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

1. A.$6\mathrm{kg}\cdot {\mathrm{m}}^2/s$
2. B.$8\mathrm{kg}\cdot {\mathrm{m}}^2/s$
3. C.$2\mathrm{kg}\cdot {\mathrm{m}}^2/s$
4. D.$3\mathrm{kg}\cdot {\mathrm{m}}^2/s$

1. A.
2. B.
3. C.
4. D.

A long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. If the radius of vessel is 5 cm and its rotational speed is 2 rotations per second, then the difference in the heights between the centre and the sides, in cm, will be :

1. A.2.0
2. B.1.2
3. C.0.1
4. D.0.4

The magnitude of torque on a particle of mass $1\text{ kg}$ is $2.5\text{ Nm}$ about the origin. If the force acting on it is $1\text{ N}$, and the distance of the particle from the origin is $5\text{ m}$, the angle between the force and the position vector is (in radians) :

1. A.$\frac{\pi }{8}$
2. B.$\frac{\pi }{6}$
3. C.$\frac{\pi }{4}$
4. D.$\frac{\pi }{3}$

1. A.$3{\mathrm{MR}}^2$
2. B.${\mathrm{MR}}^2$
3. C.$\frac{2}{3}{\mathrm{MR}}^2$
4. D.$\frac{4}{5}{\mathrm{MR}}^2$

1. A.$16{\text{ rad/s}}^2$
2. B.$20{\text{ rad/s}}^2$
3. C.$10{\text{ rad/s}}^2$
4. D.$12{\text{ rad/s}}^2$

Three charges $+Q$, $q$, $+Q$ are placed respectively, at distance, 0, $d/2$ and $d$ from the origin, on the x-axis. If the net force experienced by $+Q$, placed at $x=0$, is zero, then value of $q$ is :

1. A.$-\frac{Q}{4}$
2. B.$+\frac{Q}{2}$
3. C.$+\frac{Q}{4}$
4. D.$-\frac{Q}{2}$

1. A.$45^{\circ }\mathrm{C}$
2. B.$75^{\circ }\mathrm{C}$
3. C.$60^{\circ }\mathrm{C}$
4. D.$35^{\circ }\mathrm{C}$

1. A.$\frac{3k{\theta }_0^2}{l}$
2. B.$\frac{2k{\theta }_0^2}{l}$
3. C.$\frac{k{\theta }_0^2}{l}$
4. D.$\frac{k{\theta }_0^2}{2l}$

If the angular momentum of a planet of mass m, moving around the Sun in a circular orbit is L, about the center of the Sun, its areal velocity is :

1. A.$\frac{L}{m}$
2. B.$\frac{4L}{m}$
3. C.$\frac{L}{2m}$
4. D.$\frac{2L}{m}$

A rod, of length L at room temperature and uniform area of cross section A, is made of a metal having coefficient of linear expansion $\alpha {/}^{\circ }\mathrm{C}$. It is observed that an external compressive force F, is applied on each of its ends, prevents any change in the length of the rod, when its temperature rises by $\Delta T\text{ K}$. Young's modulus, Y, for this metal is :

1. A.$\frac{F}{A\alpha \Delta T}$
2. B.$\frac{F}{A\alpha (\Delta T-273)}$
3. C.$\frac{F}{2A\alpha \Delta T}$
4. D.$\frac{2F}{A\alpha \Delta T}$

Suppose that the angular velocity of rotation of earth is increased. Then, as a consequence :

1. A.Weight of the object, everywhere on the earth, will increase.
2. B.Weight of the object, everywhere on the earth, will decrease.
3. C.There will be no change in weight anywhere on the earth.
4. D.Except at poles, weight of the object on the earth will decrease.

