A rectangular loop has mass M and sides a and b. An axis O O ' passes through the centre C of the loop and is parallel to side a (lie in the plane of the loop). Then the radius of gyration of the loop, for the axis O O prime is

Question

A rectangular loop has mass M and sides a and b. An axis O O ' passes through the centre C of the loop and is parallel to side a (lie in the plane of the loop). Then the radius of gyration of the loop, for the axis O O prime is (A) (b/2) √(b+3 a/3(b+a)) (B) √((a2+b2)/12) (C) √(b2+3 a2/12) (D) none of these

Tardigrade Admin · Accepted Answer

Let λ is linear mass density. Mass of side a: m1=λ a, Mass of side b: m2=λ b IOO' =2[m1((b/2))2+(m2 b2/12)] Now KO O prime=√(Io o prime/2(m1+m2))

Q. A rectangular loop has mass $M$ and sides $a$ and $b$ . An axis $OO$ ' passes through the centre $C$ of the loop and is parallel to side $a$ (lie in the plane of the loop). Then the radius of gyration of the loop, for the axis $O O^{'}$ is

$\frac{b}{2} \frac{b + 3 a}{3 ( b + a )}$

$\frac{( a ^{2} + b ^{2} )}{12}$

$\frac{b ^{2} + 3 a ^{2}}{12}$

none of these

Let $λ$ is linear mass density.
Mass of side $a : m_{1} = λa$ ,
Mass of side $b : m_{2} = λb$
$I_{O O^{'}} = 2 [m_{1} (\frac{b}{2})^{2} + \frac{m _{2} b ^{2}}{12}]$
Now $K_{O O^{'}} = \frac{I _{o o^{'}}}{2 ( m _{1} + m _{2} )}$

Q. A rectangular loop has mass M and sides a and b. An axis OO ' passes through the centre C of the loop and is parallel to side a (lie in the plane of the loop). Then the radius of gyration of the loop, for the axis OO′ is

2b​3(b+a)b+3a​​

12(a2+b2)​​

12b2+3a2​​