Question

If the body has a weight of \[60\,N\] on the earth, how far must the body go away from the center of the earth so that body weight \[30\,N\].
A. \[25,500\,km\]
B. \[26,500\,km\]
C. \[24,500\,km\]
D. \[24,900\,km\]

Answer 1

Hint:Here, we have been asked to calculate the distance where the body weighs \[30N\] away from the center of the earth which is half the weight of the body on earth. We have to use the given data in some of the formulas which can help us to calculate as given \[g = \dfrac{{GMm}}{{{R^2}}}\].

Complete step by step answer:
 Let us first see the given data or the data which we have to use further.
Weight of the body on the earth, \[mg = 60\,N\]
Weight of the body away from the center of the earth, \[mg' = 30\,N\]
Radius of earth, \[R = 6400 \times {10^3}m\]
Now, we have to find the distance \[h\].
Now we have to use formula \[g = \dfrac{{GMm}}{{{R^2}}}\] because weight can be changed depending on acceleration due to gravity as it changes as the body moves away from the center of the earth.

Let \[g'\] be the acceleration due to gravity at the height \[h\]and is given by:
\[g' = \dfrac{{GMm}}{{{{(R + h)}^2}}}\]
So, mass of the body remains same so we have
\[\dfrac{{mg}}{{mg'}} = \dfrac{{60N}}{{30N}}\] …. \[(1)\]
From this we can find relation between \[g\] and \[g'\] as:
\[ \Rightarrow \dfrac{g}{{g'}} = \dfrac{{\dfrac{{{G}{M}{m}}}{{{R^2}}}}}{{\dfrac{{{G}{M}{m}}}{{{{(R + h)}^2}}}}} = \dfrac{{{{(R + h)}^2}}}{{{R^2}}}\]
\[\therefore \dfrac{{{{(R + h)}^2}}}{{{R^2}}} = 2\] …. From \[(1)\]
\[ \Rightarrow \dfrac{{R + h}}{R} = \sqrt 2 \] …. Taking the square root of the above equation.

Now putting the required values from the given data in the above equation.
\[ \Rightarrow \dfrac{{\left( {6400 \times {{10}^3}} \right) + h}}{{\left( {6400 \times {{10}^3}} \right)}} = \sqrt 2 \]
\[ \Rightarrow h = \sqrt 2 \left( {6400 \times {{10}^3}} \right) - \left( {6400 \times {{10}^3}} \right)\]
\[ \Rightarrow h = 6400 \times {10^3}\left( {\sqrt 2 - 1} \right)\]
\[ \Rightarrow h = 6400 \times {10^3}\left( {1.414 - 1} \right)\]
\[ \Rightarrow h = 6400 \times {10^3}\left( {0.414} \right)\]
\[ \therefore h = 26,500\,km\]
On calculating we have got that value of the height from the center of the earth where the body will weigh \[30\,N\] is \[26,500\,km\].

Thus, the correct answer is option B.

Note:Here, we must remember that the mass of any body is constant all over the universe but the weight changes differently just because of the gravity that the surface possesses. We have used only formulas to carry out relation between two different accelerations due to gravities. And reached our answer.