Rotational Mechanics — JEE Advanced PYQs

Consider a large disk of radius R and two smaller disks, each of radius $\mathrm{r}=\mathrm{R}/50$, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation $\Delta \theta$ between their centers. …

A solid cylinder of radius $R$ rolls without slipping with a center of mass speed $v_0=\sqrt{\frac{\mathrm{gR}}{3}}$ on a horizontal surface with a vertical edge, as shown in the figure. Here, $g$ is the acceleration due to the gravity. At the moment when the cylinder loses con …

List-I shows four planar structures made of uniform solid rods each of mass $m$ and length $l$. In the List-II the possible moment of inertia of these structures about an axis ${\mathrm{OCO}}^{\prime }$, which lies in the plane of the structures, are given.
Choose the option that de …

A uniform circular disk of radius 0.2 m and mass 1 kg is pivoted at its top point $C$ such that it can rotate freely around $C$ in the $XY$ plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass 20 g, travelling along negative $x$ direction …

A disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R/2$ is fixed to the moto …

A thin uniform rod of length $L$ and certain mass is kept on a frictionless horizontal table with a massless string of length $L$ fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O . If a horizontal impulse $P$ is imparte …

A bar of mass $M=1.00\mathrm{kg}$ and length $L=0.20\mathrm{m}$ is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m=0.10\mathrm{kg}$ is moving on the same horizontal surface with …

An annular disk of mass $M$, inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$, as shown in the figure. At some time, an impulse $J_0\hat{x}$ is applied at a height h above the center of the disk. If $h=h_m$ then the d …

A thin circular coin of mass 5 gm and radius 4/3 cm is initially in a horizontal $xy$-plane. The coin is tossed vertically up ( $+z$ direction) by applying an impulse of $\sqrt{\frac{\pi }{2}}\times 10^{-2}\mathrm{N}$-s at a distance $2/3\mathrm{cm}$ from its center. The co …

At time $t=0$, a disk of radius 1 m starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha =\frac{2}{3}rad{s}^{-2}$. A small stone is stuck to the disk. At $t=0$, it is at the contact point of the disk and the plane. Later, at time …

A solid sphere of mass $1kg$ and radius $1m$ rolls without slipping on a fixed inclined plane with an angle of inclination $\theta =30^{\circ }$ from the horizontal. Two forces of magnitude $1N$ each, parallel to the incline, act on the sphere, both at distance …

A particle of mass 1 kg is subjected to a force which depends on the position as $\vec{F}=-k(x\hat{i}+y\hat{j})kgm{s}^{-2}$ with $k=1kg{s}^{-2}$. At time $t=0$, the particle's position $\vec{r}=\left(\frac{1}{\sqrt{2}}\hat{i}+\sqrt{2}\hat{j}\right)m$ and its velocit …

A flat surface of a thin uniform disk $A$ of radius $R$ is glued to a horizontal table. Another thin uniform disk $B$ of mass $M$ and with the same radius $R$ rolls without slipping on the circumference of $A$, as shown in the figure. A flat surface of $B$ also lies …

One end of a horizontal uniform beam of weight $W$ and length $L$ is hinged on a vertical wall at point $O$ and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point $Q$, at a height $L$ above the hinge at point $O$. A bl …

A pendulum consists of a bob of mass $m=0.1\mathrm{kg}$ and a massless inextensible string of length $L=1.0\mathrm{m}$. It is suspended from a fixed point at height $H=0.9\mathrm{m}$ above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on …

A pendulum consists of a bob of mass $m=0.1\mathrm{kg}$ and a massless inextensible string of length $L=1.0\mathrm{m}$. It is suspended from a fixed point at height $H=0.9\mathrm{m}$ above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on …

A horizontal force $F$ is applied at the center of mass of a cylindrical object of mass m and radius $R$, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $\mu$. The center of mass of the object has an acceleration …

A particle of mass $M=0.2$ kg is initially at rest in the $xy$-plane at a point ($\mathrm{x}=-l,\mathrm{y}=-h$), where $l=10$ m and $h=1$ m. The particle is accelerated at time $\mathrm{t}=0$ with a constant acceleration $\mathrm{a}=10$ m/s${}^2$ along the positive x-direction. …

A thin rod of mass M and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point O. A thin circular disc of mass M and of radius $a/4$ is pivoted on this rod with its center at a distance $a/4$ from the free end so that it can rotate fre …

A football of radius R is kept on a hole of radius r (r<R) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle θ from the horizontal as shown in the figure below. The maximum value of θ so that the football does not start r …

