Question

Two circular loops of radii R and nR are made of the same wire. If their M.I about their normal axis through centre are in the ratio 1:8, the value of n is:
A. 6
B. 1
C. 2
D. 4

Answer 1

Hint: Moment of inertia is defined as the resistance offered by the object to the angular acceleration. The angular acceleration of any object is the product of the mass of the object and the square of distance from the axis of its rotation.

Formula Used:
Mathematically, the formula for moment of inertia is given by,
\[I = m{r^2}\]
Where ‘m’ is the sum of the product of mass and ‘r’ is the distance from the axis of rotation.

Complete step by step solution:
Given that there are two circular loops that are made of the same wire. Suppose,
Mass of one loop = M
Mass of ‘n’ loop = nM
Given that the radius of one loop = R
Radius of ‘n’ loop = nR
The moment of inertia about the normal axis through the centre is given in the ratio of 1:8. Therefore it can be written that,
\[\dfrac{{{I_n}}}{{{I_{nm}}}} = \dfrac{1}{8}\]

Using the formula of moment of inertia, the above equation can be written as
\[\dfrac{{M{R^2}}}{{nM \times {{(nR)}^2}}} = \dfrac{1}{8}\]
Solving the above equation, we get
\[\dfrac{1}{{{n^3}}} = \dfrac{1}{8}\]
Or it can be written that,
\[{n^3} = 8\]
\[\Rightarrow {n^3} = {(2)^3}\]
$\therefore n=2$
Therefore, for two circular loops of the same wire the value of ‘n’ is 2.

Hence, Option C is the correct answer.

Note: It is important to remember that moment of inertia is also known as rotational inertia. This is because it measures or signifies the amount of torque that is required to have angular acceleration across an axis for some object. It depends on the axis of rotation that is chosen for the object and also on how the mass is distributed while the object is in rotation.