Rotational Motion · Physics · MHT CET

Three point masses, each of mass ' $m$ ' are placed at the corners of an equilateral triangle of side ' $L$ '. The moment of inertia of the system about an axis passing through one of the vertices and parallel to the side joining other two vertices will be

Two spheres of equal masses, one of which is a thin spherical shell and the other solid sphere, have the same moment of inertia about their respective diameters. The ratio of their radii is

Two loops P and Q of radii $\mathrm{R}_1$ and $\mathrm{R}_2$ are made from uniform metal wire of same material. $I_p$ and $\mathrm{I}_{\mathrm{Q}}$ be the moment of inertia of loop P and Q respectively then ratio $R_1 / R_2$ is $\left(\right.$ Given $\left.I_P / I_Q=27\right)$

A body of mass $m$ slides down an incline and reaches the bottom with a velocity V . If the same mass were in the form of a disc which rolls down this incline, the velocity of the disc at bottom would have been

The radius of gyration of a circular disc of radius R and mass m rotating about diameter as axis is

A thin uniform metal rod of mass ' $M$ ' and length ' $L$ ' is swinging about a horizontal axis passing through its end. Its maximum angular velocity is ' $\omega$ '. Its centre of mass rises to a maximum height of ( $\mathrm{g}=$ Acceleration due to gravity)

Three thin rods, each of mass ' $M$ ' and length ' $L$ ' are placed along $\mathrm{X}, \mathrm{Y}$ and Z axes which are mutually perpendicular. One end of each rod is at origin. M. I. of the system about Z axis is

If the angular velocity of a body rotating about the given axis increases by $20 \%$, then its kinetic energy of rotation will increase by

Four thin metal rods each of mass ' $M$ ' and length ' $L$ ', are welded end to end to form a square. The moment of inertia of the system about an axis passing through the centre of the square and perpendicular to its plane is

Moment of inertia of a disc about an axis passing through its centre and perpendicular to its plane is 'I'. The ratio of moment of inertia about a parallel axis tangential to its rim to passing through a point midway between the centre and the rim is

An inclined plane makes an angle $30^{\circ}$ with horizontal. A solid sphere rolls down from the top of the inclined plane from rest without slipping has a linear acceleration along the plane equal to (where $g$ is acceleration due to gravity) (given $\sin 30^{\circ}=0.5$)

Two bodies A and B have their moments of inertia $I_1$ and $I_2$ respectively about their axis of rotation. If their kinetic energies of rotation are equal and their angular momenta $\mathrm{L}_1$ and $\mathrm{L}_2$ respectively are in the ratio $1: \sqrt{3}$, then $I_2$ will be

The moment of inertia of uniform circular disc is maximum about an axis perpendicular to the disc and passing through point

A ring and a disc roll on horizontal surface without slipping with same linear velocity. If both have same mass and total kinetic energy of the ring is 6 J then total kinetic energy of the disc is

A solid metallic sphere of radius ' $R$ ' having moment of inertia '$I$' about diameter is melted and recast into a solid disc of radius ' $r$ ' of a uniform thickness. The moment of inertia of a disc about an axis passing through its edge and perpendicular to its plane is also equal to '$I$'. The ratio $\frac{r}{R}$ is

A particle of mass ' m ' is rotating in a circular path of radius ' $r$ '. Its angular momentum is ' $L$ ' The centripetal force acting on it is ' $F$ '. The relation between ' $F$ ', ' $L$ ', ' $r$ ' and ' $m$ ' is

Three thin rods, each mass ' 2 M ' and length ' L ' are placed along $\mathrm{x}, \mathrm{y}$ and z axis which are mutually perpendicular. One end of each rod is at origin. Moment of inertia of the system about x - axis is

A thin uniform rod of length ' $L$ ' and mass ' $M$ ' is swinging freely along a horizontal axis passing through its centre. Its maximum angular speed is ' $\omega$ '. Its centre of mass rises to a maximum height of [ $\mathrm{g}=$ gravitational acceleration]

