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( x/L ) where  n  can 0 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

Option 1)

Option 2)

Option 3)

Option 4)

Answers (1)

x_{C.M}= \frac{\int_{0}^{L}\left ( \frac{k}{L^{n}}\cdot x^{n}\cdot dx \right )x}{\int_{0}^{L}\frac{k}{L^{n}}\cdot x^{n}\cdot dx}=\frac{n+1}{n+2}L=L\left (1-\frac{1}{n+2} \right )

\frac{dx}{dn}= L\left \{ \frac{\left ( n+2 \right )1-\left ( n+1 \right )}{\left ( n+2 \right )^{2}} \right \}= \frac{L}{\left ( n+2 \right )^{2}}

If the rod has the same density as  at x = 0  ie ;  n = 0

therefore uniform, the centre of mass would have been at L/2 .  As the density increases with length, the centre of mass shifts towards the right 

Therefore it can only be  (2)


Option 1)

It is an incorrect option.

Option 2)

It is the correct option.

Option 3)

It is an incorrect option.

Option 4)

It is an incorrect option.

Preparation Products

Knockout JEE Main July 2020

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 12999/- ₹ 6999/-
Buy Now
Rank Booster JEE Main 2020

This course will help student to be better prepared and study in the right direction for JEE Main..

₹ 9999/- ₹ 4999/-
Buy Now
Test Series JEE Main July 2020

Take chapter-wise, subject-wise and Complete syllabus mock tests and get in depth analysis of your test..

₹ 4999/- ₹ 1999/-
Buy Now
Knockout JEE Main April 2021

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 17999/- ₹ 11999/-
Buy Now
Knockout JEE Main April 2022

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 34999/- ₹ 19999/-
Buy Now
Exams
Articles
Questions