If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube law-

(a) Planets will not have elliptic orbits.

(b) Circular orbits of planets is not possible.

(c) Projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic.

(d) There will be no gravitational force inside a spherical shell of uniform density.

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alok kumar singh | Contributor-Level 10

4 months ago

This is a Multiple Choice Questions as classified in NCERT Exemplar

Answer-a, c

Explanation- as we know F= $\frac{G M m}{a^{2}} = \frac{m v^{2}}{a}$

V= orbital speed= $\frac{\sqrt[]{G m}}{a}$

Time period of revolution of planet T = $\frac{2 π a}{v} = \frac{2 π a}{\frac{\sqrt[]{G m}}{a}}$ = $\frac{2 π a^{2}}{\sqrt[]{G M}}$

T² $\propto a^{4}$ ……….1

Hence orbit will be elliptical.

F= $\frac{G M}{a^{3}} m$ =g’m

It is clear from equation 1 that its path is parabola.

This is a Multiple Choice Questions as classified in NCERT ExemplarAnswer-a, cExplanation- as we know F= <math> <mfrac> <mrow> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>m</mi> <msup> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> </mfrac> </math> V= orbital speed= <math> <mfrac> <mrow> <mrow> <mroot> <mrow> <mrow> <mi>G</mi> <mi>m</mi> </mrow> </mrow> <mrow></mrow> </mroot> </mrow> </mrow> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> </mfrac> </math> Time period of revolution of planet T = <math> <mfrac> <mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mfrac> <mrow> <mrow> <mroot> <mrow> <mrow> <mi>G</mi> <mi>m</mi> </mrow> </mrow> <mrow></mrow> </mroot> </mrow> </mrow> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfrac> </math> = <math> <mfrac> <mrow> <mrow> <mn>2</mn> <mi>π</mi> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mroot> <mrow> <mrow> <mi>G</mi> <mi>M</mi> </mrow> </mrow> <mrow></mrow> </mroot> </mrow> </mrow> </mfrac> </math> T2 <math> <mo>∝</mo> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </msup> </math> ……….1Hence orbit will be elliptical.F= <math> <mfrac> <mrow> <mrow> <mi>G</mi> <mi>M</mi> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> <mi>m</mi> </math> =g’mIt is clear from equation 1 that its path is parabola.

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