Rotational Motion

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 $X Y$ plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass 20 g , trave...

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 $X Y$ plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass 20 g , trave...

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 $OCO'$, which lies in the plane of the structures, are given.
Ch...

A solid cylinder of radius R rolls without slipping with a center of mass speed v0 = $\sqrt{\frac{gR}{3}}$ on a horizontal surface with a vertical edge, as shown in the figure. Here, g is the acceleration due to gravity. At the moment when ...

Consider a large disk of radius R and two smaller disks, each of radius r = 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 Δθ between their ...

The center of a disk of radius $r$ and mass $m$ is attached to a spring of spring constant $k$, inside a ring of radius $R>r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and th...

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 $\mathrm...

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 rad...

A thin circular coin of mass $5 \mathrm{gm}$ and radius $4 / 3 \mathrm{~cm}$ is initially in a horizontal $x y$-plane. The coin is tossed vertically up ( $+z$ direction) by applying an impulse of $\sqrt{\frac{\pi}{2}} \times 10^{-2} \mathrm...

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 cen...

Two point-like objects of masses $20 ~\mathrm{gm}$ and $30 ~\mathrm{gm}$ are fixed at the two ends of a rigid massless rod of length $10 \mathrm{~cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to i...

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 th...

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 fla...

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $m s^{-1}$ ) at time $t=\frac{\pi}{3} s$.

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

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

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 ...

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 ...

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 ...

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...

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 statem...

A ring and disc are initially at rest, side by side, at the top of an inclined plane which makes an angle \({60^ \circ }\) with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the...

The potential energy of a particle of mass \(m\) at a distance \(r\) from a fixed point \(O\) is given by \(V\left( r \right) = k{r^2}/2,\) where \(k\) is a positive constant of appropriate dimensions. This particle is moving in a circular ...

Two vectors \(\overrightarrow A\) and \(\overrightarrow B\) are defined as \(\overrightarrow A\) \(=\) \(a\widehat i\) and \(\overrightarrow B = a\) \(\left( {\cos \,\omega T\widehat i + \sin \,\omega t\,\widehat j} \right),\) where $$...

Consider a body of mass \(1.0\) \(kg\) at rest at the origin at time \(t=0.\) A force \(\overrightarrow F = \left( {\alpha t \widehat i + \beta \widehat j} \right)\) is applied on the body, where \(\alpha = 1.0N{s^{ - 1}}\) and $$\beta =...

Consider regular polygons with number of sides \(n=3,4,5....\) 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 an...

The minimum value of \(\omega\)0 below which the ring will drop down is

The total kinetic energy of the ring is

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 statement...

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 t...

A block of mass \(M\) has a circular cut with a frictionless surface as shown. The block resets on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at \(x=0,\) in a co-ordinate system fixed to t...

A frame of the reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular veloci...

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 rol...

A frame of the reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular veloci...

The position vector \(\overrightarrow r\) of a particle of mass m is given by the following equation
\($\overrightarrow r \left( t \right) = \alpha {t^3}\widehat i + \beta {t^2}\widehat j,\)$where \(\alpha = {{10} \over 3}m{s^{ - 3}}\), $...

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 stic...

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

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 \({M \over 8}\) at rest at O. These masses can move radially outwards along ...

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds v1 and v2, respectively, and always remain in contact with the surfaces. If they reach B and D with th...

A horizontal circular platform of radius 0.5 m and mass 0.45 kg is free to rotate about its axis. Two massless spring toy-guns, each carrying a steel ball of mass 0.05 kg are attached to the platform at a distance 0.25 m from the centre on...

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

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

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 th...

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 $...

Two identical discs of same radius R are rotating about their axes in
opposite directions with the same constant angular speed \(\omega\). The
discs are in the same horizontal plane. At time t = 0, the points P
and Q are facing each other ...

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 concentrat...

Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?

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 IO and IP respectively. Bo...

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 towar...