A uniform rod of length L pivoted at one end P is freely rotated in a horizontal plane with an angular velocity ω about a vertical axis passing through P. If the temperature of the system is increased by Δ T, angular velocity becomes (ω/2). If coefficient of linear expansion of the rod is α(α

Question

A uniform rod of length L pivoted at one end P is freely rotated in a horizontal plane with an angular velocity ω about a vertical axis passing through P. If the temperature of the system is increased by Δ T, angular velocity becomes (ω/2). If coefficient of linear expansion of the rod is α(α < < 1), then Δ T will be (A) (1/α) (B) (1/2 α) (C) (1/4 α) (D) α

Tardigrade Admin · Accepted Answer

I ω=I1 (ω/2) I1=2 I (m l12/3)=2 (m l02/3) l02(1+α Δ T) 2=2 l02, ∴ 1+2 α Δ T=2 ∴ Δ T=(1/2 α)

$\frac{1}{α}$

$\frac{1}{2 α}$

$\frac{1}{4 α}$

$α$

$I ω = I_{1} \frac{ω}{2}$
$I_{1} = 2 I$
$\frac{m l _{1}^{2}}{3} = 2 \frac{m l _{0}^{2}}{3}$
${l_{0}^{2} (1 + α Δ T)}^{2} = 2 l_{0}^{2}$ ,
$∴ 1 + 2 α Δ T = 2$
$∴ Δ T = \frac{1}{2 α}$

α1​

2α1​

4α1​

α

Iω=I1​2ω​ I1​=2I 3ml12​​=23ml02​​ {l02​(1+αΔT)}2=2l02​, ∴1+2αΔT=2 ∴ΔT=2α1​

$\frac{1}{α}$

$\frac{1}{2 α}$

$\frac{1}{4 α}$

$α$

$I ω = I_{1} \frac{ω}{2}$
$I_{1} = 2 I$
$\frac{m l _{1}^{2}}{3} = 2 \frac{m l _{0}^{2}}{3}$
${l_{0}^{2} (1 + α Δ T)}^{2} = 2 l_{0}^{2}$ ,
$∴ 1 + 2 α Δ T = 2$
$∴ Δ T = \frac{1}{2 α}$