Supposing Newton’s law of gravitation for gravitation forces \(F_{1}\) and \(F_{2}\) between two masses \(m_{1}\) and \(m_{2}\) at positions \(r_{1}\) and \(r_{2}\) read\(F_{1} = - F_{2} = - \dfrac{r_{12}}{r_{12}^{3}} G M^{2}_{0} \left(\dfrac{m_{1} m_{2}}{M_{0}^{2}}\right)^{n} \)

where, \(M_{0}\) is a constant of the dimension of mass, \(r_{12} = r_{1} - r_{2}\) and \(n\) is a number. In such a case,

(a) the acceleration due to gravity on the earth will be different for different objects.
(b) none of the three laws of Kepler will be valid.
(c) only the third law will become invalid.
(d) for \(n\) negative, an object lighter than water will sink into the water.
Choose the correct alternatives:
1. (a), (b), (c) 2. (a), (d)
3. (b), (c), (d) 4. (a), (c), (d)
() Hint: The acceleration due to gravity depends on the gravitational force.
Step 1: Find the value of acceleration due to gravity.
Given,
F1=-F2=-r12r123GM02m1m2M02n
r12=r1-r2
Acceleration due to gravity,
g=|F|mass=1r122GM02m1m2M02n×1mass
Since g depends upon the position vector, hence it will be different for different objects. As g is not constant, hence constant of proportionality will not be constant in Kepler's third law. Hence, Kepler's third law will not be valid.
As the force is of central nature.                   force1r2
Hence, the first two of Kepler's laws will be valid.
Step 2: Find the acceleration due to gravity in case the value of n is negative.
For negative n,
g=1r122GM02m1m2M02-n×1mass=1r122GM021+nm1m2-n×1mass
g=GM02r122M02nm1m2n×1mass
M0>m1 or m21m2)-
g>0, hence in this case situation will reverse i.e., an object lighter than water will sink in water.