   Question Answers

# The value of acceleration due to gravity at Earth’s surface is $9.8m{{s}^{-2}}$. The altitude above its surface at which the acceleration due to gravity decreases to $4.9m{{s}^{-2}}$, is close to:(Radius of earth = $6.4\times {{10}^{6}}m$)(a) $1.6\times {{10}^{6}}m$(b) $6.4\times {{10}^{6}}m$(c) $9.0\times {{10}^{6}}m$(d) $2.6\times {{10}^{6}}m$  Hint: The relation between value of acceleration, radius of earth and height can be given by the formula, ${{g}_{h}}=\dfrac{GM}{{{\left( R+h \right)}^{2}}}$, so using this relation we will find the value of h by equating the values of gravitational acceleration.

Formula used: ${{g}_{h}}=\dfrac{GM}{{{\left( R+h \right)}^{2}}}$

In question we are given that the value of acceleration due to gravity at Earth’s surface is $9.8m{{s}^{-2}}$. Now, we are asked to find the altitude above its surface at which the acceleration due to gravity decreases to $4.9m{{s}^{-2}}$. The relation between gravitational acceleration, height h and radius R can be given by the formula,
${{g}_{h}}=\dfrac{GM}{{{\left( R+h \right)}^{2}}}$ ……………….(ii)
Where, G is the gravitational constant of earth, m is mass of object, R is radius of earth and h is height of earth.
Now, the gravitational acceleration at surface of earth can be given as,
${{g}_{h}}=\dfrac{GM}{{{\left( R \right)}^{2}}}$ ……………….(ii)
As, height is zero on the surface of the earth.
Now, we are given that value of gravitational acceleration on surface of earth is $9.8m{{s}^{-2}}$ and it is $4.9m{{s}^{-2}}$ at height h, so on substituting these values in expression (ii) and (i) we will get,
$4.9=\dfrac{GM}{{{\left( R+h \right)}^{2}}}\Rightarrow GM=4.9{{\left( R+h \right)}^{2}}$
$9.8=\dfrac{GM}{{{\left( R \right)}^{2}}}\Rightarrow GM=9.8{{\left( R \right)}^{2}}$
On equating the equations, we will get,
$4.9{{\left( R+h \right)}^{2}}=9.8{{\left( R \right)}^{2}}$
$\dfrac{{{\left( R+h \right)}^{2}}}{{{\left( R \right)}^{2}}}=\dfrac{9.8}{4.9}\Rightarrow \dfrac{{{\left( R+h \right)}^{2}}}{{{\left( R \right)}^{2}}}=2$
On taking square roots on both the sides we will get,
$\dfrac{\left( R+h \right)}{\left( R \right)}=\sqrt{2}\Rightarrow R+h=\sqrt{2}R$
$h=\sqrt{2}R-R\Rightarrow h=\left( \sqrt{2}-1 \right)R$
$\Rightarrow h=\left( 1.41-1 \right)6.4\times {{10}^{6}}=\left( 0.41 \right)6.4\times {{10}^{6}}$
$\Rightarrow h=2.6\times {{10}^{6}}m$
Thus, the value of altitude will be $2.6\times {{10}^{6}}m$.
Hence, option (d) is the correct answer.

Note: This problem can also be solved by using the formula of gravitational acceleration at height h which can be given as, ${{g}_{h}}={{g}_{o}}\dfrac{R}{{{\left( R+h \right)}^{2}}}$, where ${{g}_{h}}={{g}_{o}}=9.8m{{s}^{-2}}$ for value for gravitation at surface of earth and ${{g}_{h}}=4.9m{{s}^{-2}}$, for acceleration at height h. And again, equating the expressions we will get the same answer. So, we can say that this is an alternative method to solve this type of problem.
View Notes
Electric Field Due to Point Charge  Derivation of Centripetal Acceleration  Acceleration Time Graph  CBSE Class 11 Physics Law of Motion Formulas  CBSE Class 11 Physics Thermal Properties of Matter Formulas  CBSE Class 11 Physics Mechanical Properties of Solids Formulas  CBSE Class 11 Physics Mechanical Properties of Fluids Formulas  Value of g  CBSE Class 11 Physics Systems of Particles and Rotational Motion Formulas  Gravity Formula  Important Questions for CBSE Class 11 Physics Chapter 11 - Thermal Properties of Matter  Important Questions for CBSE Class 11 English Snapshots Chapter 4 - Albert Einstein at School  Important Questions for CBSE Class 11 Physics Chapter 5 - Law of Motion  Important Questions for CBSE Class 11 Physics Chapter 9 - Mechanical Properties of Solids  Important Questions for CBSE Class 11 Physics Chapter 10 - Mechanical Properties of Fluids  Important Questions for CBSE Class 6 Social Science The Earth Our Habitat Chapter 3 - Motions of the Earth  Important Questions for CBSE Class 11 Physics  Important Questions for CBSE Class 11 Physics Chapter 7 - Systems of Particles and Rotational Motion  Important Questions for CBSE Class 12 Physics Chapter 11 - Dual Nature of Radiation and Matter  Important Questions for CBSE Class 11 Physics Chapter 4 - Motion in a Plane  CBSE Class 12 Physics Question Paper 2020  Previous Year Question Paper for CBSE Class 12 Physics  Previous Year Question Paper for CBSE Class 12 Physics - 2015  Previous Year Question Paper for CBSE Class 12 Physics - 2018  Previous Year Question Paper for CBSE Class 12 Physics - 2014  Previous Year Question Paper for CBSE Class 12 Physics - 2013  Previous Year Question Paper for CBSE Class 12 Physics - 2019  Previous Year Physics Question Paper for CBSE Class 12 - 2017  Physics Question Paper for CBSE Class 12 - 2016 Set 1 C  Previous Year Question Paper of CBSE Class 10 English  NCERT Solutions Class 11 English Woven Words Prose Chapter 7 Glory at Twilight  NCERT Solutions for Class 12 English Vistas Chapter 3 - Journey to the end of the Earth  NCERT Exemplar for Class 11 Physics Chapter 11 - Thermal Properties of Matter (Book Solutions)  NCERT Solutions for Class 11 Physics Chapter 11 Thermal Properties of Matter in Hindi  NCERT Solutions for Class 10 English First Flight Chapter 10 - The Sermon at Benares  NCERT Solutions for Class 12 English Kaliedoscope Poetry Chapter 6 The Wild Swans at Coole  RS Aggarwal Class 11 Solutions Chapter-16 Conditional Identities Involving the Angles of a Triangle  Class 11 Physics NCERT solutions  NCERT Solutions for Class 11 Physics Chapter 5 Law of Motion in Hindi  Textbooks Solutions for CBSE & ICSE Board of Class 6 to 12 Maths & Science  