Question

Two circular loops P and Q are made from a uniform wire. The radii of P and Q are $R_1$ and $R_2$ respectively. The moments of inertia about their own central axis are $I_P$ and $I_Q$ respectively. If $\frac{I_P}{I_Q} = \frac{1}{8}$ then the ratio $\frac{R_2}{R_1}$ is

• A. $4$
• B. $3$
• C. $2$
• D. $5$

Hint:

Be careful! For a solid disc or a pre-existing object where mass is held constant, $I \propto R^2$. But when loops are bent from a continuous wire, increasing the radius automatically adds more wire length (and thus more mass), causing the relationship to change to $I \propto R^3$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question relates the moments of inertia of two circular loops made from a single uniform wire to their respective radii. Given the ratio of their moments of inertia, we need to calculate the ratio of their radii.

Step 2: Key Formula or Approach:
The moment of inertia $I$ of a thin circular ring or loop of mass $M$ and radius $R$ about its central axis perpendicular to its plane is:
$$I = MR^2$$ Since the loops are constructed from a uniform wire, the mass of each loop is directly proportional to its total length (circumference):
$$M = \lambda \cdot L = \lambda \cdot (2\pi R)$$ where $\lambda$ represents the mass per unit length of the wire. Substituting this into the moment of inertia formula:
$$I = (2\pi \lambda R) \cdot R^2 = 2\pi \lambda R^3$$ This shows that for loops formed from a uniform wire, the moment of inertia scales with the cube of the radius:
$$I \propto R^3$$

Step 3: Detailed Explanation:
Using the derived proportionality relation, set up the ratio for loops P and Q:
$$\frac{I_P}{I_Q} = \left(\frac{R_1}{R_2}\right)^3$$ We are given that $\frac{I_P}{I_Q} = \frac{1}{8}$. Substitute this into our expression:
$$\frac{1}{8} = \left(\frac{R_1}{R_2}\right)^3$$ Take the cube root of both sides to isolate the ratio of radii:
$$\frac{R_1}{R_2} = \sqrt[3]{\frac{1}{8}} = \frac{1}{2}$$ The question explicitly asks for the reciprocal ratio, $\frac{R_2}{R_1}$:
$$\frac{R_2}{R_1} = \frac{2}{1} = 2$$

Step 4: Final Answer:
The ratio $\frac{R_2}{R_1}$ is $2$, which is option (C).