A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slo …

A rod of mass $m$ and length $L$, pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed $v$ strikes the rod horizontally at a distance $x$ from its pivoted end and gets embedded in it. The combined system now rotates with angular speed …

A thin and uniform rod of mass M and length L is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod make …

In the List-I below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \ne \beta$. In each case, the force acting on the particle is either zero or conser …

Consider a body of mass $1.0kg$ at rest at the origin at time $t=0$. A force $\vec{F}=(\alpha t\hat{ı}+\beta \hat{ȷ})$ is applied on the body, where $\alpha =1.0N{s}^{-1}$ and $\beta =1.0N$. The torque acting on the body about the origin at time $t=1.0s$ is …

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle 60° with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground i …

Consider regular polygons with number of sides $n=3,4,5\dots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum in …

A rigid uniform bar AB of length $L$ is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $\theta$. Which of the following statements about its motion is/are correct?

A wheel of radius $R$ and mass $M$ is placed at the bottom of a fixed step of height $R$ as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis …

One twirls a circular ring (of mass $M$ and radius $R$) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the pa …

One twirls a circular ring (of mass $M$ and radius $R$) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the pa …

A uniform wooden stick of mass 1.6 kg and length $l$ rests in an inclined manner on a smooth, vertical wall of height $h(<l)$ such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick m …

The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation

where $\alpha =10/3\mathrm{m}{\mathrm{s}}^{-3}$, $\beta =5\mathrm{m}{\mathrm{s}}^{-2}$ and $m=0.1\mathrm{kg}$. At …

Two thin circular discs of mass $m$ and $4m$, having radii of $a$ and $2a$, respectively, are rigidly fixed by a massless, rigid rod of length $l=\sqrt{24}a$ through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on …

A ring of mass $M$ and radius $R$ is rotating with angular speed $\omega$ about a fixed vertical axis passing through its centre O with two point masses each of mass $\frac{\mathrm{M}}{8}$ at rest at O . These masses can move radially outwards along two massless rods fixed on the …

The densities of two solid spheres $A$ and $B$ of the same radii $R$ vary with radial distance $r$ as ${\rho }_A(r)=k\left(\frac{r}{R}\right)$ and ${\rho }_B(r)=k{\left(\frac{r}{R}\right)}^5$, respectively, where $k$ is a constant. The moments of inertia of the individual spheres a …

A uniform circular disc of mass $1.5kg$ and radius $0.5m$ is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $F=0.5N$ are applied simultaneously along the three sides of an equilateral triangle XYZ with its vertices on the perimet …

A uniform circular disc of mass $50kg$ and radius $0.4m$ is rotating with an angular velocity of $10rad{s}^{-1}$ about its own axis, which is vertical. Two uniform circular rings, each of mass 6.25 kg and radius $0.2m$, are gently placed symmetrically on the disc …

A thin uniform rod, pivoted at $O$, is rotating in the horizontal plane with constant angular speed $\omega$, as shown in the figure. At time $t=0$, a small insect starts from $O$ and moves with constant speed $v$ with respect to the rod towards the other end. It reache …

A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion in the $x-y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated …

A lamina is made by removing a small disc of diameter $2R$ from a bigger disc of uniform mass density and radius $2R$, as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_O$ and $I_P$, respectively. Both thes …

Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $O$. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown …

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin unifo …

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin unifo …

Two solid cylinders $P$ and $Q$ of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder $P$ has most of its mass concentrated near its surface, while $Q$ has most of its mass concentrated near the axis. Which …

The figure shows a system consisting of (i) a ring of outer radius $3R$ rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius $2R$ rotating anti-clockwise with angular speed $\omega /2$. The ring and dis …

A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of $0.3\mathrm{m}/{\mathrm{s}}^2$. The coefficient of friction between the ground and t …

Four solid spheres each of diameter $\sqrt{5}\mathrm{cm}$ and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is $\mathrm{N}\times 10^{-4}\mathrm{kg}-{\mathrm{m}}^2$, …

A binary star consists of two stars A (mass $2.2{\mathrm{M}}_{\mathrm{s}}$) and B (mass $11{\mathrm{M}}_{\mathrm{s}}$), where ${\mathrm{M}}_{\mathrm{s}}$ is the mass of the sun. They are separated by distance $d$ and are rotating about their centre of mass, which is stationary. The r …

STATEMENT-1
Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow cylinder will reach the bottom of the inclined plane fi …