The moment of inertia of thin square plate PQRS of uniform thickness, about an axis passing through centre ' O ' and perpendicular to the plane of the plate is $\left(\mathrm{I}_1, \mathrm{I}_2, \mathrm{I}_3, \mathrm{I}_4\right.$ are respectively the moments of inertia about axis $1,2,3,4$ which are in the plane of the plate as shown in figure)

A circular disc of radius ' $R$ ' and thickness $\frac{R}{8}$ has moment of inertia 'I' about an axis passing through its centre and perpendicular to its plane. It is melted and recasted into a solid sphere then moment of inertia of sphere about an axis passing through diameter is

Two solid spheres ( A and B ) are made of metals having densities $\rho_A$ and $\rho_B$ respectively. If there masses are equal then ratio of their moments of inertia $\left(\frac{\mathrm{I}_{\mathrm{B}}}{\mathrm{I}_{\mathrm{A}}}\right)$ about their respective diameter is

A thin uniform circular disc of mass ' $M$ ' and radius ' $R$ ' is rotating with angular velocity ' $\omega$ ' in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of same radius but of mass $\left(\frac{\mathrm{M}}{3}\right)$ is placed gently on the first disc co-axially. The new angular velocity will be

A solid cylinder of mass ' $M$ ' and radius ' $R$ ' rolls down an inclined plane of height ' $h$ '. When it reaches the foot of the plane, its rotational kinetic energy is ( $\mathrm{g}=$ acceleration due to gravity)

A disc and a ring both have same mass and radius. The ratio of moment of inertia of the disc about its diameter to that of a ring about a tangent in its plane is

A rotating body has angular momentum ' $L$ '. If its frequency is doubled and kinetic energy is halved, its angular momentum will be

A solid cylinder of mass M and radius R is rotating about its geometrical axis. A solid sphere of the same mass and same radius is also rotating about its diameter with an angular speed half that of the cylinder. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder will be

The earth is assumed to be a sphere of radius ' $R$ ' and mass ' $M$ ' having period of rotation ' $T$ '. The angular momentum of earth about its axis of rotation is

Two loops ' $A$ ' and ' $B$ ' of radii ' $R_1$ ' and ' $R_2$ ' are made from uniform wire. If moment of inertia of ' A ' is ' $\mathrm{I}_{\mathrm{A}}$ ' and that ' B ' is ' $\mathrm{I}_{\mathrm{B}}$ ', then $\mathrm{R}_2 / \mathrm{R}_1$ is $\left[\frac{\mathrm{I}_{\mathrm{A}}}{\mathrm{I}_{\mathrm{B}}}=27\right]$

In case of rotational dynamics, which one of the following statements is correct?

[$\vec{\omega}=$ angular velocity, $\overrightarrow{\mathrm{v}}=$ linear velocity

$\overrightarrow{\mathbf{r}}=$ radius vector, $\vec{\alpha}=$ angular acceleration

$\overrightarrow{\mathrm{a}}=$ linear acceleration, $\overrightarrow{\mathrm{L}}=$ angular momentum

$\overrightarrow{\mathrm{p}}=$ linear momentum, $\bar{\tau}=$ torque,

$\overrightarrow{\mathrm{f}}=$ centripetal force]

Ratio of radius of gyration of a circular disc to that of circular ring each of same mass and radius around their respective axes is

Two circular loops P and Q of radii ' r ' and ' nr ' are made respectively from a uniform wire. Moment of inertia of loop Q about its axis is four times that of loop P about its axis. The value of ' $n$ ' is

A solid sphere of mass ' $m$ ', radius ' $R$ ', having moment of inertia about an axis passing through center of mass as 'I' is recast into a disc of thickness ' $t$ ' whose moment of inertia about an axis passing through the rim (edge) \& perpendicular to plane remains 'I'. Then the radius of disc is

An inclined plane makes an angle of $30^{\circ}$ with the horizontal. A solid sphere rolling down this inclined plane from rest without slipping has a linear acceleration ( $\mathrm{g}=$ acceleration due to gravity, $\sin 30^{\circ}=0.5$ )

