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System of Particles and Rotational Motion

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A thin rod of length L is lying along the x-axis with its ends

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A thin rod of length $L$ is lying along the $x$ -axis with its ends at $x = 0$ and $x = L$ . Its linear density (mass/length) varies with $x$ as $k (\frac{x}{L})^{n}$ , where $n$ can be zero or any positive number. If the position $x_{CM}$ of the centre of mass of the rod is plotted against $n$ , which of the following graphs best approximates the dependence of $x_{CM}$ on $n$ ?

Name: Video solution 1: A thin rod of length L is lying along the x-axis with its ends at x=0 and x=L. Its linear density (mass/length) varies with x as k(Lx)n, where n can be zero or any positive number. If the position xCM of the centre of mass of the rod is plotted against n, which of the following graphs best approximates the dependence of xCM on n ?
Uploaded: 2022-03-26T08:12:59.000Z
Duration: 10 min
Description: Video solution 1: A thin rod of length L is lying along the x-axis with its ends at x=0 and x=L. Its linear density (mass/length) varies with x as k(Lx)n, where n can be zero or any positive number. If the position xCM of the centre of mass of the rod is plotted against n, which of the following graphs best approximates the dependence of xCM on n ?

Views: 5,603 students

Updated on: Mar 26, 2022

Book:

Arihant Physics JEE Main Chapterwise Solutions (2019-2002) (Arihant)

Exam:

JEE Mains 2008

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Harper Discussed

A thin rod of length

L

is lying along the

x

-axis with its ends at

x = 0

and

x = L

. Its linear density (mass/length) varies with

x

k (\frac{x}{L})^{n}

, where

n

can be zero or any positive number. If the position

x_{CM}

of the centre of mass of the rod is plotted against

n

, which of the following graphs best approximates the dependence of

x_{CM}

n

10 mins ago

Discuss this question LIVE

10 mins ago

Text solutionVerified

Exp. (a)

x_{CM} = \frac{\int x d m}{\int d m} x = 0

If n=0,then

x_{CM} = \frac{L}{2}

n

increases, the centre of mass shift away from

x = \frac{L}{2}

towards

x = L

which only option (a) is satisfying.
Alternately, you can use basic concept.

x_{CM} = \frac{\int _{0}^{L} k ( \frac{x}{L} ) ^{n} \times x d x}{\int _{0}^{L} k ( \frac{x}{L} ) ^{n} d x} = L [\frac{n + 1}{n + 2}]

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Question Text	A thin rod of length $L$ is lying along the $x$ -axis with its ends at $x = 0$ and $x = L$ . Its linear density (mass/length) varies with $x$ as $k (\frac{x}{L})^{n}$ , where $n$ can be zero or any positive number. If the position $x_{CM}$ of the centre of mass of the rod is plotted against $n$ , which of the following graphs best approximates the dependence of $x_{CM}$ on $n$ ?
Updated On	Mar 26, 2022
Topic	System of Particles and Rotational Motion
Subject	Physics
Class	Class 11
Answer Type	Text solution:1 Video solution: 1
Upvotes	193
Avg. Video Duration	10 min

Text solutionVerified

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Text solutionVerified

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