Question

A satellite is launched into a circular orbit of radius \( R \) around the earth. A second satellite is launched into an orbit of radius \( 1.03R \). The time period of revolution of the second satellite is larger than the first one approximately by:

• A. 3 %
• B. 4.5 %
• C. 9 %
• D. 2.5 %

Hint:

The time period of a satellite in orbit is related to the radius of the orbit by \( T^2 \propto R^3 \). This allows us to calculate how changes in the orbital radius affect the time period.

Answer 1

The Correct Option is B

To resolve this problem, understanding the relationship between a satellite's orbital period and its orbital radius is essential. Kepler's Third Law of Planetary Motion posits that the square of a satellite's orbital period (\( T \)) is directly proportional to the cube of its semi-major axis (which equates to the radius \( R \) for a circular orbit):

\(T^2 \propto R^3\)

This relationship can be expressed for an orbiting satellite as:

\(T^2 = kR^3\)

where \( k \) represents a constant of proportionality, contingent upon the gravitational constant and the Earth's mass.

Let \( T_1 \) be the time period of the first satellite and \( R \) be its orbital radius. Therefore:

\(T_1^2 = kR^3\)

For a second satellite with an orbital radius of \( 1.03R \), its time period \( T_2 \) is determined by:

\(T_2^2 = k(1.03R)^3\)

The ratio of the squares of the two time periods is:

\(\left(\frac{T_2}{T_1}\right)^2 = \frac{k(1.03R)^3}{kR^3}\)

Simplification yields:

\(\left(\frac{T_2}{T_1}\right)^2 = (1.03)^3\)

The calculation of \((1.03)^3\) is:

\(1.03^3 = 1.03 \times 1.03 \times 1.03 \approx 1.092727\\)

Consequently:

\(\left(\frac{T_2}{T_1}\right)^2 = 1.092727\)

Taking the square root of both sides provides:

\(\frac{T_2}{T_1} = \sqrt{1.092727} \approx 1.045\)

This indicates that \( T_2 \) is approximately 1.045 times the period of the first satellite, signifying a 4.5% increase.

Therefore, the time period of revolution for the second satellite is approximately 4.5% greater than that of the first. The correct answer is:

4.5%