Question

The moment of inertia of a thin rod about an axis passing through its mid point and perpendicular to the rod is 2400 g cm². The length of the 400 g rod is nearly :

• A. 8.5 cm
• B. 17.5 cm
• C. 20.7 cm
• D. 72.0 cm

Answer 1

The Correct Option is A

The moment of inertia \( I \) for a thin rod, about an axis through its midpoint and perpendicular to its length, is defined as:

$$ I = \frac{1}{12} M L^2 $$

Here, \( M \) represents the rod's mass, and \( L \) is its length.

Given values:

• \( I = 2400 \) g cm\(^2\)
• \( M = 400 \) g

Substitute the given values into the equation:

$$ 2400 = \frac{1}{12} \times 400 \times L^2 $$

Multiply both sides by 12 to eliminate the fraction:

$$ 2400 \times 12 = 400 \times L^2 $$

$$ 28800 = 400 \times L^2 $$

Divide both sides by 400 to isolate \( L^2 \):

$$ L^2 = \frac{28800}{400} $$

$$ L^2 = 72 $$

Take the square root of both sides to determine \( L \):

$$ L = \sqrt{72} $$

$$ L \approx 8.5 \text{ cm} $$

Consequently, the rod's length is approximately 8.5 cm.