Problem 837 - JEE Physics

Question

Match the following Table-1 \(\quad\) Table-2 (A) kinetic energy (P) \([(-\mathrm{GMm}) /(2 \mathrm{r})]\) (B) Potential energy (Q) \(\sqrt{(\mathrm{GM} / \mathrm{r})}\) (C) Total energy (R) - [(GMm) / r] (D) orbital velocity (S) \([(\mathrm{GMm}) /(2 \mathrm{r})]\) Copyright (O StemEZ.com. All rights reserved.

Step-by-Step Solution

Answer

The correct matching for the given properties and equations is: (A) Kinetic energy → (S) \(\frac{GMm}{2r}\) (B) Potential energy → (R) -\(\frac{GMm}{r}\) (C) Total energy → (P) -\(\frac{GMm}{2r}\) (D) Orbital velocity → (Q) \(\sqrt{\frac{GM}{r}}\)

Step 1: Identify the formula for kinetic energy

Kinetic energy (KE) is given by the formula: KE = \(\frac{1}{2}\)mv² In the context of orbital mechanics, the velocity "v" can be represented by the formula: v = \(\sqrt{\frac{GM}{r}}\) Thus, the kinetic energy (KE) can be written as: KE = \(\frac{1}{2}\)m\(\left(\sqrt{\frac{GM}{r}}\right)^2\) = \(\frac{GMm}{2r}\) This matches with option (S) in Table-2.

Step 2: Identify the formula for potential energy

Potential energy (PE) in the context of gravitation is given by the formula: PE = -\(\frac{GMm}{r}\) This matches with option (R) in Table-2.

Step 3: Identify the formula for total energy

Total energy (TE) of an object in orbital motion is the sum of its kinetic energy (KE) and potential energy (PE). We can write: TE = KE + PE Substituting the formulas for KE and PE from previous steps, we have: TE = \(\frac{GMm}{2r}\) - \(\frac{GMm}{r}\) = -\(\frac{GMm}{2r}\) This matches with option (P) in Table-2.

Step 4: Identify the formula for orbital velocity

Orbital velocity (v) of an object in orbital motion around a mass M is given by the formula: v = \(\sqrt{\frac{GM}{r}}\) This matches with option (Q) in Table-2. So, the correct matching is as follows: (A) kinetic energy → (S) \(\frac{GMm}{2r}\) (B) Potential energy → (R) -\(\frac{GMm}{r}\) (C) Total energy → (P) -\(\frac{GMm}{2r}\) (D) Orbital velocity → (Q) \(\sqrt{\frac{GM}{r}}\)

Key Concepts

Kinetic Energy

Kinetic energy is the energy of motion. When an object moves, it gains kinetic energy according to the formula \( KE = \frac{1}{2}mv^2 \). In orbital mechanics, this plays a crucial role.

For objects in orbit, like satellites, we use the orbital velocity to determine kinetic energy. The velocity \( v \) for an orbiting object is \( \sqrt{\frac{GM}{r}} \). Substituting this into the kinetic energy formula, we get:

• \( KE = \frac{1}{2}m\left(\sqrt{\frac{GM}{r}}\right)^2 \)
• Simplifying gives \( KE = \frac{GMm}{2r} \).

This formula reveals how the motion of an object in orbit relates to the gravitational force from the massive body it orbits.

Potential Energy

Potential energy in the context of gravity is about the stored energy due to an object's position. When dealing with gravity, the formula is \( PE = -\frac{GMm}{r} \). This negative sign indicates that energy is released as two bodies move closer.

In orbital mechanics, potential energy is important because it contributes to the total energy of an object in orbit. As the distance between the object and the massive body decreases, potential energy becomes more negative. This change reflects how gravity influences objects through their positions relative to each other.

Orbital Velocity

Orbital velocity is the speed an object needs to maintain its orbit around a larger body, like a planet. The formula \( v = \sqrt{\frac{GM}{r}} \) tells us this velocity depends on the gravitational constant \( G \), the mass of the central body \( M \), and the radius of the orbit \( r \).

Understanding orbital velocity helps explain:

• Why satellites stay in space without falling back to Earth.
• How the balance between gravity and forward motion works.

This balance is what keeps the object moving in a perfect circle or ellipse around its orbital path.

Gravitational Potential Energy

Gravitational potential energy refers to the potential energy stored in an object due to its position in a gravitational field. The formula \( PE = -\frac{GMm}{r} \) captures this concept for celestial objects.

Key things to understand about gravitational potential energy:

• It is crucial for calculating the total energy in an object’s orbit.
• It becomes increasingly negative as objects move closer, releasing energy.
• It helps balance kinetic energy to maintain stable orbits.

Thus, gravitational potential energy is a major player in explaining how and why objects follow certain paths in the universe.