A charge q is spread uniformly over an insulated loop of radius r. If it is rotated with an angular velocity $\omega$ with resect to normal axis then the magnetic moment of the loop is :

1. A.$q\omega r^2$
2. B.$\frac{4}{3}q\omega r^2$
3. C.$\frac{3}{2}q\omega r^2$
4. D.$\frac{1}{2}q\omega r^2$

1. A.1 : 3
2. B.1 : 4
3. C.1 : 5
4. D.1 : 2

1. A.$m,x$
2. B.$m,\frac{1}{x}$
3. C.$m,\frac{1}{x^2}$
4. D.$m,x^2$

1. A.$\frac{181}{2}M{R}^2$
2. B.$\frac{55}{2}M{R}^2$
3. C.$\frac{19}{2}M{R}^2$
4. D.$\frac{73}{2}M{R}^2$

1. A.$\frac{37}{9}M{R}^2$
2. B.$4M{R}^2$
3. C.$\frac{40}{9}M{R}^2$
4. D.$10M{R}^2$

A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the ${\mathrm{n}}^{\mathrm{th}}$ power of R. If the period of rotation of the particle is T, then :

1. A.$\mathrm{T}\propto {\mathrm{R}}^{\mathrm{n}/2}$
2. B.$\mathrm{T}\propto {\mathrm{R}}^{3/2}$ for any n
3. C.$\mathrm{T}\propto {\mathrm{R}}^{\mathrm{n}/2+1}$
4. D.$\mathrm{T}\propto {\mathrm{R}}^{(\mathrm{n}+1)/2}$

A granite rod of 60 cm length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is $2.7\times 10^{3}kg/{\mathrm{m}}^3$ and its Young’s modulus is $9.27\times 10^{10}\mathrm{Pa}$. What will be the fundamental frequency of the longitudinal vibrations ?

1. A.$7.5\mathrm{kHz}$
2. B.$5\mathrm{kHz}$
3. C.$2.5\mathrm{kHz}$
4. D.$10\mathrm{kHz}$

1. A.Constant speed
2. B.decreasing speed
3. C.increasing speed
4. D.speed which is $\frac{3}{4}\mathrm{th}$ of that of the roller when the weight is 0.4 m above the ground

1. A.$\frac{219M{R}^2}{256}$
2. B.$\frac{237M{R}^2}{512}$
3. C.$\frac{19M{R}^2}{512}$
4. D.$\frac{197M{R}^2}{256}$

1. A.$\frac{7}{8}{\mathrm{I}}_0$
2. B.$\frac{15}{16}{\mathrm{I}}_0$
3. C.$\frac{3{\mathrm{I}}_o}{4}$
4. D.$\frac{31{\mathrm{I}}_o}{32}$

A compressive force, F is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $\Delta \mathrm{T}$. The net change in its length is zero. Let $\ell$ be the length of the rod, A its area of cross-section,Y its Young's modulus, and $\alpha$ its coefficient of linear expansion. Then, F is equal to :

1. A.${\ell }^2\mathrm{Y}\alpha \Delta \mathrm{T}$
2. B.$\ell \mathrm{A}\mathrm{Y}\alpha \Delta \mathrm{T}$
3. C.$\mathrm{A}\mathrm{Y}\alpha \Delta \mathrm{T}$
4. D.$\frac{\mathrm{AY}}{\alpha \Delta \mathrm{T}}$

1. A.$\frac{2mg}{2m+M}$
2. B.$\frac{2Mg}{2m+M}$
3. C.$\frac{2mg}{2M+m}$
4. D.$\frac{2Mg}{2M+m}$

If the Earth has no rotational motion, the weight of a person on the equator is W. Determine the speed with which the earth would have to rotate about its axis so that the person at the equator will weigh $\frac{3}{4}\mathrm{W}$. Radius of the Earth is 6400 km and $\mathrm{g}=10m/{\mathrm{s}}^2$.

1. A.$1.1\times 10^{-3}rad/s$
2. B.$0.83\times 10^{-3}rad/s$
3. C.$0.63\times 10^{-3}rad/s$
4. D.$0.28\times 10^{-3}rad/s$