Centre of Mass & Rotational Motion · JEE Main previous year questions

A cylindrical tube $AB$ of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to $AB$ and passing through the edge at end $A$, as shown in the figure. If $P_A$ and $P_B$ are the pressures at $A$ and $B$ respectively, then (Consider the temperature is same at all points in the tube)

1. A.$P_B=P_A{e}^{\left(\frac{M{\omega }^2{l}^2}{3RT}\right)}$
2. B.$P_B=P_A$
3. C.$P_B=P_A{e}^{\left(\frac{M{\omega }^2{l}^2}{2RT}\right)}$
4. D.$P_B=P_A{e}^{\left(\frac{M{\omega }^2{l}^2}{RT}\right)}$

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm . The moment of inertia of this pair of spheres about the tangent passing through the point of contact is ____ $\mathrm{kg}\cdot {\mathrm{m}}^2$.

1. A.0.18
2. B.0.72
3. C.0.36
4. D.0.63

Three identical coils $C_1,C_2$ and $C_3$ are closely placed such that they share a common axis. $C_2$ is exactly midway. $C_1$ carries current $I$ in anti-clockwise direction while $C_3$ carries current $I$ in clockwise direction. An induced current flows through $C_2$ will be in clockwise direction when

1. A.$C_1$ and $C_3$ move with equal speeds away from $C_2$
2. B.$C_1$ moves towards $C_2$ and $C_3$ moves away from $C_2$
3. C.$C_1$ and $C_3$ move with equal speeds towards $C_2$
4. D.$C_1$ moves away from $C_2$ and $C_3$ moves towards $C_2$

A circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2}=2$ then $\frac{T_1}{T_2}=\frac{1}{\alpha }$. The value of $\alpha$ is ____.

An aluminium and steel rods having same lengths and cross-sections are joined to make total length of $120\mathrm{cm}$ at $30^{\circ }\mathrm{C}$. The coefficient of linear expansion of aluminium and steel are $24\times 10^{-6}{/}^{\circ }\mathrm{C}$ and $1.2\times 10^{-5}{/}^{\circ }\mathrm{C}$, respectively. The length of this composite rod when its temperature is raised to $100^{\circ }\mathrm{C}$, is ____ cm.

1. A.120.20
2. B.120.06
3. C.120.15
4. D.120.03

A large drum having radius $R$ is spinning around its axis with angular velocity $\omega$, as shown in figure. The minimum value of $\omega$ so that a body of mass $M$ remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass $M$ as $\mu$, is :

1. A.$\sqrt{\frac{2g}{\mu R}}$
2. B.$\sqrt{\frac{g}{2\mu R}}$
3. C.$\sqrt{\frac{\mu g}{R}}$
4. D.$\sqrt{\frac{g}{\mu R}}$

Two cars A and B each of mass $10^3$ kg are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is ____ J·s.

1. A.$3.6\times 10^5$
2. B.$10^5$
3. C.$3\times 10^5$
4. D.$2\times 10^5$

A rod of linear mass density '$\lambda$' and length 'L' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is.

1. A.$\frac{\lambda L^3}{8{\pi }^2}$
2. B.$\frac{\lambda L^3}{16{\pi }^2}$
3. C.$\frac{\lambda L^3}{4{\pi }^2}$
4. D.$\frac{\lambda L^3}{12}$

A thin solid disk of $1\mathrm{kg}$ is rotating along its diameter axis at the speed of $1800\mathrm{rpm}$. By applying an external torque of $25\pi \mathrm{Nm}$ for $40\mathrm{s}$, the speed increases to $2100\mathrm{rpm}$. The diameter of the disk is $\underline{}\mathrm{m}$.

A cube having a side of $10\mathrm{cm}$ with unknown mass and $200\mathrm{gm}$ mass were hung at two ends of a uniform rigid rod of $27\mathrm{cm}$ long. The rod along with masses was placed on a wedge keeping the distance between wedge point and $200\mathrm{gm}$ weight as $25\mathrm{cm}$. Initially the masses were not at balance. A beaker is placed beneath the unknown mass and water is added slowly to it. At given point the masses were in balance and half volume of the unknown mass was inside the water. (Take the density of unknown mass is more than that of the water, the mass did not absorb water and water density is $1{gm/cm}^3$.) The unknown mass is $\underline{}\mathrm{kg}$.

