Rotational Mechanics — JEE Main PYQs

The position vector of 1 kg object is $\overrightarrow{\mathrm{r}}=(3\hat{\mathrm{i}}-\hat{\mathrm{j}})\mathrm{m}$ and its velocity $\overrightarrow{\mathrm{v}}=(3\hat{\mathrm{j}}+\hat{\mathrm{k}}){\mathrm{ms}}^{-1}$. The magnitude of its angular momentum is …

The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is ${\mathrm{I}}_1$. The same rod is bent into a ring and its moment of inertia about a diameter is ${\mathrm{I}}_2$. If $\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}$ is …

Solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is :-

A spherical shell of 1 kg mass and radius R is rolling with angular speed $\omega$ on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin O is $\frac{a}{3}{R}^2\omega$. The value of a will be:

A thin circular ring of mass $M$ and radius $R$ is rotating with a constant angular velocity $2{\mathrm{rads}}^{-1}$ in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass $m$ be attached gently to …

A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be $x\times 10^{-2}$ kg. The value of $x$ …

A fly wheel is accelerated uniformly from rest and rotates through 5 rad in the first second. The angle rotated by the fly wheel in the next second, will be :

Moment of Inertia (M.I.) of four bodies having same mass 'M' and radius '$2\mathrm{R}$' are as follows:
${\mathrm{I}}_1=$ M.I. of solid sphere about its diameter
${\mathrm{I}}_2=$ M.I. of solid cylinder about its axis
${\mathrm{I}}_3=$ M.I. of solid circular disc about its diame …

If force $\vec{F}=3\hat{i}+4\hat{j}-2\hat{k}$ acts on a particle having position vector $2\hat{i}+\hat{j}+2\hat{k}$ then, the torque about the origin will be :-

Match List-I with List-II:
List-I: (A) M.I. of a solid sphere of radius R about any tangent (B) M.I. of a hollow sphere of radius R about any tangent (C) M.I. of a circular ring of radius R about its diameter (D) M.I. of a circular disc of radius R about any diameter
List-II …

Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badminton racket are same (M) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle an …

Moment of inertia (M.I) of four bodies, having same mass and radius, are reported as:
${\mathrm{I}}_1=$ M.I. of thin circular ring about its diameter
${\mathrm{I}}_2=$ M.I. of circular disc about an axis perpendicular to disc and going through the centre,
${\mathrm{I}}_3=$ M.I of …

The solid cylinder of length 80 cm and mass M has a radius of 20 cm . Calculate the density of the material used if the moment of inertia of the cylinder about an axis $CD$ parallel to $AB$ as shown in figure is $2.7\mathrm{kg}{\mathrm{m}}^2$.

A circular disc reaches from top to bottom of an inclined plane of length 'L' when it slips down the plane, it takes time '${\mathrm{t}}_1$'. When it rolls down the plane, it takes time ${\mathrm{t}}_2$. The value of $\frac{t_1}{t_2}$ is $\sqrt{\frac{3}{x}}$. The valu …

A uniform thin bar of mass 6 kg and length 2.4 meter is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is ____ $\times 10^{-1}\mathrm{kg}{\mathrm{m}}^2$.

A body rolls down an inclined plane without slipping. The kinetic energy of rotation is $50\%$ of its translational kinetic energy. The body is :

Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is $\frac{\sqrt{x}}{2}$. …

Two discs have moments of intertia $I_1$ and $I_2$ about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, ${\omega }_1$ and ${\omega }_2$ respectively and are brought into contact face to face with their axe …

Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.
The correct statement for this situation is :

The centre of a wheel rolling on a plane surface moves with a speed $v_0$. A particle on the rim of the wheel at the same level as the centre will be moving at a speed $\sqrt{x}{v}_0$. Then the value of $x$ is ______.

Four identical solid spheres each of mass ' $m$ ' and radius ' $a$ ' are placed with their centres on the four corners of a square of side ' $b$ '. The moment of inertia of the system about side of square where the axis of rotation is parallel to the plane of the square is …

Moment of inertia of a square plate of side / about the axis passing through one of the corner and perpendicular to the plane of square plate is given by:

A system consists of two identical spheres each of mass 1.5 kg and radius 50 cm at the end of light rod. The distance between the centres of the two spheres is 5 m. What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoin …

Angular momentum of a single particle moving with constant speed along circular path :

Match the moment of inertia (MI) of rods with different configurations:

Choose the correct answer from the options given below:

The figure shows two solid discs with radius $R$ and $r$ respectively. If mass per unit area is same for both, what is the ratio of MI of bigger disc around axis AB (which is ⟂ to the plane of the disc and passing through its centre) to MI of smaller disc around one of its diamet …

