Problem 773 - JEE Physics

Question

orbital velocity of earth's satellite near the surface is \(7 \mathrm{kms}^{-1}\). when the radius of orbit is 4 times that of earth's radius, then orbital velocity in that orbit is \(=\ldots \ldots \ldots \ldots \mathrm{kms}^{-1}\) (A) \(3.5\) (B) 17 (C) 14 (D) 35

Step-by-Step Solution

Answer

The orbital velocity of the satellite when the orbit radius is 4 times Earth's radius is (A) \(3.5\, km/s\).

Step 1: Recall the formula for orbital velocity

The formula for orbital velocity of a satellite is given by: \[v = \sqrt{\frac{GM}{r}}\] where \(v\) is the orbital velocity, \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(r\) is the radius of the orbit.

Step 2: Set up the given orbital velocity

The given orbital velocity near Earth's surface (\(r = R_{earth}\)) is \(7\, km/s\). We can plug this information into the formula from Step 1: \(7 = \sqrt{\frac{GM}{R_{earth}}}\)

Step 3: Find the orbital velocity at 4 times Earth's radius

Now, we need to find the orbital velocity when the orbit radius is 4 times Earth's radius. Let this orbital velocity be \(v'\), then, we have: \(v' = \sqrt{\frac{GM}{4 \times R_{earth}}}\)

Step 6: Find the ratio of the two orbital velocities

To find the relationship between the two orbital velocities, we can take the ratio: \(\frac{v'}{7} = \frac{\sqrt{\frac{GM}{4 \times R_{earth}}}}{\sqrt{\frac{GM}{R_{earth}}}}\)

Step 7: Simplify the ratio

Simplify the ratio: \(\frac{v'}{7} = \frac{\sqrt{\frac{1}{4}}}{\sqrt{1}}\)

Step 8: Solve for the new orbital velocity

Now we can solve for the new orbital velocity, \(v'\): \(v' = 7 \times \frac{1}{2}\) Therefore, the orbital velocity of the satellite when the orbit radius is 4 times Earth's radius is: \(v' = 3.5\, km/s\) So, the correct answer is: (A) \(3.5\, km/s\)

Key Concepts

Satellite Motion

Satellite motion is a fascinating aspect of space exploration and celestial mechanics that involves satellites orbiting planets, including Earth. A satellite is any object that moves around another object in a circular or elliptical path.

• Artificial satellites are human-made objects launched into orbit, such as communication satellites and research satellites.
• Natural satellites include entities like our Moon, which orbits around Earth naturally.

The motion of satellites is determined by gravitational forces that act between the satellite and the planet it orbits. The gravitational force serves as the centripetal force, keeping the satellite moving in its path.
To maintain this motion, satellites need a specific speed known as orbital velocity. If a satellite travels slower than the required orbital speed, it will fall back to Earth. Conversely, if it travels too fast, it might escape the Earth's gravitational pull. Understanding satellite motion is critical for various applications like GPS, weather prediction, scientific observation, and telecommunications.

Gravitational Constant

The gravitational constant, denoted by the symbol \(G\), is a key figure in understanding the force of gravity. It is a universal constant that appears in Newton's law of universal gravitation, which states:
\[ F = \frac{G \, m_1 \, m_2}{r^2} \]where:

• \(F\) is the gravitational force between two masses, \(m_1\) and \(m_2\).
• \(r\) is the distance between the centers of the two masses.

\(G\) has a value of approximately \(6.674 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\).
In the context of satellite motion, \(G\) helps calculate the orbital velocity \(v\) of a satellite, ensuring it stays in its intended orbit. Newton's law of universal gravitation explains the attraction between the satellite and the Earth due to their masses, with \(G\) indicating the strength of this gravitational pull.
This constant is crucial in physics because it enables us to understand how objects in space attract each other and how planets, stars, and galaxies form and interact.

Radius of Orbit

The radius of orbit plays a vital role in determining satellite dynamics. In simple terms, it is the radius of the circular path that the satellite follows around the Earth or any celestial body.

• For low Earth orbit (LEO) satellites, the radius of the orbit is relatively small, often just a few hundred kilometers above Earth's surface.
• This proximity allows satellites to move quickly, requiring higher orbital velocities compared to satellites which orbit at greater distances.

The relationship between the orbital velocity \(v\) and the radius of orbit \(r\) is inverse in nature. As the radius increases, the orbital velocity decreases. This relationship is described by the formula:
\[ v = \sqrt{\frac{GM}{r}} \]Thus, when the orbit radius becomes larger, such as being four times the Earth's radius, the velocity needs to be recalculated based on this new radius. It's important because any change in radius impacts the conditions under which the satellite can remain in stable orbit.
With satellites orbiting at different radii, it's possible to utilize a variety of applications, from low-orbit weather monitoring to geostationary satellites for communication that remain at a constant position relative to Earth.