The question asks for the relationship between acceleration due to gravity (\(g\)), the universal gravitational constant (\(G\)), the mass of the planet (\(M\)), and its radius (\(R\)).
Gravity Formula Derivation
We can derive the formula for acceleration due to gravity (\(g\)) using Newton's Law of Universal Gravitation and Newton's Second Law of Motion.
- Newton's Law of Universal Gravitation states that the force (\(F\)) between two masses \(M\) and \(m\) separated by a distance \(R\) is:
$F = \frac{GMm}{R^2}
- Newton's Second Law of Motion states that the force acting on a mass \(m\) is:
$F = mg
- For an object on the surface of a planet (or other celestial body) of mass \(M\) and radius \(R\), the gravitational force experienced by the object is responsible for its acceleration \(g\). Therefore, we can equate the two expressions for force:
$mg = \frac{GMm}{R^2}
- By canceling the mass of the object (\(m\)) from both sides, we get the relationship for acceleration due to gravity:
$g = \frac{GM}{R^2}
Resulting Gravity Equation
The derived equation shows the direct relationship between acceleration due to gravity (\(g\)) and the universal gravitational constant (\(G\)), the mass (\(M\)) of the central body, and its radius (\(R\)). This matches Option 3.
Similar Questions
- If three particles, each of mass \(M\) are placed at the three corners of an equilateral triangle of side \(a\), the force exerted by this system on another particle of mass \(M\) placed at the midpoint of a side is
- Imagine that there exists a planet that has half the radius of the earth and half its mass. Then, the acceleration due to gravity on the surface of that planet in relation to the acceleration due to gravity g on the surface of the earth is
Important Questions from Universal law of gravitation
Which of the following is NOT correct ?
Three point masses each of mass m are placed at the three corners of an equilateral triangle of side x. Find the resultant force acting on any one particle at the corner.
Which of the following is NOT correct ?
- Which phenomenon, successfully explained by the universal law of gravitation, relies on the combined and differential gravitational attraction exerted by both the moon and the sun on the earth?
- The law of gravitation helped to prove that:
