Problem 782 - JEE Physics

Question

Potential energy of a satellite having mass \(\mathrm{m}\) and rotating at a height of \(6.4 \times 10^{6} \mathrm{~m}\) from the surface of earth (A) \(-0.5 \mathrm{mg} \operatorname{Re}\) (B) \(-\mathrm{mg} \mathrm{Re}\) (C) \(-2 \mathrm{mgRe}\) (D) \(4 \mathrm{mgRe}\)

Step-by-Step Solution

Answer

First, calculate the satellite's distance from the center of the earth: \[r = \text{height} + \text{Re} = (6.4 \times 10^{6} ) + (6.371 \times 10^{6})\] Next, calculate the potential energy using the formula: \[U = -\frac{GMm}{r}\] Finally, compare the calculated potential energy with the given options to find the correct answer.

Step 1: Calculate the satellite's distance from the center of the earth

Since the satellite is at a height of \(6.4 \times 10^{6}\) m from the surface and Earth's radius is approximately \(6.371 \times 10^{6}\) m, the distance from the satellite to the center of the earth is: \[r = \text{height} + \text{Re} = (6.4 \times 10^{6} ) + (6.371 \times 10^{6})\]

Step 2: Calculate the potential energy

Using the given mass of the satellite, and the calculated value of `r`, use the potential energy formula: \[U = -\frac{GMm}{r}\] Plug in the values for the gravitational constant \(G\), mass of the earth \(M\), and mass of the satellite \(m\).

Step 3: Compare the result with the given options

Once the potential energy is calculated, compare the result with the four given options (A, B, C, and D), to find which option matches the calculated value of the potential energy.

Key Concepts

Satellite Motion

Satellite motion involves the movement and path a satellite takes as it orbits a celestial body such as Earth. Satellites are objects that depend on gravitational forces and tangential velocity to maintain their motion around Earth. They follow precise paths known as orbits, which can be circular or elliptical.

• Types of Orbits: Satellites in geostationary orbit remain above the same point on Earth's surface. Polar orbits pass over the poles, covering the entire Earth as it rotates.
• Forces at Play: A satellite in orbit is subject to Earth's gravitational pull, which acts as a centripetal force holding the satellite in its path.

The balance between centrifugal force due to the satellite's speed and the gravitational pull keeps it from falling to Earth or flying away into space. Understanding this harmony is key to keeping a satellite in stable motion. Moreover, the energy dynamics, including gravitational potential energy, play an essential role in determining satellite motion.

Gravitational Constant

The gravitational constant, denoted by \(G\), is a fundamental constant in physics. It plays a crucial role in the laws of gravitation and helps us quantify gravitational forces between masses.

• Value of \(G\): The gravitational constant is approximately \(6.674 imes 10^{-11} \, \text{Nm}^2\,\text{kg}^{-2}\).


• Role in Universal Gravitation: It measures the strength of gravitational attraction between two objects of mass. The formula used is \(F = \frac{GMm}{r^2}\), where \(F\) is the gravitational force, \(M\) and \(m\) are the masses involved, and \(r\) is the distance between their centers.

The gravitational constant is vital for calculating gravitational forces in satellite motion. It helps determine the potential energy surrounding any satellite in Earth's orbit.
Understanding and using \(G\) accurately ensures the ability to estimate gravitational influence, which affects how and where satellites need to be positioned.

Distance from Earth's Center

Distance from Earth's center refers to how far an object, like a satellite, is from the exact middle point of our planet. This distance significantly impacts the gravitational forces and potential energy experienced by the satellite.

• Calculating Distance: When a satellite is a known height above Earth's surface, the distance from Earth's center is the sum of Earth's radius and the satellite's altitude. This is calculated by \(r = \text{height} + \text{Earth's radius}\).


• Importance in Calculations: Distance from the Earth's center (\(r\)) is fundamental for determining gravitational potential energy with the formula \(U = -\frac{GMm}{r}\).

Gravity decreases with an increase in \(r\), since distant objects are less influenced by Earth's gravity. Having an accurate measurement ensures precise calculations for orbit stability and energy requirements.