JEE MAIN - Physics (2020 - 8th January Morning Slot - No. 1)
Consider two solid spheres of radii R1 = 1m,
R2 = 2m and masses M1 and M2, respectively.
The gravitational field due to sphere (1) and (2) are shown. The value of $${{{M_1}} \over {{M_2}}}$$ is :
_8th_January_Morning_Slot_en_1_1.png)
$${2 \over 3}$$
$${1 \over 6}$$
$${1 \over 2}$$
$${1 \over 3}$$
Explanation
Gravitational field on the surface of a solid
sphere
E = $${{GM} \over {{R^2}}}$$
By the graph
E1 = $${{G{M_1}} \over {R_1^2}}$$
and E2 = $${{G{M_2}} \over {R_2^2}}$$
$$ \therefore $$ $${{{E_1}} \over {{E_2}}} = {\left( {{{{R_2}} \over {{R_1}}}} \right)^2}\left( {{{{M_1}} \over {{M_2}}}} \right)$$
$$ \Rightarrow $$ $${2 \over 3} = {\left( {{2 \over 1}} \right)^2}\left( {{{{M_1}} \over {{M_2}}}} \right)$$
$$ \Rightarrow $$ $${{{{M_1}} \over {{M_2}}}}$$ = $${1 \over 6}$$
E = $${{GM} \over {{R^2}}}$$
By the graph
E1 = $${{G{M_1}} \over {R_1^2}}$$
and E2 = $${{G{M_2}} \over {R_2^2}}$$
$$ \therefore $$ $${{{E_1}} \over {{E_2}}} = {\left( {{{{R_2}} \over {{R_1}}}} \right)^2}\left( {{{{M_1}} \over {{M_2}}}} \right)$$
$$ \Rightarrow $$ $${2 \over 3} = {\left( {{2 \over 1}} \right)^2}\left( {{{{M_1}} \over {{M_2}}}} \right)$$
$$ \Rightarrow $$ $${{{{M_1}} \over {{M_2}}}}$$ = $${1 \over 6}$$
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