Find the gravitational force of attraction between the ring and sphere as shown in the diagram where the plane of the ring is perpendicular to the joining the centres. If $\sqrt[]{8} R$ is the distance between the centres of a ring (of mass ‘m’) and a sphere (mass ‘M’) where both have equal radius ‘R’.

Option 1 -

$\frac{\sqrt[]{8}}{9} . \frac{G m M}{R}$

Option 2 -

$\frac{1}{3 \sqrt[]{8}} . \frac{G M m}{R^{2}}$

Option 3 -

$\frac{\sqrt[]{8}}{27} . \frac{G m M}{R^{2}}$

Option 4 -

$\frac{2 \sqrt[]{2}}{3} . \frac{G M m}{R^{2}}$

2 Views | Posted a month ago

Asked by Shiksha User

1 Answer

Answered by

Payal Gupta | Contributor-Level 10

a month ago

Correct Option - 3

Detailed Solution:

Method – I:

$F = M_{s p h e r e} E_{r i n g} = M \times \frac{G m x}{{(R^{2} + x^{2})}^{\frac{3}{2}}} = M \times \frac{G m x \sqrt[]{8} R}{{(R^{2} + 8 R^{2})}^{\frac{3}{2}}} = \frac{\sqrt[]{8} G M m}{27 R^{2}}$

Method – II:

$F = \int d F c o s θ = c o s θ \int (d m) \frac{G M}{(R^{2} + 8 R^{2})} = \frac{\sqrt[]{8} G M m}{27 R^{2}}$

0 Upvotes 0 Downvotes

Similar Questions for you

A wire can be broken by a load of 20 kg-wt. The force required to break wire of same material with twice the diameter will be

alok kumar singh

The radii of circular orbits of two satellites A and B of the earth, are 4R and R, respectively. If speed of satellite A is 2V, then the speed of satellite B will be

alok kumar singh

At a point away from planet of radius 6400 km, the gravitational potential and field are –6.4 × 10⁷ SI units and 6.4 SI units respectively. Find height of that point above surface of planet.

alok kumar singh

$\frac{G M}{r} = 6.4 * 1 0^{7}$ . (i)

$\frac{G M}{r^{2}} = 6.4$ . (ii)

$r = \frac{6.4 * 1 0^{7}}{6.4}$

= 10⁷ m

= 10,000 km

R + h = 10,000

h = 10,000 – 6400 = 3600 km

Mass of moon is $\frac{1}{81}$ times the mass of a planet and radius is $\frac{1}{9}$ times the radius of the planet. The ratio of escape speed from planet to escape speed from moon is ________.

alok kumar singh

-> $V_{e s c} = \sqrt[]{\frac{2 G M}{R}}$

$R a t i o = \sqrt[]{\frac{81}{9}} = 3$

If the gravitational acceleration at the surface of earth is g, then increase in potential energy in lifting an object of mass m from earth surface to a height equal to half of radius of earth from surface will be:

alok kumar singh