Rotational Motion

The position vectors of two 1 kg particles, (A) and (B), are given by

$$
\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m} \text { and } \...

The torque due to the force $(2 \hat{i}+\hat{j}+2 \hat{k})$ about the origin, acting on a particle whose position vector is $(\hat{i}+\hat{j}+\hat{k})$, would be

A string is wrapped around the rim of a wheel of moment of inertia \(0.40 \mathrm{~kgm}^2\) and radius \(10 \mathrm{~cm}\). The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of $...

A circular disc reaches from top to bottom of an inclined plane of length \(l\). When it slips down the plane, if takes \(t \mathrm{~s}\). When it rolls down the plane then it takes \(\left(\frac{\alpha}{2}\right)^{1 / 2} t \mathrm{~s}\), w...

A circular table is rotating with an angular velocity of \(\omega \mathrm{~rad} / \mathrm{s}\) about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of $$1 \...

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 dimens...

Three balls of masses \(2 \mathrm{~kg}, 4 \mathrm{~kg}\) and \(6 \mathrm{~kg}\) respectively are arranged at centre of the edges of an equilateral triangle of side \(2 \mathrm{~m}\). The moment of inertia of the system about an axis through...

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis \(A B\) as shown in figure is \(\sqrt{8 / x}\). The value of \(x\) is :

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 _________.

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed \(v\). The sphere and the cylinder reaches upto maximum heights \(h_1\) and \(h_2\) respectively, above the initial level. The rati...

Two identical spheres each of mass \(2 \mathrm{~kg}\) and radius \(50 \mathrm{~cm}\) are fixed at the ends of a light rod so that the separation between the centers is \(150 \mathrm{~cm}\). Then, moment of inertia of the system about an axi...

A body of mass '\(m\)' is projected with a speed '\(u\)' making an angle of \(45^{\circ}\) with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as \(\frac{\sqrt{2} m u^3}{X g}\)....

Consider a Disc of mass \(5 \mathrm{~kg}\), radius \(2 \mathrm{~m}\), rotating with angular velocity of \(10 \mathrm{~rad} / \mathrm{s}\) about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotat...

A particle of mass \(\mathrm{m}\) is projected with a velocity '\(\mathrm{u}\)' making an angle of \(30^{\circ}\) with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at...

Two discs of moment of inertia \(I_1=4 \mathrm{~kg} \mathrm{~m}^2\) and \(I_2=2 \mathrm{~kg} \mathrm{~m}^2\), about their central axes & normal to their planes, rotating with angular speeds \(10 \mathrm{~rad} / \mathrm{s}\) & $$4 \mathrm{~r...

A cylinder is rolling down on an inclined plane of inclination \(60^{\circ}\). It's acceleration during rolling down will be \(\frac{x}{\sqrt{3}} m / s^2\), where \(x=\) ________ (use \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\)).

A body of mass \(5 \mathrm{~kg}\) moving with a uniform speed \(3 \sqrt{2} \mathrm{~ms}^{-1}\) in \(X-Y\) plane along the line \(y=x+4\). The angular momentum of the particle about the origin will be _________ $$\mathrm{kg} \mathrm{~m}^2 \m...

Four particles each of mass \(1 \mathrm{~kg}\) are placed at four corners of a square of side \(2 \mathrm{~m}\). Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is _____ $$\mathrm{k...

A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical and the ratio of their kinetic energies is \(\frac{7}{x}\), where \(x\) is _________.

A heavy iron bar of weight \(12 \mathrm{~kg}\) is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle \(60^{\circ}\) with the horizontal, the weight experienced by the man is :

A uniform rod $A B$ of mass $2 \mathrm{~kg}$ and length $30 \mathrm{~cm}$ at rest on a smooth horizontal surface. An impulse of force $0.2 \mathrm{~Ns}$ is applied to end B. The time taken by the rod to turn through at right angles will be ...

A disc of radius $\mathrm{R}$ and mass $\mathrm{M}$ is rolling horizontally without slipping with speed $v$. It then moves up an inclined smooth surface as shown in figure. The maximum height that the disc can go up the incline is :

The moment of inertia of a semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is \(\frac{1}{x} \mathrm{MR}^{2}\), where \(\mathrm{R}\) is the radius and \(M\) is the mass of the semicircular ...

A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity \(3 \mathrm{~m} / \mathrm{s}\) (as shown in figure). Maximum height with respect to the initial position covered by it will be __________ cm.

Two identical solid spheres each of mass \(2 \mathrm{~kg}\) and radii \(10 \mathrm{~cm}\) are fixed at the ends of a light rod. The separation between the centres of the spheres is \(40 \mathrm{~cm}\). The moment of inertia of the system ab...

A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is \(\sqrt{\frac{2}{x}}\). The...