An annular ring has mass 10 kg and inner and outer radii are 10 m and 5 m respectively. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is

Four identical uniform solid spheres each of same mass '$$M$$' and radius '$$R$$' are placed touching each other as shown in figure, with centres A, B, C, D. $$\mathrm{I}_{\mathrm{A}}, \mathrm{I}_{\mathrm{B}}, \mathrm{I}_{\mathrm{C}}$$ and $$\mathrm{I}_{\mathrm{D}}$$ are the moment of inertia of these spheres respectively about an axis passing through centre and perpendicular to the plane. The difference in $$\mathrm{I}_{\mathrm{A}}$$, and $$\mathrm{I}_{\mathrm{B}}$$ is

A solid cylinder and a solid sphere having same mass and same radius roll down on the same inclined plane. The ratio of the acceleration of the cylinder '$$a_c$$' to that of sphere '$$a_s$$' is

A mass '$$M$$' is moving with constant velocity parallel to $$\mathrm{X}$$-axis. Its angular momentum with respect to the origin is

A thin uniform rod of mass '$$m$$' and length '$$P$$' is suspended from one end which can oscillate in a vertical plane about the point of intersection. It is pulled to one side and then released. It passes through the equilibrium position with angular speed '$$\omega$$'. The kinetic energy while passing through mean position is

Four identical uniform solid spheres each of same mass $$M$$ and radius $$R$$ are placed touching each other as shown in figure with centres $$A, B, C, D. I_A, I_B, I_C, I_D$$ are the moment of inertia of these spheres respectively about an axis passing through centre and perpendicular to the plane, then

A thin uniform $$\operatorname{rod} A B$$ of mass $$m$$ and length $$l$$ is hinged at one end $$A$$ to the ground level. Initially the rod stands vertically and is allowed to fall freely to the ground in the vertical plane. The angular velocity of the rod when its end $$B$$ strikes the ground is ( $$g=$$ acceleration due to gravity)

A rigid body rotates with an angular momentum L. If its rotational kinetic energy is made four times, its angular momentum will become

The rotational kinetic energy and translational kinetic energy of a rolling body are same, the body is

A solid cylinder of mass $$3 \mathrm{~kg}$$ is rolling on a horizontal surface with velocity $$4 \mathrm{~m} / \mathrm{s}$$. It collides with a horizontal spring whose one end is fixed to rigid support. The force constant of material of spring is $$200 \mathrm{~N} / \mathrm{m}$$. The maximum compression produced in the spring will be (assume collision between cylinder & spring be elastic)

A thin wire of length '$$L$$' and uniform linear mass density '$$m$$' is bent into a circular coil. The moment of inertia of this coil about tangential axis and in plane of the coil is

In $$\mathrm{P}^{\text {th }}$$ second, a particle describes angular displacement of '$$\beta$$' rad. If it starts from rest, the angular acceleration is

$$I_1$$ is the moment of inertia of a circular disc about an axis passing through its centre and perpendicular to the plane of disc. $$I_2$$ is its moment of inertia about an axis $$A B$$ perpendicular to plane and parallel to axis $$\mathrm{CM}$$ at a distance $$\frac{2 R}{3}$$ from centre. The ratio of $$I_1$$ and $$I_2$$ is $$x: 17$$. The value of '$$x$$' is (R = radius of the disc)

A thin uniform circular disc of mass '$$\mathrm{M}$$' and radius '$$R$$' is rotating with angular velocity '$$\omega$$', in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of same radius but of mass $$\left(\frac{M}{2}\right)$$ is placed gently on the first disc co-axially. The new angular velocity will be

Two bodies have their moments of inertia I and 2I respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the ratio

From a disc of mass '$$M$$' and radius '$$R$$', a circular hole of diameter '$$R$$' is cut whose rim passes through the centre. The moment of inertia of the remaining part of the disc about perpendicular axis passing through the centre is

A particle moves along a circular path with decreasing speed. Hence

Radius of gyration of a thin uniform circular disc about the axis passing through its centre and perpendicular to its plane is $$\mathrm{K}_{\mathrm{c}}$$. Radius of gyration of the same disc about a diameter of the disc is $$K_d$$. The ratio $$K_c: K_d$$ is

A disc has mass $$M$$ and radius $$R$$. How much tangential force should be applied to the rim of the disc, so as to rotate with angular velocity '$$\omega$$' in time $$\mathrm{t}$$ ?

