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Chapter 6: Problem 837

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.

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.
04

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) \(\sqrt{\frac{GM}{r}}\)

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