A solid sphere of mass \(1 \mathrm{~kg}\) rolls without slipping on a plane surface. Its kinetic energy is \(7 \times 10^{-3} \mathrm{~J}\). The speed of the centre of mass of the sphere is __________ \(\operatorname{cm~s}^{-1}\)

Two discs of same mass and different radii are made of different materials such that their

thicknesses are $1 \mathrm{~cm}$ and $0.5 \mathrm{~cm}$ respectively. The densities of materials are in the ratio $3: 5$. The

moment of inertia of ...

A thin uniform rod of length \(2 \mathrm{~m}\), cross sectional area '\(A\)' and density '\(\mathrm{d}\)' is rotated about an axis passing through the centre and perpendicular to its length with angular velocity \(\omega\). If value of $$\o...

A uniform disc of mass $0.5 \mathrm{~kg}$ and radius $r$ is projected with velocity $18 \mathrm{~m} / \mathrm{s}$ at $\mathrm{t}=0$ s on a rough horizontal surface. It starts off with a purely sliding motion at $\mathrm{t}=0 \mathrm{~s}$. A...

A solid sphere of mass 2 kg is making pure rolling on a horizontal surface with kinetic energy 2240 J. The velocity of centre of mass of the sphere will be _______ ms\(^{-1}\).

A particle of mass 100 g is projected at time t = 0 with a speed 20 ms\(^{-1}\) at an angle 45\(^\circ\) to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time t = 2s...

\(\mathrm{I_{CM}}\) is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc. \(\mathrm{I_{AB}}\) is it's moment of inertia about an axis AB perpendicular to plane and ...

An object of mass 8 kg is hanging from one end of a uniform rod CD of mass 2 kg and length 1 m pivoted at its end C on a vertical wall as shown in figure. It is supported by a cable AB such that the system is in equilibrium. The tension in ...

If a solid sphere of mass 5 kg and a disc of mass 4 kg have the same radius. Then the ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be \(\frac{x}{7}\). The ...

Solid sphere A is rotating about an axis PQ. If the radius of the sphere is 5 cm then its radius of gyration about PQ will be \(\sqrt x\) cm. The value of \(x\) is ________.

A uniform solid cylinder with radius R and length L has moment of inertia I\(_1\), about the axis of the cylinder. A concentric solid cylinder of radius \(R'=\frac{R}{2}\) and length \(L'=\frac{L}{2}\) is carved out of the original cylinder...

A solid cylinder is released from rest from the top of an inclined plane of inclination \(30^{\circ}\) and length \(60 \mathrm{~cm}\). If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is _____...

Moment of inertia of a disc of mass '\(M\)' and radius '\(R\)' about any of its diameter is \(\frac{M R^{2}}{4}\). The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $$\frac{...

A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively $\left(k_{\text {sph }}: k_{\text {cyl }}\right)$ is $2: \sqrt{x}$. The va...

A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is \(\pi: 22\) then, the value of its angular speed will be _______...

A disc is rolling without slipping on a surface. The radius of the disc is \(R\). At \(t=0\), the top most point on the disc is \(\mathrm{A}\) as shown in figure. When the disc completes half of its rotation, the displacement of point A fro...

A light rope is wound around a hollow cylinder of mass 5 kg and radius 70 cm. The rope is pulled with a force of 52.5 N. The angular acceleration of the cylinder will be _________ rad s\(^{-2}\).

For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is \(\frac{x}{5}\). The value of \(x\) is ___________.

A solid sphere of mass \(500 \mathrm{~g}\) and radius \(5 \mathrm{~cm}\) is rotated about one of its diameter with angular speed of \(10 ~\mathrm{rad} ~\mathrm{s}^{-1}\). If the moment of inertia of the sphere about its tangent is $$x \time...

A circular plate is rotating in horizontal plane, about an axis passing through its center and perpendicular to the plate, with an angular velocity \(\omega\). A person sits at the center having two dumbbells in his hands. When he stretches...

A force of \(-\mathrm{P} \hat{\mathrm{k}}\) acts on the origin of the coordinate system. The torque about the point \((2,-3)\) is \(\mathrm{P}(a \hat{i}+b \hat{j})\), The ratio of \(\frac{a}{b}\) is \(\frac{x}{2}\). The value of \(x\) is -

Given below are two statements: one is labelled as Assertion \(\mathbf{A}\) and the other is labelled as Reason \(\mathbf{R}\)
Assertion A : An electric fan continues to rotate for some time after the current is switched off.
Reason R : Fan...

Four particles with a mass of 1 kg, 2 kg, 3 kg and 4 kg are situated at the corners of a square with side 1 m (as shown in the figure). The moment of inertia of the system, about an axis passing through the point O and perpendicular to the ...

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 \({a \over 3}\) R2\(\omega\). The value of a...