Question

If the mass of a planet is 6 x10²⁶ kg and its radius is 6.4 x 10³ km, what is the estimated acceleration due to gravity on the surface of the planet? *​

Answer 1

→ To Find :

The Acceleration due to gravity on the Planet's surface.

→ We Know :

Acceleration due to gravity :

$\purple{\sf{\underline{\boxed{g = \dfrac{G\:M}{r^{2}}}}}}$

Where ,

• g = Acceleration due to gravity

• r = Radius of the Planet

• M = Mass of the planet

• G = Universal Gravitational Constant

→ Solution :

Value of Universal Gravitational Constant :

$\green{\sf{G = 6.67 \times 10^{-11}}}$

Given :

• Mass of the Planet = 6 × 10²⁶ kg

• Radius of the Planet = 6.4 × 10³ km

Converting 6.4 × 10³ km in m ,we get :

$\sf{\Rightarrow 6.4 \times 10^{3} \times 1000 m}$

$\sf{\Rightarrow 6.4 \times 10^{3} \times 10^{3})m}$

[Using the exponential law :

$\sf{x^{n} \times x^{m} = x^{n + m}}$

$\sf{\Rightarrow 6.4 \times 10^{3 + 3} m}$

$\purple{\sf{\Rightarrow 6.4 \times 10^{6} m}}$

Hence , the Radius of the Planet is 6.4 × 10⁶ m.

Using the formula and substituting the values in it , we get :

$\purple{\sf{g = \dfrac{G\:M}{r^{2}}}}$

$\sf{\Rightarrow g = \dfrac{6.67 \times 10^{-11} \times 6 \times 10^{26}}{\left(6.4 \times 10^{6}\right)}}$

$\sf{\Rightarrow g = \dfrac{6.67 \times 10^{-11} \times 6 \times 10^{26}}{6.4 \times 6.4 \times 10^{6} \times 10^{6}}}$

$\sf{\Rightarrow g = \dfrac{6.67 \times 10^{-11} \times 6 \times 10^{26}}{40.96 \times 10^{12}}}$

$\sf{\Rightarrow g = \dfrac{6.67 \times 10^{-11} \times 6 \times 10^{26} \times 10^{-12}}{40.96}}$

$\sf{\Rightarrow g = \dfrac{6.67 \times 10^{-23} \times 6 \times 10^{26}}{40.96}}$

$\sf{\Rightarrow g = \dfrac{6.67 \times 6 \times 10^{2}}{40.96}}$

$\sf{\Rightarrow g = \dfrac{40.02 \times 10^{2}}{40.96}}$

$\sf{\Rightarrow g = \dfrac{4002}{40.96}}$

$\sf{\Rightarrow g = 97.7 ms^{-2}}$

Hence ,the value of acceleration due to gravity on that planet is 97.7 m/s².

» Additional information :

• Force = $\sf{F = G\dfrac{m_{1}m_{2}}{r^{2}}}$

• First Equation of Motion (under Gravity) = v = u + gt

• Second Equation of Motion (under Gravity) = h = ut + ½gt²

• Third Equation of Motion (under Gravity) = v² = u² + 2gh