If the value of the Universal Gravitational Constant (G), the mass of the Earth (M), and the radius of the Earth (R) are known, which equation is used to calculate the acceleration due to gravity (g) near the Earth's surface?
- AOption (a)
\( g = GR/M^2 \)
- BOption (b)
\( R^2/(GM) \)
- COption (c)
\( g = GM/R \)
- DOption (d)
\( g = GM/R^2 \)
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Correct answer
\( g = GM/R^2 \)
Why this answer is correct
The correct answer is g = GM/R^2.
Key Points
- The relationship between acceleration due to gravity (g) and the Universal Gravitational Constant (G) is derived from Newton’s Law of Universal Gravitation and Newton’s Second Law of Motion.
- According to the Law of Universal Gravitation, the force of attraction (F) between the Earth (mass M) and an object (mass m) on its surface is given by F = G(Mm/R^2), where R is the radius of the Earth.
- According to Newton’s Second Law, the force acting on the object is also its weight, expressed as F = mg, where g is the acceleration due to gravity.
- By equating the two formulas for force (mg = GMm/R^2), the mass of the object (m) cancels out from both sides, leaving the equation g = GM/R^2. Hence, option 4 is correct.
- This formula shows that the value of g depends only on the mass of the Earth (M) and the radius of the Earth (R), and is independent of the mass of the falling object.
- The options g = GR/M^2, R^2/(GM), and g = GM/R are mathematically and dimensionally inconsistent with the physical laws of gravitation. Hence, options 1, 2, and 3 are incorrect.
Additional Information
- Universal Gravitational Constant (G): It is a scalar quantity with a constant value of approximately 6.674 x 10^-11 N m^2/kg^2. It was first experimentally determined by Henry Cavendish in 1798.
- Mass of the Earth (M): The Earth's mass is approximately 5.972 x 10^24 kg. This value, combined with G and R, results in the standard gravity we experience.
- Radius of the Earth (R): The mean radius of the Earth is about 6,371 kilometers (or 6.4 x 10^6 meters). Because the Earth is an oblate spheroid, the radius is larger at the Equator and smaller at the Poles.
- Value of g: On the surface of the Earth, the standard value of g is taken as 9.8 m/s^2. However, since g is inversely proportional to R^2, the value of g is maximum at the Poles (smallest radius) and minimum at the Equator (largest radius).
- Variation with Altitude and Depth: The value of g decreases as we go higher above the Earth's surface (altitude increases R) and also decreases as we go down into the Earth (towards the center), becoming zero at the center of the Earth.
Important Points
- Difference between G and g: G is a universal constant that remains the same everywhere in the universe, while g is a vector quantity that changes depending on the location (planet, altitude, or latitude).
- Units and Dimensions: The SI unit of g is m/s^2 (same as acceleration), and its dimensional formula is [M^0 L^1 T^-2].
- Weight vs. Mass: While mass is the amount of matter in an object and remains constant, weight (W = mg) is a force that changes if the value of g changes (e.g., you would weigh less on the Moon where g is about 1.62 m/s^2).
- Applications: This equation is fundamental in astrophysics and space science to calculate the escape velocity of a planet and to determine the orbital velocity of satellites.