Which of the following are correct expression for torque acting on a body? A. $\vec{\tau }=\vec{r}\times \vec{L}$ B. $\vec{\tau }=\frac{d}{dt}(\vec{r}\times \vec{p})$ C. $\vec{\tau }=\vec{r}\times \frac{d\vec{p}}{dt}$ D. $\vec{\tau }=I\vec{\alpha }$ E. $\vec{\tau }=\vec{r}\times \vec{F}$ ($\vec{r}=\text{position vector}$; $\vec{p}=\text{linear momentum}$; $\vec{L}=\text{angular momentum}$; $\vec{\alpha }=\text{angular acceleration}$; $I=\text{moment of inertia}$; $\vec{F}=\text{force}$; $t=\text{time}$) Choose the correct answer from the options given below:

1. A.A, B, D and E Only
2. B.C and D Only
3. C.B, C, D and E Only
4. D.B, D and E Only

If $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass ' $m$ ' having position vector as $\vec{r}=a(\hat{i}\cos \omega t+\hat{j}\sin \omega t)$. The direction of force is

1. A.Opposite to the direction of $\vec{L}$
2. B.Opposite to the direction of $\vec{L}\times \vec{P}$
3. C.Opposite to the direction of $\vec{r}$
4. D.Opposite to the direction of $\vec{P}$

A wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is $8m/s$. The speed of the particle on the rim of the wheel at the same level as the centre of wheel, will be :

1. A.$4\sqrt{2}m/s$
2. B.$8m/s$
3. C.$4m/s$
4. D.$8\sqrt{2}m/s$

A cylindrical rod of length $1\mathrm{m}$ and radius $4\mathrm{cm}$ is mounted vertically. It is subjected to a shear force of $10^5\mathrm{N}$ at the top. Considering infinitesimally small displacement in the upper edge, the angular displacement $\theta$ of the rod axis from its original position would be : (shear moduli, $G=10^{10}N/{\mathrm{m}}^2$ )

1. A.$1/160\pi$
2. B.$1/2\pi$
3. C.$1/4\pi$
4. D.$1/40\pi$

A solid sphere with uniform density and radius $R$ is rotating initially with constant angular velocity $({\omega }_1)$ about its diameter. After some time during the rotation it starts losing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become $\mathrm{R}/2$ is $x{\omega }_1$. The value of $x$ is ____ .

Moment of inertia of a rod of mass ' M ' and length ' L ' about an axis passing through its center and normal to its length is ' $\alpha$ '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is :

1. A.$\alpha /4$
2. B.$\alpha /8$
3. C.$\alpha$
4. D.$\alpha /2$

A light wave is propagating with plane wave fronts of the type $x+y+z=\text{constant}$. The angle made by the direction of wave propagation with the $x$-axis is :

1. A.${\cos }^{-1}(2/3)$
2. B.${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
3. C.${\cos }^{-1}\left(\frac{1}{3}\right)$
4. D.${\cos }^{-1}\left(\sqrt{\frac{2}{3}}\right)$

1. A.20 rad/s
2. B.30 rad/s
3. C.10 rad/s
4. D.0 rad/s

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is :

1. A.$\frac{3}{8}\mathrm{M}{\mathrm{r}}^2$
2. B.$2\mathrm{M}{\mathrm{r}}^2$
3. C.$\frac{1}{2}\mathrm{M}{\mathrm{r}}^2$
4. D.$\frac{3}{2}\mathrm{M}{\mathrm{r}}^2$

1. A.$\frac{1}{2}\mathrm{a}\mathrm{m}{\mathrm{V}}_0$
2. B.$3\mathrm{a}\mathrm{m}{\mathrm{V}}_0$
3. C.$\frac{3}{2}\mathrm{a}\mathrm{m}{\mathrm{V}}_0$
4. D.$\frac{\sqrt{3}}{2}\mathrm{a}\mathrm{m}{\mathrm{V}}_0$

Match List - I with List - II. $$\begin{array}{ll}\text{List - I}& \text{List - II}\\ \text{(A) Young’s Modulus}& \text{(I) }\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}\\ \text{(B) Torque}& \text{(II) }\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}\\ \text{(C) Coefficient of Viscosity}& \text{(III) }{\mathrm{M}}^{-1}{\mathrm{L}}^3{\mathrm{T}}^{-2}\\ \text{(D) Gravitational Constant}& \text{(IV) }\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-2}\end{array}$$ Choose the correct answer from the options given below :

1. A.(A)-(I), (B)-(III), (C)-(II), (D)-(IV)
2. B.(A)-(IV), (B)-(II), (C)-(III), (D)-(I)
3. C.(A)-(II), (B)-(I), (C)-(IV), (D)-(III)
4. D.(A)-(II), (B)-(IV), (C)-(I), (D)-(III)

Two planets, $A$ and $B$ are orbiting a common star in circular orbits of radii $R_A$ and $R_B$, respectively, with $R_B=2R_A$. The planet $B$ is $4\sqrt{2}$ times more massive than planet $A$. The ratio $\left(\frac{L_B}{L_A}\right)$ of angular momentum $(L_B)$ of planet $B$ to that of planet $A(L_A)$ is closest to integer ____.

1. A.$\left(\frac{2}{3}a,\frac{2}{3}b\right)$
2. B.$\left(\frac{2}{3}a,\frac{b}{2}\right)$
3. C.$\left(\frac{1}{3}a,\frac{b}{2}\right)$
4. D.$\left(\frac{a}{2},\frac{b}{2}\right)$