Problem 797 - JEE Physics

Question

The orbital speed of jupiter is (A) greater than the orbital speed of earth (B) less than the orbital speed of earth (C) equal to the orbital speed of earth (D) zero

Step-by-Step Solution

Answer

The orbital speed of Jupiter (\(13,100 \, m/s\)) is less than the orbital speed of Earth (\(29,500 \, m/s\)). Therefore, the correct answer is (B) less than the orbital speed of Earth.

Step 1: Understand orbital speed

Orbital speed is the speed at which an object, in this case, a planet, moves in its orbit around another object, usually the Sun. The orbital speed depends on the mass of the object being orbited and the distance between the two objects.

Step 2: Calculate Earth's orbital speed

The Earth's orbital speed can be found using the formula for the orbital speed of an object around another object, given by: \[v_{Earth} = \sqrt{\frac{G * M_{Sun}}{r_{Earth}}}\] Where: - \(v_{Earth}\) is the orbital speed of Earth - \(G\) is the gravitational constant (\(6.674 \times 10^{-11} \, N \cdot m^2/kg^2\)) - \(M_{Sun}\) is the mass of the Sun (\(1.989 \times 10^{30} \, kg\)) - \(r_{Earth}\) is the average distance from Earth to the Sun (about \(1.496 \times 10^{11}\, m\)) Plugging in the values and calculating Earth's orbital speed: \[v_{Earth} = \sqrt{\frac{6.674 \times 10^{-11} * 1.989 \times 10^{30}}{1.496 \times 10^{11}}} \approx 29,500 \, m/s\]

Step 3: Calculate Jupiter's orbital speed

Similarly, we can find Jupiter's orbital speed using the same formula: \[v_{Jupiter} = \sqrt{\frac{G * M_{Sun}}{r_{Jupiter}}}\] Where: - \(v_{Jupiter}\) is the orbital speed of Jupiter - \(r_{Jupiter}\) is the average distance from Jupiter to the Sun (about \(7.785 \times 10^{11}\, m\)) Plugging in the values and calculating Jupiter's orbital speed: \[v_{Jupiter} = \sqrt{\frac{6.674 \times 10^{-11} * 1.989 \times 10^{30}}{7.785 \times 10^{11}}} \approx 13,100 \, m/s\]

Step 4: Compare the orbital speeds of Earth and Jupiter

Now that we have calculated the orbital speeds of both Earth and Jupiter, we can compare them: - Earth's orbital speed: \(29,500 \, m/s\) - Jupiter's orbital speed: \(13,100 \, m/s\) As we can see, Jupiter's orbital speed is lower than Earth's. So, our answer is: (B) less than the orbital speed of earth

Key Concepts

Gravitational Constant

The gravitational constant, often symbolized as \( G \), is a fundamental constant in physics that quantifies the strength of the gravitational force between two bodies. It is a key component in Newton's law of universal gravitation. Mathematically, this law is represented as:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]where \( F \) is the gravitational force between two masses \( m_1 \) and \( m_2 \), and \( r \) is the distance between the centers of these two masses.
The value of \( G \) is approximately \( 6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \). This very small number reflects the relative weakness of the gravitational force compared to other fundamental forces, like electromagnetism.
In the context of planetary motion, \( G \) is essential to calculate the gravitational pull of the Sun on a planet, which directly influences the planet's orbital speed.

Mass of the Sun

The Sun's mass plays a decisive role in determining the dynamics of the solar system. The mass of the Sun, denoted as \( M_{Sun} \), is about \( 1.989 \times 10^{30} \text{ kg} \).
Given that the Sun accounts for over 99.8% of the total mass of the solar system, its gravitational influence shapes the orbits of the planets. This immense mass provides the gravitational force necessary to keep the planets, including Earth and Jupiter, in orbit.

• It affects the orbital speed of planets.
• It ensures the planets remain in a stable orbit.

Thus, understanding the mass of the Sun is crucial for comprehending the dynamics of planetary motion and calculating orbital speeds.

Average Distance from the Sun

The average distance of a planet from the Sun is a fundamental factor in determining its orbital speed. This distance is represented as \( r \) in the formula for calculating orbital speed:
\[ v = \sqrt{\frac{G \cdot M_{Sun}}{r}} \]The greater the average distance, the slower the orbital speed. This is why Jupiter, which is much farther from the Sun than Earth, has a lower orbital speed.
Here are some key points:

• Earth's average distance from the Sun is about \( 1.496 \times 10^{11} \text{ m} \).
• Jupiter's average distance is significantly greater, at approximately \( 7.785 \times 10^{11} \text{ m} \).

Understanding these distances helps us comprehend why different planets move at different speeds around the Sun.

Planetary Motion

Planetary motion refers to the movement of planets around the Sun or another star. This motion is governed by Kepler's laws of planetary motion, which describe the elliptical orbits of planets.
**Key aspects of planetary motion include:**

• Elliptical Orbits: Planets move in elliptical orbits, with the Sun at one of the focal points.
• Varying Speeds: A planet moves faster when it is closer to the Sun due to the stronger gravitational pull (Kepler's second law).
• Orbital Speed Relation: As you move further from the Sun, the orbital speed decreases because the gravitational pull weakens with distance.

These principles allow us to understand why planets like Jupiter, which are farther from the Sun than Earth, have slower orbital speeds, emphasizing the interconnectedness of gravity and motion in our solar system.