What is the moment of inertia of the electron moving in second Bohr orbit of hydrogen atom? [ $$\mathrm{h}=$$ Planck's constant, $$\mathrm{m}=$$ mass of electron, $$\varepsilon_0=$$ permittivity of free space, $$\mathrm{e}=$$ charge on electron]

Two spheres each of mass '$$M$$' and radius $$\frac{R}{2}$$ are connected at the ends of massless rod of length '$$2 R$$'. What will be the moment of inertia of the system about an axis passing through centre of one of the spheres and perpendicular to the rod?

The moment of inertia of a uniform square plate about an axis perpendicular to its plane and passing through the centre is $$\frac{\mathrm{Ma}^2}{6}$$, where '$$M$$' is the mass and '$$a$$' is the side of square plate. Moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

If 'I' is moment of inertia of a thin circular disc about an axis passing through the tangent of the disc and in the plane of disc. The moment of inertia of same circular disc about an axis perpendicular to plane and passing through its centre is

Seven identical discs each of mass $$M$$ and radius $$\mathrm{R}$$ are arranged in a hexagonal plane pattern so as to touch each neighbour disc as shown in the figure. The moment of inertia of the system of seven discs about an axis passing through the centre of central disc and normal to the plane of all discs is

A particle of mass '$$\mathrm{m}$$' is rotating along a circular path of radius '$$r$$' having angular momentum '$$L$$'. The centripetal force acting on the particle is given by

A disc of radius $$R$$ and thickness $$\frac{R}{6}$$ has moment of inertia I about an axis passing through its centre and perpendicular to its plane. Dise is melted and recast into a solid sphere. The moment of inertia of a sphere about its diameter is

A square lamina of side '$$b$$' has same mass as a disc of radius '$$R$$' the moment of inertia of the two objects about an axis perpendicular to the plane and passing through the centre is equal. The ratio $$\frac{b}{R}$$ is

A solid sphere rolls without slipping on an inclined plane at an angle $$\theta$$. The ratio of total kinetic energy to its rotational kinetic energy is

Two discs of same mass and same thickness (t) are made from two different materials of densities '$$d_1$$' and '$$d_2$$' respectively. The ratio of the moment of inertia $$I_1$$ to $$I_2$$ of two discs about an axis passing through the centre and perpendicular to the plane of disc is

The moment of inertia of a thin uniform rod of mass 'M' and length 'L' about an axis passing through a point at a distance $$\frac{L}{4}$$ from one of its ends and perpendicular to the length of the rod is

Two identical particles each of mass '$$m$$' are separated by a distance '$$d$$'. The axis of rotation passes through the midpoint of '$$\mathrm{d}$$' and is perpendicular to the length $$\mathrm{d}$$. If '$$\mathrm{K}$$' is the average rotational kinetic energy of the system, then the angular frequency is

A body of mass '$$\mathrm{m}$$' and radius of gyration '$$\mathrm{K}$$' has an angular momentum $$\mathrm{L}$$. Its angular velocity is

The moment of inertia of a body about the given axis, rotating with angular velocity 1 rad/s is numerically equal to 'P' times its rotational kinetic energy. The value of 'P' is

Two circular loops P and Q are made from a uniform wire. The radii of P and Q are R$$_1$$ and R$$_2$$ respectively. The momentsw of inertia about their own axis are $$\mathrm{I_P}$$ and $$\mathrm{I_Q}$$ respectively. If $$ \frac{\mathrm{I}_{\mathrm{P}}}{\mathrm{I}_Q}=\frac{1}{8}$$ then $$\mathrm{\frac{R_2}{R_1}}$$ is

