Question

Two spherical balls of mass 1kg and 4kg are seperated by a distance of 12cm. The distance from 1kg at which at which the gravitational force on any mass becomes zero is

Answer 1

Given data:-

• Two spherical balls of mass 1kg and 4kg are seperated by a distance of 12cm.
• The distance from 1kg at which the gravitational force on any mass becomes zero.

Assumption:-

• Let, a be the spherical balls of mass 1kg and b be the spherical balls of mass 4kg.
• Let, x the distance from 1kg of sphere, at which gravitational force on any mass becomes zero.

Solution:-

According to assumption,

Distance for mass of 4kg sphere is 12 - x

Now, we use formula to find distance of x the distance from 1kg.

Gravitational force between two particles,

${ \bf{ \dashrightarrow{F = G \frac{m_{1}m_{2}}{ {r}^{2} } }}}$

Now, according to assumption

${ \bf{ \dashrightarrow{ \huge {\frac{1G}{ {x}^{2} } = \frac{4G}{ {(12 - x)}^{2} }} }}}$

as we know G is gravitational constant

${ \bf{ \dashrightarrow{ \huge {\frac{1}{ {x}^{2} } = \frac{4}{ ({12 - x})^{2} }} }}}$

${ \dashrightarrow {\bf{ {(12 - x)}^{2} = 4 {x}^{2} }}}$

${ \dashrightarrow {\bf{ 144 - 24x + {x}^{2} = 4 {x}^{2} }}}</p><p>$

i.e.

${ \dashrightarrow {\bf{ {x}^{2} - 24x + 144 = 4 {x}^{2} }}}$

${ \dashrightarrow {\bf{ {x}^{2} - 4 {x}^{2} - 24x + 144 = 0}}}$

${ \dashrightarrow {\bf{ - 3 {x}^{2} - 24x + 144 = 0}}}$

i.e.

${ \dashrightarrow {\bf{ 3 {x}^{2} + 24x - 144 = 0}}}$

compair above equation with

ax² + bx + c = 0

Here, a = 3, b = 24 and c = - 144

Now,

${ \bf{ \dashrightarrow{x \: = \frac{ - b \: ± \: \sqrt{ {b}^{2} - 4ac} }{2a} }}}$

${ \bf{ \dashrightarrow{x \: = \frac{ - 24 \: ± \: \sqrt{ {24}^{2} - 4 \times 3 \times ( - 144)} }{2 \times3 } }}}$

${ \bf{ \dashrightarrow{x \: = \frac{ - 24 \: ± \: \sqrt{ {576} + 1728} }{6 } }}}$

${ \bf{ \dashrightarrow{x \: = \frac{ - 24 \: ± \: \sqrt{ 2304} }{6 } }}}$

${ \bf{ \dashrightarrow{x \: = \frac{ - 24 \: ± \: 48 }{6 } }}}$

Now,

${ \bf{ \dashrightarrow{x \: = \frac{ - 24 \: + \: 48 }{6 } }} \: \: \: \: or \: \: \: \: x \: = \frac{ - 24 \: - \: 48 }{6 }}$

${ \bf{ \dashrightarrow{x \: = \frac{ 24}{6 } }} \: \: \: \: or \: \: \: \: x \: = - \frac{ 72 }{6 }}$

${ \bf{ \dashrightarrow{x \: = 4 }} \: \: \: \: or \: \: \: \: x \: = - 12}$

We, know distance is not in negative hence, x = 4 cm.

Hence, the distance from 1kg at which the gravitational force on any mass becomes zero is 4 cm.