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

Two iron solid discs of negligible thickness have radii $R_1$ and $R_2$ and moment of inertia $I_1$ and $I_2$, respectively. For $R_2=2R_1$, the ratio of $I_1$ and $I_2$ would be $1/x$, where $x=$ ____ .

The moment of inertia of a solid disc rotating along its diameter is $2.5$ times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is $n$ times higher than the moment of inertia of the given ring. Here, $n=$ ____ Consider all the bodies have equal masses.

An object of mass ' m ' is projected from origin in a vertical xy plane at an angle $45^{\circ }$ with the $x-$ axis with an initial velocity $v_0$. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [ g is acceleration due to gravity]

1. A.$\frac{m{v}_0^3}{2\sqrt{2}g}$ along negative $z$-axis
2. B.$\frac{m{v}_0^3}{2\sqrt{2}g}$ along positive $z$-axis
3. C.$\frac{m{v}_0^3}{4\sqrt{2}g}$ along positive $z$-axis
4. D.$\frac{m{v}_0^3}{4\sqrt{2}g}$ along negative $z$-axis

A uniform solid cylinder of mass ' m ' and radius ' r ' rolls along an inclined rough plane of inclination $45^{\circ }$. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder's axis will be

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

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

A solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is :

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

A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that $\theta (t)=5t^2-8t$, where $\theta (t)$ is the angular position of the rotating disc as a function of time $t$. How much power is delivered by the applied torque, when $t=2\text{ s}$ ?

1. A.$60{\mathrm{MR}}^2$
2. B.$72{\mathrm{MR}}^2$
3. C.$8{\mathrm{MR}}^2$
4. D.$108{\mathrm{MR}}^2$

1. A.$\frac{17}{32}{\mathrm{MR}}^2$
2. B.$\frac{13}{32}{\mathrm{MR}}^2$
3. C.$\frac{9}{32}{\mathrm{MR}}^2$
4. D.$\frac{7}{32}{\mathrm{MR}}^2$

The position vectors of two $1\text{ kg}$ particles, (A) and (B), are given by ${\vec{r}}_{\mathrm{A}}=\left({\alpha }_1{t}^2\hat{i}+{\alpha }_{2}t\hat{j}+{\alpha }_{3}t\hat{k}\right)\text{m}$ and ${\vec{r}}_{\mathrm{B}}=\left({\beta }_{1}t\hat{i}+{\beta }_2{t}^2\hat{j}+{\beta }_{3}t\hat{k}\right)\text{m}$, respectively; $\left({\alpha }_1=1{\text{ m/s}}^2,{\alpha }_2=3n\text{ m/s},{\alpha }_3=2\text{ m/s},{\beta }_1=2\text{ m/s},{\beta }_2=-1{\text{ m/s}}^2,{\beta }_3=4p\text{ m/s}\right)$, where $t$ is time, $n$ and $p$ are constants. At $t=1\text{ s}$, $\left|{\vec{V}}_{\mathrm{A}}\right|=\left|{\vec{V}}_{\mathrm{B}}\right|$ and velocities ${\vec{V}}_A$ and ${\vec{V}}_B$ of the particles are orthogonal to each other. At $t=1\text{ s}$, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{\mathrm{L}}\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. The value of $\mathrm{L}$ is _________.

A string is wrapped around the rim of a wheel of moment of inertia $0.40{\text{ kgm}}^2$ and radius $10\text{ cm}$. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of $40\text{ N}$. The angular velocity of the wheel after $10\text{ s}$ is $x\text{ rad/s}$, where $x$ is ______.

1. A.$1\mathrm{N}$
2. B.$10\mathrm{N}$
3. C.$\frac{10}{\sqrt{2}}N$
4. D.$\frac{1}{10\times \sqrt{2}}N$

Two planets $A$ and $B$ having masses $m_1$ and $m_2$ move around the sun in circular orbits of $r_1$ and $r_2$ radii respectively. If angular momentum of $A$ is $L$ and that of $B$ is $3L$, the ratio of time period $\left(\frac{T_A}{T_B}\right)$ is:

1. A.${\left(\frac{r_2}{r_1}\right)}^{\frac{3}{2}}$
2. B.$27{\left(\frac{m_1}{m_2}\right)}^3$
3. C.${\left(\frac{r_1}{r_2}\right)}^3$
4. D.$\frac{1}{27}{\left(\frac{m_2}{m_1}\right)}^3$

A thin circular disc of mass $\mathrm{M}$ and radius $\mathrm{R}$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity $\omega$. If another disc of same dimensions but of mass $\mathrm{M}/2$ is placed gently on the first disc co-axially, then the new angular velocity of the system is :

1. A.$\frac{4}{5}\omega$
2. B.$\frac{5}{4}\omega$
3. C.$\frac{3}{2}\omega$
4. D.$\frac{2}{3}\omega$

Three balls of masses $2\text{ kg}$, $4\text{ kg}$ and $6\text{ kg}$ respectively are arranged at centre of the edges of an equilateral triangle of side $2\text{ m}$. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be ________ ${\text{kg m}}^2$.

The angular momentum of an electron in a hydrogen atom is proportional to : (Where $r$ is the radius of orbit of electron)

1. A.$\frac{1}{r}$
2. B.$\frac{1}{\sqrt{r}}$
3. C.$\sqrt{r}$
4. D.$r$

A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is $\frac{x}{5}$. The value of $x$ is ____.

In a system two particles of masses $m_1=3\text{ kg}$ and $m_2=2\text{ kg}$ are placed at certain distance from each other. The particle of mass $m_1$ is moved towards the center of mass of the system through a distance $2\text{ cm}$. In order to keep the center of mass of the system at the original position, the particle of mass $m_2$ should move towards the center of mass by the distance ________ $\text{cm}$.

A rod of length $60\text{ cm}$ rotates with a uniform angular velocity $20\text{ rad}{\text{s}}^{-1}$ about its perpendicular bisector, in a uniform magnetic field $0.5T$. The direction of magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is ________ V.

A ball of mass $0.5\mathrm{kg}$ is attached to a string of length $50\mathrm{cm}$. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is $400\mathrm{N}$. The maximum possible value of angular velocity of the ball in $rad/s$ is :

1. A.1600
2. B.20
3. C.40
4. D.1000

Three identical spheres each of mass $2\mathrm{M}$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to $4\mathrm{m}$ each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is $\frac{4\sqrt{2}}{x}$, where the value of $x$ is ____ .

1. A.$\frac{3}{4}\frac{v^2}{g}$
2. B.$\frac{v^2}{g}$
3. C.$\frac{2}{3}\frac{v^2}{g}$
4. D.$\frac{1}{2}\frac{v^2}{g}$

A particle initially at rest starts moving from reference point $x=0$ along $x$-axis, with velocity $v$ that varies as $v=4\sqrt{x}\text{ m/s}$. The acceleration of the particle is ________ $\mathrm{m}{\mathrm{s}}^{-2}$.

A uniform rod $AB$ of mass $2\text{ kg}$ and length $30\text{ cm}$ is at rest on a smooth horizontal surface. An impulse of force $0.2\text{ Ns}$ is applied to end B. The time taken by the rod to turn through at right angles will be $\frac{\pi }{x}\text{ s}$, where $x=$ ________ .

A coin is placed on a disc. The coefficient of friction between the coin and the disc is $\mu$. If the distance of the coin from the center of the disc is $r$, the maximum angular velocity which can be given to the disc, so that the coin does not slip away, is :

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