One project after deviation from its path, starts moving...

One project after deviation from its path, starts moving round the earth in a circular path at radius equal to nine times the radius at earth $R$, its time period will be (a) $2 \pi \sqrt{\frac{R}{g}}$ (b) $27 \times 2 \pi \sqrt{\frac{R}{g}}$ (c) $\pi \sqrt{\frac{R}{g}}$ (d) $8 \times 2 \pi \sqrt{\frac{R}{g}}$

One project after deviation from its path, starts moving round the earth in a circular path at radius equal to nine times the radius at earth $R$, its time period will be
(a) $2 \pi \sqrt{\frac{R}{g}}$
(b) $27 \times 2 \pi \sqrt{\frac{R}{g}}$
(c) $\pi \sqrt{\frac{R}{g}}$
(d) $8 \times 2 \pi \sqrt{\frac{R}{g}}$

Key Concepts

Scaling Laws in Orbital Mechanics

The orbital period scales with the orbital radius to the power of 3/2, meaning that if the radius increases, the period increases by a factor of the radius raised to the 3/2 power. This scaling law is crucial for understanding how changes in the orbital radius affect the time period of the motion.

Newton's Law of Gravitation

Newton's Law of Gravitation explains that every mass attracts every other mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This law is fundamental in analyzing orbital motions where gravitational force provides the necessary centripetal force for circular motion.

Centripetal Force and Circular Motion

Centripetal force is the inward force required to keep an object moving in a circular path. In orbital mechanics, the gravitational force acts as the centripetal force that keeps a satellite in orbit, which leads to the derivation of the formula for the orbital period of a satellite moving in a circular orbit.

Orbital Period Calculation

The orbital period of an object in circular orbit is given by T = 2??(r³/GM), where r is the orbital radius, G is the gravitational constant, and M is the mass of the central body. This formula is derived from equating the gravitational force with the necessary centripetal force for circular motion.

Relationship between g and the Gravitational Parameter

At the surface of a planet like Earth, the acceleration due to gravity (g) is related to the gravitational constant and the mass of the Earth by the relation g = GM/R², where R is the Earth's radius. This relationship allows us to substitute GM in the orbital period formula to express the period in terms of measurable quantities like g and R.

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