Question 227: Gravitation

Question

Two spherical bodies of mass $M$ and $5M$ & radii $R$ & $2R$ respectively are released in free space with initial separation between their centers equal to $12R$. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is

Accepted Answer

Let $t$ be the time taken for the two masses to collide and ${x_{5M,}}\,{x_M}$ be the distance travelled by the mass $5M$ and $M$ respectively.
The gravitational force acting between two sphere when the distance between them (12R - x) where x is a variable,
$F = {{GM 	imes 5M} \over {{{\left( {12R - x} ight)}^2}}}$
Acceleration of mass M, ${a_M} = {{G 	imes 5M} \over {{{\left( {12R - x} ight)}^2}}}$
Acceleration of mass 5M, ${a_{5M}} = {{GM} \over {{{\left( {12R - x} ight)}^2}}}$
For mass $5M$
$u = 0,\,\,S = {x_{5M}},\,\,t = t,\,\,a{ = a_{5M}}$
$S = ut + {1 \over 2}a{t^2}$
$	herefore$ ${x_{5M}} = {1 \over 2}{a_{5M}}{t^2}\,\,\,\,\,\,\,\,\,\,...\left( {i} ight)$
For mass $M$
$u = 0,\,\,s = {x_M},\,\,t = t,\,\,a = {a_M}$
$	herefore$ $s = ut + {1 \over 2}a{t^2} \Rightarrow $
${x_M} = {1 \over 2}{a_M}{t^2}\,\,\,\,\,\,\,\,\,...\left( {ii} ight)$
Dividing $(ii)$ by $(iii)$
${{{x_{5M}}} \over {{x_M}}} = {{{1 \over 2}{a_5}_M{t^2}} \over {{1 \over 2}{a_M}{t^2}}}$
$ = {{{a_{5M}}} \over {{a_M}}} = {1 \over 5}$
$	herefore$ $5{x_{5M}} = {x_M}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left....( {iii} ight)$
From the figure,
${x_{5M}} + {x_M} = 12R - 2R - R = 9R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( iv ight)$
From $(iii)$ and $(iv)$
${{{x_M}} \over 5} + {x_M} = 9R$
$	herefore$ $6{x_M} = 45R$
$	herefore$ ${x_M} = {{45} \over 6}R = 7.5R$
So two sphere collide when the sphere of mass M covered the distance of 7.5R.

Answer

$2.5$ $R$

Answer

$4.5$ $R$

Answer

$7.5$ $R$

Answer

$1.5$ $R$

Gravitation

Two spherical bodies of mass M and 5M & radii R & 2R respectively are released in free space with initial separation between their centers equal to 12R. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is

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