A, B, C and Dlie in a same horizontal line. The apparatus cannot be disturbed by air, so that the room is kept closed. The two telescopes can be fixed inthe room to observe the deflection of rod RS.


Method of Determining the Gravitational Constant


  • Byrotating the rod PQ, the centre of the large balls C and D are at rightangles to the torsion rod RS, there is no twist in the wire W. The position of the rod RS can be note down.

  • The PQ is turned, so that the large balls C and D brought in position C1 and D1, then AC1 = BD1.The position of the rod RS is noted. The larger balls can be placed in the position of C2 and D2 with distance of two nearer balls we did above. The position of the rod RS in noted again.

     

  • From the untwisted position of the wire W, mean deflection of the rod RS is found.
  • Time period of oscillations of rod is measured.

     


Derivation of the Formula for Gravitational Constant


Let us take M be the mass of the large ball C and D, and m be the mass of each small balls, and d be the distance between the centers of each  pair of nearer balls.

AC1 = BD1 = AC2 = BD2 = d

     

Force of attraction between pairs of balls = `(GMm)/(d^(2))`

Then l is the length of torsion rod RS, then the deflecting couple,

`tau` = `(GMm)/(d^(2))`(l)

       

 C is said to be trosional rigidity of wire W, restoring couple per unittwist (in terms of radians). Restoring couple produced for twist `theta`, = C`theta`

      

For equilibrium, Deflecting couple = Restoring couple

`(GMm)/(d^(2))`(l) = C`theta`

G = `(Cd^(2)theta)/(Mml)`


 If Time period of oscillation of rod Rs is T, then

T = 2`pi` `sqrt((I)/(C))`


I is the moment of inertia of rod RS ( containing 2 spheres A and B)

C = `(4pi^(2)I)/(T^(2))`


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Neglecting mass of rod, moment of inertia of oscillation system is

I = m`((l)/(2))^(2)` + m `((l)/(2))^(2)`     =  `(ml^(2))/(2)`

C = `(4pi^(2))/(T^(2))``(ml^(2))/(2)`) = `(2pi^(2)ml^(2))/(T^(2))`


Put the value of c in  (1), we get,

G = `(2pi^(2)ml^(2))/(T^(2))``(d^(2)theta)/(Mml)`

G = `(2pi^(2)ld^(2))/(MT^(2))``theta`


This is the Gravitational Constant observed by Cavendish apparatus.

Cavendish took number of observations and found G = 6.6754 × `10^(-11)` N`m^(2)` `kg^(-2)` .