SATHEE: Rotation 1 Question 19

Rotation 1 Question 19

22. A lamina is made by removing a small disc of diameter $2 R$ from a bigger disc of uniform mass density and radius $2 R$ , as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_{O}$ and

$I_{P}$ , respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\frac{I_{P}}{I_{O}}$ to the nearest integer is

(2012)

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Answer:

Correct Answer: 22. (3)

Solution:

$T =$ Total portion

$R =$ Remaining portion and

$C =$ Cavity and let $σ =$ mass per unit area.

Then, $m_{T} = π (2 R)^{2} σ = 4 π R^{2} σ$

For $I_{P}$

$m_{C} = π (R)^{2} σ = π R^{2} σ$

$\begin{aligned} I_{R} & = I_{T} - I_{C} \\ = \frac{3}{2} m_{T} (2 R)^{2} - \frac{1}{2} m_{C} R^{2} + m_{C} r^{2} \\ = \frac{3}{2} (4 π R^{2} σ) (4 R^{2}) - \frac{1}{2} (π R^{2} σ) + (π R^{2} σ) (5 R^{2}) \\ = (18.5 π R^{4} σ) \end{aligned}$

For $I_{O} I_{R} = I_{T} - I_{C}$

$= \frac{1}{2} m_{T} (2 R)^{2} - \frac{3}{2} m_{C} R^{2}$

$= \frac{1}{2} (4 π R^{2} σ) (4 R^{2}) - \frac{3}{2} (π R^{2} σ) (R^{2}) = 6.5 π R^{4} σ$

$∴ \frac{I_{P}}{I_{O}} = \frac{18.5 π R^{4} σ}{6.5 π R^{4} σ} = 2.846$

Therefore, the nearest integer is 3 .

Rotation 1 Question 19

22. A lamina is made by removing a small disc of diameter $2 R$ from a bigger disc of uniform mass density and radius $2 R$ , as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_{O}$ and

Answer:

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Rotation 1 Question 19

22. A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and P is IO and

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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22. A lamina is made by removing a small disc of diameter $2 R$ from a bigger disc of uniform mass density and radius $2 R$ , as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_{O}$ and