First slide

Gravitational field intensity

Question

Consider two solid spheres of radii $R_{1} = 1 m, R_{2} = 2 m$ and masses $M_{1}$ and $M_{2}$ , respectively. The gravitational field due to sphere 1 and 2 are shown. The value of $\frac{M_{1}}{M_{2}}$
is :

Moderate

A

$\frac{1}{2}$
B

$\frac{2}{3}$
C

$\frac{1}{6}$
D

$\frac{1}{3}$

Solution

$\begin{array}{l} A t s u r f a c e (p e a k o f t h e g r a p h), g = \frac{G M}{R^{2}} \\ F o r S p h e r e 2 f i e l d a t s u r f a c e i s 3 = \frac{G M_{2}}{2^{2}} - - - - (1) \\ F o r S p h e r e 1 f i e l d a t s u r f a c e i s 2 = \frac{G M_{1}}{1^{2}} - - - - (2) \\ d i v i d i n g (2) b y (1) w e g e t \\ ∴ \frac{3}{2} = \frac{\frac{G M_{2}}{4}}{G M_{1}} \\ ∴ \frac{3}{2} = \frac{M_{2}}{4 M_{1}} \\ \Rightarrow \frac{M_{1}}{M_{2}} = \frac{1}{6} \end{array}$

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