A uniform cylindrical rod of mass m and length L is rotating with an angular velocity ω. The axis of rotation is ⊥ to its axis of symmatry and passes through one of its edge faces . If the room temperature increases by 't' and the coefficient of linear expansion is α, the change in its angular velocity is

Question

A uniform cylindrical rod of mass m and length L is rotating with an angular velocity ω. The axis of rotation is ⊥ to its axis of symmatry and passes through one of its edge faces . If the room temperature increases by 't' and the coefficient of linear expansion is α, the change in its angular velocity is (A) 2ωα t (B) ωα t (C) (3/2)ωα t (D) (ωα t/2)

Tardigrade Admin · Accepted Answer

As no external torque is acting on the rod, its angular momentum about the axis of rotation should remain constant .

\Rightarrow

I ω = I^{1} ω^{1}

I = \frac{M L ^{2}}{3}

and

I^{1} = \frac{M L ^{2}}{3}

∴ L^{2} ω = L^{2} ω^{1}

here

L^{1} = L (1 + \propto t)

∴

ω^{1} = \frac{ω}{( 1 + \propto t ) ^{2}} = ω (1 - 2 \propto t)

∴

change in angular speed

Δ ω = 2 \propto ω t

$2 ω α t$

$ω α t$

$\frac{3}{2} ω α t$

$\frac{ω α t}{2}$

2ωαt

ωαt

23​ωαt

2ωαt​

As no external torque is acting on the rod, its angular momentum about the axis of rotation should remain constant . ⇒ Iω=I1ω1 I=3ML2​ and I1=3ML2​ ∴L2ω=L2ω1 here L1=L(1+∝t) ∴ ω1=(1+∝t)2ω​=ω(1−2∝t) ∴ change in angular speed Δω=2∝ωt

$2 ω α t$

$ω α t$

$\frac{3}{2} ω α t$

$\frac{ω α t}{2}$