Question

Two rods of same material and same cross-section
have the ratio of their lengths as 2:3. The ratio of
their moment of inertia is....

Plz explain the answer

Answer is 8:27​

Answer 1

Answer:

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Answer 2

Given that,

The ratio of their length = 2:3

We know that,

The moment of inertia of the rod  about end is

$M.I=\dfrac{1}{3}ML^2$

Two rods of same material and same cross-section but has different length.

Mass of rod will be different but mass per unit length will be same.

So, The moment of inertia of rod is

$M.I=\dfrac{1}{3}(\dfrac{M}{L})L^3$

Now, Moment of inertia of first rod is

$M.I=\dfrac{1}{3}m(2L)^3$....(I)

Moment of inertia of second rod is

$M.I=\dfrac{1}{3}m(3L)^3$.....(II)

We need to calculate the ratio of their moment of inertia

Dividing equation (I) by equation (II)

$\dfrac{M.I_{1}}{M.I_{2}}=\dfrac{\dfrac{1}{3}m(2L)^3}{\dfrac{1}{3}m(3L)^3}$

$\dfrac{M.I_{1}}{M.I_{2}}=\dfrac{8}{27}$

Hence, The ratio of their moment of inertia is 8:27.