A metre scale is supported on a wedge at its centre of gravity. A body of weight 'w'. is suspended from the $$20 \mathrm{~cm}$$ mark and another weight of 25 gram is suspended from $$74 \mathrm{~cm}$$ mark balance it and the metre scale remains perfectly horizontal. Neglecting the weight of the metre scale, the weight of the body is

A body of mass 'm' and radius of gyration 'K' has an angular momentum 'L'. Then its angular velocity is

A molecule consists of two atoms each of mass '$$m$$' and separated by a distance '$$\mathrm{d}$$'. At room temperature, if the average rotational kinetic energy is '$$E$$' then the angular frequency is

Three solid spheres each of mass '$$M$$' and radius '$$R$$' are arranged as shown in the figure. The moment of inertia of the system about YY' will be

Figure shows triangular lamina which can rotate about different axes moment of inertia is maximum, about the axis

A particle with position vector $$\overrightarrow{\mathrm{r}}$$ has a linear momentum $$\overrightarrow{\mathrm{P}}$$. Which one of the following statements is true in respect of its angular momentum 'L' about the origin?

A child is standing with folded hands at the centre of the platform rotating about its central axis. The kinetic energy of the system is '$$K$$'. The child now stretches his arms so that the moment of inertia of the system becomes double. The kinetic energy of the system now is

Two rings of radius 'R' and 'nR' made of same material have the ratio of moment of inertia about an axis passing through its centre and perpendicular to the plane is $$1: 8$$. The value of '$$n$$' is (mass per unit length $$=\lambda$$)

Two rotating bodies $$P$$ and $$Q$$ of masses '$$\mathrm{m}$$' and '$$2 \mathrm{~m}$$' with moment of inertia $$I_P$$ and $$I_Q\left(I_Q > I_P\right)$$ have equal Kinetic energy of rotation. If $$\mathrm{L}_P$$ and $$\mathrm{L}_Q$$ be their angular momenta respectively then

A solid sphere of mass '$$M$$' and radius '$$R$$' is rotating about its diameter. A solid cylinder of same mass and same radius is also rotating about its geometrical axis with an angular speet twice that of the sphere. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder is

A particle performs rotational motion with an angular momentum 'L'. If frequency of rotation is doubled and its kinetic energy becomes one fourth, the angular momentum becomes.

Three rings each of mass 'M' and radius 'R' are arranged as shown in the figure. The moment of inertia of system about axis YY' will be

The moment of inertia of a circular disc of radius $$2 \mathrm{~m}$$ and mass $$1 \mathrm{~kg}$$ about an axis XY passing through its centre of mass and perpendicular to the plane of the disc is $$2 \mathrm{~kg} \mathrm{~m}^2$$. The moment of inertia about an axis parallel to the axis $$\mathrm{XY}$$ and passing through the edge of the disc is

The moment of inertia of a body about a given axis is $$1.2 \mathrm{~kg} / \mathrm{m}^3$$. Initially the body is at rest. In order to produce rotational kinetic energy of $$1500 \mathrm{~J}$$, an angular acceleration of $$25 \mathrm{rad} / \mathrm{s}^2$$ must be applied about an axis for a time duration of

A disc of radius 0.4 metre and mass 1 kg rotates about an axis passing through its centre and perpendicular to its plane. The angular acceleration is 10 rad s$$^{-2}$$. The tangential force applied to the rim of the disc is

The ratio of radii of gyration of a circular ring and circular disc of the same mass and radius, about an axis passing through their centres and perpendicular to their planes is

A solid sphere of mass $$\mathrm{M}$$, radius $$\mathrm{R}$$ has moment of inertia '$$\mathrm{I}$$' about its diameter. It is recast into a disc of thickness 't' whose moment of inertia about an axis passing through its edge and perpendicular to its plane remains 'I'. Radius of the disc will be

Two bodies rotate with kinetic energies 'E$$_1$$' and 'E$$_2$$'. Moments of inertia about their axis of rotation are 'I$$_1$$' and 'I$$_2$$'. If $$\mathrm{I_1=\frac{I_2}{3}}$$ and E$$_1$$ = 27 E$$_2$$, then the ratio of angular momenta 'L$$_1$$' to 'L$$_2$$' is