A triangular plate is shown. A force $\vec{F}=4\hat{i}-3\hat{j}$ is applied at point P. The torque at point P with respect to point 'O' and 'Q' are:

A mass M hangs on a massless rod of length $\ell$ which rotates at a constant angular frequency. The mass M moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega$. The angular …

The following bodies,
(1) a ring
(2) a disc
(3) a solid cylinder
(4) a solid sphere
of same mass ' $m$ ' and radius ' $R$ ' are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclin …

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Moment of inertia of a circular disc 'M' and radius 'R' about X, Y axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be …

A particle of mass '$m$' is moving in time '$t$' on a trajectory given by

Where $\alpha$ and $\beta$ are dimensional constants.
The angular momentum of the particle become the same as it was for $t=0$ at time t = ____ secon …

A solid disc of radius 20 cm and mass 10 kg is rotating with an angular velocity of 600 rpm, about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc at rest in 10s is _____ $\pi \times 10^{-1}$ Nm.

In the given figure, two wheels P and Q are connected by a belt B . The radius of P is three times as that of Q . In case of same rotational kinetic energy, the ratio of rotational inertias $\left(\frac{I_1}{I_2}\right)$ will be $x:1$. The value of $x$ will
be ____ …

A 2 kg steel rod of length 0.6 m is clamped on a table vertically at its lower end and is free to rotate in vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of rod when i …

Four equal masses, $m$ each are placed at the corners of a square of length $(\ell )$ as shown in the figure. The moment of inertia of the system about an axis passing through A and parallel to DB would be :

Consider a 20 kg uniform circular disk of radius 0.2 m . It is pin supported at its centre and is at rest initially. The disk is acted upon by a constant force $\mathrm{F}=20\mathrm{N}$ through a massless string wrapped around its periphery as shown in the figure. Suppose the …

A solid disc of radius 'a' and mass 'm' rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the will be $\frac{2}{b}g\sin \theta$ where $b$ is ____ (Round off to the Nearest Integer)
( $\mathrm{g}=$ acceleration …

A force $\vec{F}=4\hat{i}+3\hat{j}+4\hat{k}$ is applied on an intersection point of $x=2$ plane and $x$-axis. The magnitude of torque of this force about a point $(2,3,4)$ is ⋯---------. (Round off to the Nearest Integer)

Consider a uniform wire of mass M and length L . It is bent into a semi-circle. Its moment of inertia about a line perpendicular to the plane of the wire passing through centre is :

A solid cylinder of mass $m$ is wrapped with an inextensible light string and is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is:
[the coefficient of static friction, ${\mu }_{\mathrm{s}}$ is 0.4 …

A sphere of mass 2 kg and radius 0.5 m is rolling with an initial speed of $1{\mathrm{ms}}^{-1}$ goes up an inclined plane which makes an angle of $30^0$ with the horizontal plane, without slipping. How long will the sphere take to return to the starting point $A$?

A thin circular ring of mass M and radius r is rotating about its axis with an angular speed $\omega$. Two particles having mass $m$ each are now attached at diametrically opposite points. The angular speed of the ring will become:

The radius of gyration of a uniform rod of length $\ell$, about an axis passing through a point $\frac{\ell }{4}$ away from the centre of the rod, and perpendicular to it is:

As shown in the figure, a bob of mass $m$ is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius $r$ and mass $m$. When released from rest the bob starts falling vertically. When it has covered a distance of $h$, the angular speed of the whe …

Mass per unit area of a circular disc of radius a depends on the distance $r$ from its centre as $\sigma (r)=A+Br$. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is:

Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force $F$ at a point $b$ above its centre of mass (see figure). If the coefficient of friction is $\mu =0.4$, the maximum possible value of $100\times \frac{b}{a}$ for box n …

Consider a uniform rod of mass $\mathrm{M}=4\mathrm{m}$ and length $\ell$ pivoted about its centre. A mass m moving with velocity v making angle $\theta =\frac{\pi }{4}$ to the rod's long axis collides with one end of the rod and sticks to it. The angular sped of the rod-mass sys …

A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of $5.00\mathrm{cm}/\mathrm{s}$. Its kinetic energy is:

A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses $m_1$ and $m_2\left(m_1>m_2\right)$ are attached to the ends of the string. The syste …

Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d . The ratio ${\mathrm{I}}_0/{\mathrm{I}}_{\mathrm{A}}$ of moment of inertia ${\mathrm{I}}_0$ of the system …