Question

a frame is formed by the uniform rods having constant linear mass density .ACB part of frame is semicircular of radius 'R' and AB is straight rod. Centre of mass of the system from centre O of AB is at a distance

Answer 1

Answer:

frame is semicircular of mass of the system from centre 0 of ab is at a distance a frame is formed by the uniform rods having constant linear mass density.

Answer 2

Centre of mass of the system from centre O of AB is at a distance $\frac{2R}{\pi +2}$

Explanation:

1. On taking AB on x-axis and origin at centre O. Line OC lie on y-axis. So frame is symmetry with respect to y-axis. Centre of mass of frame is on line OC means on y-axis. Let this distance is $\bar{Y}$ on taking O as a reference point.

2. From equation of centre of mass of discrete system

     $\bar{Y}= \frac{L_{1}Y_{1}+L_{2}Y_{2}}{L_{1} +L_{2}}$   ...1)

    $L_{1}$ = length of line AB =2R

     $L_{2}$ = length of arc ACB = $\pi R$

     $Y_{1}$ = centre of mass of line AB = 0

     $Y_{2}$= centre of mass of arc ACB = $\frac{2R}{\pi }$

3.  Now from equation 1)

$\bar{Y} = \frac{2R\times 0+\pi R\times \frac{2R}{\pi }}{2R+\pi R}$

on solving above equation

$\bar{Y} = \frac{2R}{\pi +2}$ above the centre O in direction of OC.