JEE Advanced · Physics · 8. Rotational Motion

The figure shows a system consisting of (i) a ring of outer radius $3 R$ rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius $2 R$ rotating anti-clockwise with angular speed $\omega / 2$. The ring and disc are separated by frictionless ball bearings. The point $P$ on the inner disc is at a distance $R$ from the origin, where $O P$ makes an angle of $30^{\circ}$ with the horizontal. Then with respect to the horizontal surface,

A the point $O$ has linear velocity $3 R \omega \hat{i}$
B the point $P$ has linear velocity $\frac{11}{4} R \omega \hat{i}+\frac{\sqrt{3}}{4} R \omega \hat{k}$.
C the point $P$ has linear velocity $\frac{13}{4} R \omega \hat{i}-\frac{\sqrt{3}}{4} R \omega \hat{k}$
D the point $P$ has linear velocity

$\left(3-\frac{\sqrt{3}}{4}\right) R \omega \hat{i}+\frac{1}{4} R \omega \hat{k}$

Verified Solution

Answer & Solution

Correct Answer

(B) the point $P$ has linear velocity $\frac{11}{4} R \omega \hat{i}+\frac{\sqrt{3}}{4} R \omega \hat{k}$.

Step-by-step Solution

Detailed explanation

Velocity at centre 'O' $\quad \therefore \quad \vec{v}_{o}=3 R \omega \hat{i}$

$\vec{V}_{P}=3 R \omega \hat{i}-\frac{R \omega}{2} \sin 30^{\circ} \hat{i}+\frac{R \omega}{2} \cos 30^{\circ} \hat{k}$

$\therefore \quad \vec{V}_{P}=\left[3 R_{\omega} \hat{i}-\frac{R_{\omega}}{4} \hat{i}\right]+\frac{\sqrt{3} R_{\omega}}{4} \hat{k}$

or, $\quad \vec{V}_{P}=\frac{11}{4} R_{\omega} \hat{i}+\frac{\sqrt{3}}{4} R_{\omega} \hat{k}$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

	List-I		List-II
(I)	$10^{- 3} kg$ of water $100 ° C$ is converted to steam at the same temperature, at a pressure of $10^{5} Pa$ . The volume of the system changes from $10^{- 6} m^{3}$ to $10^{- 3} m^{3}$ in the process. Latent heat of water $= 2250 kJ {kg}^{- 1}$	(P)	$2 kJ$
(II)	$0.2$ moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 K$ undergoes an isobaric expansion to volume $3 V$ . Assume $8.0 J {mol}^{- 1} K^{- 1}$	(Q)	$7 kJ$
(III)	One mole of a monoatomic ideal gas is compressed adiabatically from volume $V = \frac{1}{3} m^{3}$ and pressure $2 kPa$ to volume $\frac{V}{8}$ .	(R)	$4 kJ$
(IV)	Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 kJ$ of heat and undergoes isobaric expansion.	(S)	$5 kJ$
		(T)	$3 kJ$

Which one of the following options is correct?

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Consider the given data with frequency distribution

\begin{array}{l} x_{i} & 3 & 8 & 11 & 10 & 5 & 4 \\ f_{i} & 5 & 2 & 3 & 2 & 4 & 4 \end{array}

Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$(P)$	The mean of the above data is	$(1)$	$2.5$
$(Q)$	The median of the above data is	$(2)$	$5$
$(R)$	The mean deviation about the mean of the above data is	$(3)$	$6$
$(S)$	The mean deviation about the median of the above data is	$(4)$	$2.7$
		$(5)$	$2.4$

The correct option is

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