When a body of mass 1 kg dropped from the roof of a house, it gets attracted towards the earth with a certain gravitational force. In accordance with the third law of motion, the stone also attracts the earth with the same gravitational force towards it. Is it true that we do not see any appreciable motion of the earth towards the stone? (Given mass of earth is 6×1024 kg , radius of the earth r=6.4×106 m gravitational constant G =6.67×10-11 N m 2kg-2).

Question

When a body of mass 1 kg dropped from the roof of a house, it gets attracted towards the earth with a certain gravitational force. In accordance with the third law of motion, the stone also attracts the earth with the same gravitational force towards it. Is it true that we do not see any appreciable motion of the earth towards the stone? (Given mass of earth is 6×1024 kg , radius of the earth r=6.4×106 m gravitational constant G =6.67×10-11 N m 2kg-2).

Accepted Answer

The given statement is true.It is true that we cannot see any appreciable motion of the earth towards the stone even though the stone also attracts the earth with the same gravitational force towards it.As per Newton’s law of gravitation we know that  F=Gm1m2r2 .....................(a) where G is the gravitational constant (6.67×10-11 Nm2kg-2)          m1is the mass of the 1st object         m2 is the mass of the 2nd object              r is the distance between themMass of earth  m1=6×1024 kg Mass of stone m2=1 kg Since earth has a very large radius, the distance between centers of earth and the stone is taken as the radius of the earth.Radius of the earth  r=6.4×106 m Gravitational constant G =6.67×10-11 N m 2kg-2Substituting these values in the equation for (a)   we get      F=6.67×10-11×6×1024×1(6.4×106)2⇒F=9.8 NThus 9.8 N is the gravitational force of attraction between the earth and the stone. This gravitational force is the force acting on the earth due to the stone and is equal to the force acting on the stone due to earth. As per the third law of motion these forces act in the opposite direction which is of equal value.Now we have to consider two cases: Acceleration of the stone in the downward direction and the acceleration of earth in the upward direction.Here force of attraction for both earth and stone is same, Hence F1=F2=FF=9.8 NConsidering t=1 s as time takenu=u1=u2 Initial velocity of both bodies are zero. Thus u=0Case 1: Acceleration of earth in the upward directionWe know that F=ma ...........................(b) where F is the force acting on the body          m is the mass of the body            a is the acceleration.For the case 1 equation (b) can be written as     F1=m1a1 ⇒a1=Fm1⇒a1=9.8N6×1024kg⇒a1=1.63×10-24 ms-2Now we need to find the distance traveled by the earth due to this force.We know that distance traveled S=ut+12at2 .......................(c)S is the distance traveled Here equation (c) can be written as S1=ut+12a1t2 ......................(d)Substituting the values in (d), we get      S1=0×1+121.63×10-24×12 ⇒S1=8.15×10-25m ..........................(i)Case 2:Acceleration of the stone in the downward directionFrom (b) For the case 2 equation (b) can be written as    F2=m2a2 ⇒a2=Fm2⇒a2=9.8 N1 kg⇒a2=9.8 ms-2Now we need to find the distance traveled by the stone due to this force.From (c)Here S2=ut+12a2t2---(e)Substituting the values in (e)⇒S2=0×1+129.8×12 ⇒S2=4.9 m----(ii)Comparing  (i) and (ii) we can see that the distance traveled by earth is negligibly small when it is compared with the distance traveled by the stone under the same given condition. Hence we cannot notice this very small motion of earth.

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