A sphere made of iron is rotating about its diameter as axis, α=1 × 10-5 ° C -1. If the temperature rises by 100° C , the percentage increase in its moment of inertia is

Question

A sphere made of iron is rotating about its diameter as axis, α=1 × 10-5 ° C -1. If the temperature rises by 100° C , the percentage increase in its moment of inertia is (A) 0.1 % (B) 0.2 % (C) 0.5 % (D) 0.002 %

Tardigrade Admin · Accepted Answer

For a solid sphere I=(2/5) M R2 (Δ R/R)=α Δ T=(1 × 10-5 ° C -1)(100° C )=10-3 ° C -1 As I propto R2 (Δ I/I)=(2 Δ R/R) =2 × 10-3=2 × 10-3 × 100 %=0.2 %

Q. A sphere made of iron is rotating about its diameter as axis, $α = 1 \times 1 0^{- 5}^{\circ} C^{- 1}$ . If the temperature rises by $10 0^{\circ} C,$ the percentage increase in its moment of inertia is

$0.1%$

$0.2%$

$0.5%$

$0.002%$

For a solid sphere $I = \frac{2}{5} M R^{2}$
$\frac{Δ R}{R} = α Δ T = (1 \times 1 0^{- 5}^{\circ} C^{- 1}) (10 0^{\circ} C) = 1 0^{- 3}^{\circ} C^{- 1}$
As $I \propto R^{2}$
$\frac{Δ I}{I} = \frac{2Δ R}{R}$
$= 2 \times 1 0^{- 3} = 2 \times 1 0^{- 3} \times 100% = 0.2%$

Q. A sphere made of iron is rotating about its diameter as axis, α=1×10−5∘C−1. If the temperature rises by 100∘C, the percentage increase in its moment of inertia is

0.1%

0.2%

0.5%

0.002%