A disc of radius 0.4 m and mass one kg rotates about an axis passing through its centre and perpendicular to its plane. The angular acceleration of the disc is 10 rad/s$$^2$$. The tangential force applied to the rim of the disc is

Three points masses, each of mass $m$ are placed at the corners of an equilateral triangle of side $\ell$. The moment of inertia of the system about an axis passing through one of the vertices and parallel to the side joining other two vertices, will be

A rope is wound around a solid cylinder of mass 1 kg and radius 0.4 m . What is the angular acceleration of cylinder, if the rope is pulled with a force of 25 N ? (Cylinder is rotating about its own axis.)

Two circular loop $A$ and $B$ of radii $R$ and $N R$ respectively are made from a uniform wire. Moment of inertia of $B$ about its axis is 3 times that of $A$ about its axis. The value of $N$ is

A body slides down a smooth inclined plane having angle $$\theta$$ and reaches the bottom with velocity $$v$$. If a body is a sphere, then its linear velocity at the bottom of the plane is

A thin uniform rod has mass $$M$$ and length $$L$$ The moment of inertia about an axis perpendicular to it and passing through the point at a distance $$\frac{L}{3}$$ from one of its ends, will be

The ratio of radii of gyration of a ring to a disc (both circular) of same radii and mass, about a tangential axis perpendicular to the plane is

If there is a change of angular momentum from $$1 \mathrm{j}$$-$$\mathrm{s}$$ to $$4 \mathrm{j}$$-$$\mathrm{s}$$ in $$4 \mathrm{~s}$$, then the torque

A solid cylinder of radius $$r$$ and mass $$M$$ rolls down an inclined plane of height $$h$$. When it reaches the bottom of the plane, then its rotational kinetic energy is ($$g=$$ acceleration due to gravity)

A rod $l \mathrm{~m}$ long is acted upon by a couple as shown in the figure. The moment of couple is $\tau \mathrm{~Nm}$. If the force at each end of the rod, then magnitude of each force is

$$\left(\sin 30^{\circ}=\cos 60^{\circ}=0.5\right)$$

A solid sphere rolls down from top of inclined plane, 7 m high, without slipping. Its linear speed at the foot of plane is $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$

Three identical rods each of mass ' $M$ ' and length ' $L$ ' are joined to form a symbol ' $H$. The moment of inertia of the system about one of the sides of ' $H$ ' is

Three point masses each of mass ' $m$ ' are kept at the corners of an equilateral triangle of side. The system rotates about the center of the triangle without any change in the separation of masses during rotation. The period of rotation is directly proportional to $\left(\cos 30^{\circ}=\sin 60^{\circ}=\frac{\sqrt{3}}{2}\right)$

When a 12000 joule of work is done on a flywheel, its frequency of rotation increases from 10 Hz to 20 Hz . The moment of inertia of flywheel about its axis of rotation is $\left(\pi^2=10\right)$

A rigid body is rotating with angular velocity ' $\omega$ ' about an axis of rotation. Let $v$ ' be the linear velocity of particle which is at perpendicular distance ' $r$ ' from the axis of rotation. Then the relation $v=r \omega$ ' implies that

If radius of the solid sphere is doubled by keeping its mass constant, the ratio of their moment of inertia about any of its diameter is

A uniform rod of length ' 6 L ' and mass ' 8 m ' is pivoted at its centre ' $C$ '. Two masses ' $m$ ' and ' $2 m^{\prime}$ with speed $2 v, v$ as shown strikes the rod and stick to the rod. Initially the rod is at rest. Due to impact, if it rotates with angular velocity ' $\omega$ ' then ' $\omega$ ' will be

A thin metal wire of length 'L' and uniform linear mass density '$\rho$' is bent into a circular coil with 'O' as centre. the moment of inertia of a coil about the axis XX' is

The dimensions of torque are same as that of