Rotational Mechanics — JEE Main PYQs

One end of a straight uniform 1 m long bar is pivoted on horizontal table. It is released from rest when it makes an angle $30^{\circ }$ from the horizontal (see figure). Its angular speed when it hits the table is given as $\sqrt{\mathrm{n}}{\mathrm{s}}^{-1}$, where $n$ is an inte …

A block of mass $m=1$ kg slides with velocity $v=6$ m/s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle $\theta$ before momentarily coming …

Moment of inertia of a cylinder of mass $M$, length $L$ and radius $R$ about an axis passing through its centre and perpendicular to the axis of the cylinder is $I=M\left(\frac{R^2}{4}+\frac{L^2}{12}\right)$. If such a cylinder is to be made for a given mass of a material, th …

A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm). The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the pers …

A uniform rod of length '$l$' is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed $\omega$ the rod makes an angle $\theta$ with it (see figure). To find $\theta$ equate the rate of change of angular momentum (dir …

A massless equilateral triangle EFG of side 'a' (As shown in figure) has three particles of mass m situated at its vertices. The moment of inertia of the system about the line EX perpendicular to $EG$ in the plane of $EFG$ is $\frac{N}{20}m{a}^2$ where $N$ is an integer. The val …

The liner mass density of a thin rod AB of length L varies from A to B as $\lambda (x)={\lambda }_0\left(1+\frac{x}{L}\right)$, where x is the distance from A. If M is the mass of the rod hen its moment of inertia about an axis passing through A and perpendicular to the rod is:

A wheel is rotating freely with an angular speed $\omega$ on a shaft. The moment of inertia of the wheel is I and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia 3 I initially at rest is suddenly coupled to the same shaft. The resultant frac …

A force $\vec{F}=(\hat{i}+2\hat{j}+3\hat{k})$ N acts at a point $(4\hat{i}+3\hat{j}-\hat{k})$ m. Then the magnitude of torque about the point $(\hat{i}+2\hat{j}+\hat{k})$ m will be $\sqrt{x}$ N-m. The value of $x$ is ______.

Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is M, radius of its top, R and height, H, then its moment of inertia about its axis is:

Four point masses, each of mass m, are fixed at the corners of a square of side l. The square is rotating with angular frequency $\omega$, about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum o …

ABC is a plane lamina of the shape of an equilateral triangle. $\mathrm{D}$, $\mathrm{E}$ are mid points of $\mathrm{AB}$, $\mathrm{AC}$ and G is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is …

A circular disc of mass M and radius R is rotating about its axis with angular speed ${\omega }_1$. If another stationary disc having radius $\frac{R}{2}$ and same mass $M$ is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed …

Consider two uniform discs of the same thickness and different radii ${\mathrm{R}}_1=\mathrm{R}$ and ${\mathrm{R}}_2=\alpha \mathrm{R}$ made of the same material. If the ratio of their moments of inertia ${\mathrm{I}}_1$ and ${\mathrm{I}}_2$, respectively, about their axes is …

For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through O (the centre of mass) and O' (corner point) is:

A thin circular plate of mass $M$ and radius $R$ has its density varying as $p(r)=p_{0}r$ with $P_0$ as constant and $r$ is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge …

A rectangles solid box of length 0.3 m is held horizontally, with one of its sides on the edge of a platform of height 5 m. When released, it slips off the table in a very short time $\tau =0.01$ s, remaining essentially horizontal. The angle by which it would rotate when it hits …

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal place:
(i) a ring of radius R ,
(ii) a solid cylinder of radius $\frac{\mathrm{R}}{2}$ and
(iii) a solid sphere of radius $\frac{R}{4}$.
If, in each case, the speed of the ce …

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $\theta$, where $\theta$ is the angle by which it has rotated, is given as $\mathrm{k}{\theta }^2$. If its moment of inertia is I then the angu …

To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and the floor is $\mu$, the torque, applied by the ma …

A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is:

A circular disc $D_1$ of mass $M$ and radius $R$ has two identical discs $D_2$ and $D_3$ of the same mass M and radius R attached rigidly as its opposite ends (see figure). The moment of inertia of the system about the axis ${\mathrm{OO}}^{\prime }$, passing through the centre …

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

A string is wound around a hollow cylinder of mass 5 kg and radius 0.5 m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect …

A circular disc of radius $b$ has a hole of radius $a$ at its centre (see figure). If the mass per unit area of the disc varies as $\left(\frac{{\sigma }_0}{r}\right)$, then the radius of gyration of the disc about its axis passing through the centre is :

A uniform rod of length $\ell$ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is $\mathrm{T}(\mathrm{x})$ at a distance x from the axis, then which of the …

A solid sphere of mass M and radius R is divided into two unequal parts. The first part has a mass of $\frac{7M}{8}$ and is converted into a uniform disc of radius 2R. The second part is converted into a uniform solid sphere. Let $I_1$ be the moment of inertia of the disc about …

The time dependence of the position of a particle of mass $m=2$ is given by $\vec{r}(t)=2t\hat{i}-3t^2\hat{j}$ Its angular momentum with respect to the origin at time $t=2$ is .

A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5 s is close to

The equilateral triangle $ABC$ is cut from a thin solid sheet of wood. (See figure) D, E and F are the mid points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of …

A slab is subjected to the two forces ${\vec{F}}_1$ and ${\vec{F}}_2$ of same magnitude $F$ as shown in the figure. Force ${\vec{F}}_2$ is in $XY$ - plane while force $F_1$ acts along $z-$ axis at the point $(2\vec{i}+3\vec{j})$. The moment of these forces ab …

Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm ), about its axis be I . The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is:

The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x from it, is 'I(x)'. Which one of the graphs represents the variation of I(x) with x correctly?

A particle of mass 20 g is released with an initial velocity $5\mathrm{m}/\mathrm{s}$ along the curve from the point A, as shown in the figure. The point A is at height h from point B. The particle slides along the frictionless surface. When the particle reaches point B, it …

Moment of inertia of a body about a given axis is $1.5\mathrm{kg}{\mathrm{m}}^2$ Initially the body is at rest. In order to produce a rotational kinetic energy of 1200 J , the angular acceleration of $20\mathrm{rad}/{\mathrm{s}}^2$ must be applied about the axis of rotatio …

A thin smooth rod of length $L$ and mass $M$ is rotating freely with angular speed ${\omega }_0$ about an axis perpendicular to the rod and passing through its centre. Two beads of mass m and negligible size are at the centre of the rod initially. The beads are free to slide along …

A rod of length 50 cm is pivoted at one end. It is raised such that if makes an angle of $30^{\circ }$ fro the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in rad ${\mathrm{s}}^{-1}$) will be ($\mathrm{g}=10{\mathrm{ms}}^{-2}$ …

An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If $AB=BC$, and the angle made by $AB$ with downward vertical is $\theta$, then:

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

A particle of mass $m$ is moving along a trajectory given by

The torque, acting on the particle about the origin, at $t=0$ is:

A thin disc of mass $M$ and radius $R$ has mass per unit area $\sigma (r)=k{r}^2$ where $r$ is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is:

Two coaxial discs, having moments of inertia ${\mathrm{I}}_1$ and $\frac{{\mathrm{I}}_1}{2}$, area rotating with respectively angular velocities ${\omega }_1$ and $\frac{{\omega }_1}{2}$, about their common axes. They are brought in contact with each other and thereafter they rotat …

Seven identical circular planar disks, each of mass $M$ and radius $R$ are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is :

From a uniform circular disc of radius $R$ and mass 9 M , a small disc of radius $\frac{\mathrm{R}}{3}$ is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is :

The moment of inertia of a uniform cylinder of length $l$ and radius $R$ about its perpendicular bisector is I. What is the ratio $l/R$ such that the moment of inertia is minimum?

A slender uniform rod of mass M and length $l$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it m …

A particle of mass $m$ is moving along the side of a square of side 'a', with a uniform speed v in the $\mathrm{x}-\mathrm{y}$ plane as shown in the figure :

Which of the following statements is false for the angular momentum $\overrightarrow{\mathrm{L}}$ about the origin?

A roller is made by joining together two cones at their vertices O. It is kept on two rails AB and CD which are placed asymmetrically (see figure), with its axis perpendicular to CD and its centre O at the centre of line joining AB and CD (see figure). It is given a light push so …

From a solid sphere of mass M and radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is :

A bob of mass m attached to an inextensible string of length $\ell$ is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $\omega \mathrm{rad}/\mathrm{s}$ about the vertical. About the point of suspension :