IMAT 2011 Q75 [Law of Gravitation]

The law of gravitation states that the gravitational force between two bodies of mass m1 and m2 is given by:

{{F=}\frac{Gm_1m_2}{r^2}}

G (gravitational constant) = {7 \times10^{-11} Nm^2kg^{-2}}
r (distance between the two bodies) in the case of the Earth and Moon = { 4 \times 10^8 m}
{m_1} (Earth) = { 6 \times 10^{24} kg}
{m_2} (Moon) = {7 \times 10^{22} kg}

What is the gravitational force between the earth and the moon?

A. 1.8365 x 1019N

B. 1.8365 x 1020N

C. 1.8365 x 1025N

D. 1.8365 x 1026N

E. 1.8365 x 1028N

Start by displacing the numbers into the given formula:

{{F=}\frac{Gm_1m_2}{r^2}}= {\frac {(7 \times10^{-11)} \times (6 \times 10^{24}) \times (7 \times 10^{22)}}{ (4 \times 10^8 )^2}}

Numerator calculations:

Multiply all the non-exponent integers in the numerator: {7\times 6 \times 7}= 294.

Using exponent rules, we know that numbers with the same bases multiplied means the exponents must be added together:

{x^ax^bx^c= x^{a+b+c}}

So, {10^{-11}10^{24}10^{22}}= {10^{-11+24+22}}= {10^{35}}

Which sets our numerator value at {294\times 10^{35}}

However, we know that the rules of standard form dictate:

{A \times 10^n}
where { 1\le A < 10} and n is an integer.

Therefore, we must adjust our value to become: {2.94 \times 10^{37}}

Denominator Calculations:

Once again we use exponent rules, particularly “Power to a Power” to say:

{(a^m)^n} = {a^{m\times n}}

Hence, calculating-

{(4^1)^2}= {4^{1\times 2}} = {4^2}= 16

{(10^8)^2} = {10^{8\times 2}}= {10^{16}}

Giving the answer {16\times 10^{16}}

And remembering to adjust this into appropriate Standard Form measures, it becomes {1.6\times 10^{17}}

Putting the Numerator and Denominator of our fraction together it becomes:

{F= \frac {2.94 \times 10^{37}}{1.6 \times 10^17}}

If you focus on the question, you’ll see they’ve given you the help of calculating the first part of our fraction, and we now know that {\frac{2.94}{1.6}}= 1.8365

Our final calculation then is simple, with the knowledge of the Quotient Rule:

{\frac{a^m}{a^n}}= {a^{m-n}}

Making {\frac{10^{37}}{10^{17}}}= {10^{37-17}}= {10^{20}}

Our final answer is: {1.8365\times 10^{20}} , which is B.