Two discs A and B are mounted coaxially on a vertical axle. The discs have moments of inertia and respectively about the common axis. [2007]

Question

Two discs A and B are mounted coaxially on a vertical axle. The discs have moments of inertia $I$ and $2 I$ respectively about the common axis. Disc A is imparted an initial angular velocity $2 ω$ using the entire potential energy of a spring compressed by a distance $x_{1}$ . Disc B is imparted an angular velocity $ω$ by a spring having the same spring constant and compressed by a distance $x_{2}$ . Both the discs rotate in the clockwise direction. [2007]

Q. The ratio $\frac{x_{1}}{x_{2}}$ is

Smart Achievers · Accepted Answer

(3)

$For disc A$

$\frac{1}{2} k x_{1}^{2} = \frac{1}{2} I {(2 ω)}^{2} \Rightarrow k x_{1}^{2} = 2 I ω^{2}$

$For disc B$

$\frac{1}{2} k x_{2}^{2} = \frac{1}{2} \times 2 I ω^{2} \Rightarrow k x_{2}^{2} = I ω^{2}$

$∴$ $\frac{k x_{1}^{2}}{k x_{2}^{2}} = \frac{2 I ω^{2}}{I ω^{2}} \Rightarrow \frac{x_{1}}{x_{2}} = \sqrt{2}$

Solution

1 $2$

2 $\frac{1}{2}$

3 $\sqrt{2}$

4 $\frac{1}{\sqrt{2}}$

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Solution

1 2

2 12

3 2

4 12

Ans. (3) For disc A 12kx12=12I(2ω)2 ⇒ kx12=2Iω2 For disc B 12kx22=12×2Iω2 ⇒ kx22=Iω2 ∴ kx12kx22=2Iω2Iω2 ⇒ x1x2=2

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1 $2$

2 $\frac{1}{2}$

3 $\sqrt{2}$

4 $\frac{1}{\sqrt{2}}$