Ncert Exercises & Solutions

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,

$F_{1} = - F_{2} = \frac{- \vec{r_{12}}}{r_{12}^{3}} G M_{0}^{2} {(\frac{m_{1} m_{2}}{M_{0}^{2}})}^{n}$

$r_{12} = r_{1} - r_{2}$

Acceleration due to gravity,

$g = \frac{| F |}{mass} = \frac{1}{r_{12}^{2}} {GM}_{0}^{2} {(\frac{m_{1} m_{2}}{M_{0}^{2}})}^{n} \times \frac{1}{mass}$

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.

$[∵ f o r c e \propto \frac{1}{r^{2}}]$

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,

$g = \frac{1}{r_{12}^{2}} G M_{0}^{2} {(\frac{m_{1} m_{2}}{M_{0}^{2}})}^{- n} \times \frac{1}{m a s s} = \frac{1}{r_{12}^{2}} G M_{0}^{2 (1 + n)} {(m_{1} m_{2})}^{- n} \times \frac{1}{m a s s}$

$g = \frac{G M_{0}^{2}}{r_{12}^{2}} \frac{M_{0}^{2 n}}{{(m_{1} m_{2})}^{n}} \times \frac{1}{m a s s}$

$M_{0} > m_{1} o r m_{2}$ ₁m₂)-

g>0, hence in this case situation will reverse i.e., an object lighter than water will sink in water.

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