First slide

Universal law of gravitation

Question

The magnitudes of gravitational field at distances r₁ and r₂ from the centre of a uniform sphere of radius R and mass M are F₁ and F₂, respectively. Then

Easy

A

$F_{1} / F_{2} = r_{1} / r_{2} if r_{1} < R and r_{2} < R$
B

$F_{1} / F_{2} = r_{2}^{2} / r_{1}^{2} if r_{1} > R and r_{2} > R$
C

all the above
D

none of the above

Solution

We consider the following two cases
(i) When r₁ > R and r₂ > R i.e., the points are outside the sphere. In this case, the gravitational field is given by GM /r².So
$F_{1} = \frac{GM}{r_{1}^{2}} and F_{2} = \frac{GM}{I_{2}^{2}}$
or $\frac{F_{1}}{F_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}}$ i.e., option (b) is correct
(ii) When r₁ <.R i.e .,the points are inside the sphere. In this case
Gravitational field $= \frac{{GM}_{1}}{r^{2}}$ where M₁, is the mass of the body within the sphere of radius r.
$∴ M_{1} = \frac{4}{3} {πr}^{3} ρ$ where p is the density
$F_{1} = G (\frac{4}{3} {πr}_{1}^{3} ρ) / r_{1}^{2}$
and $F_{2} = G (\frac{4}{3} {πr}_{2}^{3} ρ) / r_{2}^{2}$
or $\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}}$ 4,i.e option (a) is correct.

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