Step-by-Step Solution
Step 1
Given that the mass of first particle is $8\;{\rm{kg}}$ and the mass of second particle is $12\;{\rm{kg}}$. Also the spacing between the particles is $800\;{\rm{mm}}$
We are required to determine the gravitational force between particles and we need to compare the force with weights.
Step 2
The length $1\;{\rm{mm}}$ is equal to the $1 \times {10^{ – 3}}\;{\rm{m}}$.
Step 3
The spacing between the particles in SI units can be calculated as,\[\begin{array}{c} r = 800\;{\rm{mm}} \times \left( {\frac{{{{10}^{ – 3}}\;{\rm{m}}}}{{1\;{\rm{mm}}}}} \right)\\ = 0.8\;{\rm{m}} \end{array}\]
Step 4
The equation to calculate the force of gravity between the particles is given by,\[F = G\frac{{{m_1}{m_2}}}{{{r^2}}}\]
Here, $G$ is the universal gravitational constant, having a standard value of $6.67 \times {10^{ – 11}}\;{{{{\rm{m}}^3}} \mathord{\left/ {\vphantom {{{{\rm{m}}^3}} {{\rm{kg}} \cdot {{\rm{s}}^2}}}} \right. } {{\rm{kg}} \cdot {{\rm{s}}^2}}}$, and ${m_1}$ and ${m_2}$ are the masses of first and second particles respectively.
Step 5
Substitute all the values in the above equation.\[\begin{array}{c} F = 6.67 \times {10^{ – 11}}\;\left( {\frac{{{{\rm{m}}^3}}}{{{\rm{kg}} \cdot {{\rm{s}}^2}}}} \right) \times \frac{{8\;{\rm{kg}} \times {\rm{12}}\;{\rm{kg}}}}{{{{\left( {0.8\;{\rm{m}}} \right)}^2}}}\\ = 1.00 \times {10^{ – 8}}\;\left( {{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/ {\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}} \right)\\ = 1.00 \times {10^{ – 8}}\;\left( {{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/ {\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}} \right) \times \left( {\frac{{1\;{\rm{N}}}}{{1{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/ {\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}}}} \right)\\ = 1.00 \times {10^{ – 8}}\;{\rm{N}} \end{array}\]
Step 6
The equation for the weight of the first particle is given by,\[{w_1} = {m_{\rm{1}}} \times g\]
Here, $g$ is the acceleration due to gravity, having a standard value of $9.81\;{{\rm{m}} \mathord{\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}$.
Step 7
Substitute all the values in the above equation.\[\begin{array}{c} {w_1} = 8\;{\rm{kg}} \times 9.81\;{{\rm{m}} \mathord{\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}\\ = 78.4\;\left( {\frac{{{\rm{kg}} \cdot {\rm{m}}}}{{{{\rm{s}}^2}}}} \right)\\ = 78.4\;\left( {\frac{{{\rm{kg}} \cdot {\rm{m}}}}{{{{\rm{s}}^2}}}} \right) \times \left( {\frac{{1\;{\rm{N}}}}{{1\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/ {\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}}}} \right)\\ = 78.4\;{\rm{N}} \end{array}\]
Step 8
The ratio of weight of first particle to gravitational force is given by,\[{R_1} = \frac{{{w_1}}}{F}\]
Step 9
Substitute all the values in the above equation.\[\begin{array}{c} {R_1} = \frac{{78.4\;{\rm{N}}}}{{1.00 \times {{10}^{ – 8}}\;{\rm{N}}}}\\ = 7.84 \times {10^9} \end{array}\]
Thus, the weight of the first particle is $7.84 \times {10^9}$ greater than the gravitational force.
Step 10
The equation for the weight of the second particle is given by,\[{w_2} = {m_2} \times g\]
Here, $g$ is the acceleration due to gravity, having a standard value of $9.81\;{{\rm{m}} \mathord{\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}$.
Step 11
Substitute all the values in the above equation.\[\begin{array}{c} {w_2} = 12\;{\rm{kg}} \times 9.81\;{{\rm{m}} \mathord{\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}\\ = 117.7\;\left( {\frac{{{\rm{kg}} \cdot {\rm{m}}}}{{{{\rm{s}}^2}}}} \right)\\ = 117.7\;\left( {{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/ {\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}} \right) \times \left( {\frac{{1\;{\rm{N}}}}{{1\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/ {\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {{{\rm{s}}^2}}}} \right. } {{{\rm{s}}^2}}}}}} \right)\\ = 117.7\;{\rm{N}} \end{array}\]
Step 12
The ratio of weight of second particle to gravitational force is given by,\[{R_2} = \frac{{{w_2}}}{F}\]
Step 13
Substitute all the values in the above equation.\[\begin{array}{c} {R_2} = \frac{{117.7\;{\rm{N}}}}{{1.00 \times {{10}^{ – 8}}\;{\rm{N}}}}\\ = 1.17 \times {10^{10}} \end{array}\]
Thus, the weight of the first particle is $1.17 \times {10^{10}}$ greater than the gravitational force.