Question

h ऊंचाई पर गुरुत्वीय त्वरण सतह के मान का 50% रह जाता है यदि पृथ्वी की त्रिज्या 6400 km हो तो h का मान ज्ञात कीजिए |

Answer 1

Answer:

F =mg= G(mM/r^2)

So

g=G(M/r^2)

So gravitational acceleration is inversely proportional to square of distance from Centre of earth.

So

$\frac{g}{ (\frac{g}{2}) } = \frac{( \frac{1}{6400 {}^{2} }) }{ \frac{1}{ {(6400 + h)}^{2} } } \\ \\ \implies 2 = \frac{(6400 + h) {}^{2} }{ {6400}^{2} } \\ h + 6400 = 6400 \sqrt{2} \\ h = 6400( \sqrt{2} - 1) \\ h \approx6400 \times 0.41 421\\ h = 2650 \: \: kms$

Answer 2

The value of h is 2650 km.

Explanation:

Given that,

Radius of earth = 6400 km

$g'=\dfrac{g}{2}$

We need to calculate the gravitational force at earth surface

Using formula of gravity

$F=\dfrac{GmM}{r^2}$

$mg=\dfrac{GmM}{r^2}$

$g=\dfrac{GM}{r^2}$....(I)

The gravitational force at some height where the gravity is half

$g'=\dfrac{GM}{(r+h)^2}$

$\dfrac{g}{2}=\dfrac{GM}{(r+h)^2}$.....(II)

Divided equation (I) by equation (II)

$\dfrac{g}{\dfrac{g}{2}}=\dfrac{(r+h)^2}{r^2}$

Put the value into the formula

$2=\dfrac{(6400+h)^2}{6400^2}$

$6400+h=6400\sqrt{2}$

$h=6400\sqrt{2}-6400$

$h=2650\ km$

Hence, The value of h is 2650 km.

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Topic : gravitational force

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