Rotational Mechanics — JEE Main PYQs

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is ______ kg.m².

A circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2}=2$ then $\frac{T_1}{T_2}=\frac{1}{\alpha }$. …

Two masses m and 2m are connected by a light string going over a pulley (disc) of mass 30m with radius $r=0.1\mathrm{m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The 2m mass is released from rest and its speed when it has descended thr …

Two cars A and B each of mass $10^3\mathrm{kg}$ are moving on parallel tracks separated by a distance of 10 m , in same direction with speeds $72\mathrm{km}/\mathrm{h}$ and $36\mathrm{km}/\mathrm{h}$. The magnitude of angular momentum of car A with respect to ca …

The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M . Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in v …

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length $L$ ($R<L$) about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as M):

Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is …

Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by $\frac{X}{256}M{r}^2$. The value of x is $\_\_\_\_$.

Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm. When released from rest the heavier mass is observed to fall 81 cm in 9 s. The rotational inertia of the pulley is ______ $\mathrm{kg}.{\mathrm{m}}^2$.
( …

Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure.
If the mass of each disc is 600 gm and applied torque between two discs is $43\times 10^5$ dyne cm, the angular acceleration of the discs about the …

A solid sphere of radius 10 cm is rotating about an axis which is at a distance 15 cm from its centre. The radius of gyration about this axis is $\sqrt{n}\mathrm{cm}$. The value of $n$ is ____.

A thin uniform rod (X) of mass M and length L is pivoted at a height $\left(\frac{L}{3}\right)$ as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is ____. ( …

A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $\mathrm{R}/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio …

A fly wheel having mass 3 kg and radius 5 m is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to a 3 kg mass. The mass is kept at rest initially and released. Kinetic energy of the whe …

A uniform rod of mass m and length $l$ suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is ______. (g acceleration due to gravity)

Two identical thin rods of mass M kg and length L m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is $\frac{X}{12}M{L}^2\mathrm{kg}{\mathrm{m}}^2$. The value of …

Moment of inertia about an axis AB for a rod of mass 40 kg and length 3 m is same as that of a solid sphere of mass of 10 kg and radius R about an axis parallel to AB axis with separation of 3 m as shown in figure below. The value of R is given as $\sqrt{\frac{\alpha }{2}}$. The …

A solid sphere of mass M and radius R is divided into two unequal parts. The smaller part having mass $M/8$ is converted into a sphere of radius $r$ and the larger part is converted into a circular disc of thickness $t$ and radius $2R$. If $I_1$ is moment of inertia of a sphere …

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first 2 s it rotates through an angle ${\theta }_1$ and in the next 2 s it rotates through an angle ${\theta }_2$. The ratio $\frac{{\theta }_2}{{\theta }_1}$ is ______.

An object of uniform density rolls up the curved path with the initial velocity $v_0$ as shown in the figure. If the maximum height attained by an object is $\frac{7v_0^2}{10g}$ ($g=$ acceleration due to gravity), the object is a ______.

A solid sphere of radius 4 cm and mass 5 kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of 1200 rpm. It is brought to rest in 10 s by applying a constant torque. The torque applied and the number of rotations it made before it …

A solid sphere (A) of mass, 5 m and spherical shell (B) of mass $m$, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of A and B, they start rolling without slipping with an acceleration of …

A solid cylinder having radius R and length L is slipping on a rough horizontal plane. At time $\mathrm{t}=0$ the cylinder has a translational velocity ${\mathrm{v}}_0=49\mathrm{m}/\mathrm{s}$, perpendicular to its axis and a rotational velocity …

The position of an object having mass 0.1 kg as a function of time $t$ is given as $\vec{r}=\left(10t^2\hat{i}+5t^3\hat{j}\right)\mathrm{m}$. At $\mathrm{t}=1\mathrm{s}$, which of the following statements are correct?
A. The linear momentum …

A uniform circular disc of radius 'R' and mass 'M' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius $\mathrm{R}/2$ is removed from the original disc as shown in the figure. Find the moment of inertia of the rem …

The position vectors of two 1 kg particles, (A) and (B), are given by
${\overrightarrow{\mathrm{r}}}_{\mathrm{A}}=\left({\alpha }_1{\mathrm{t}}^2\hat{\mathrm{i}}+{\alpha }_2\hat{\mathrm{t}}\hat{\mathrm{j}}+{\alpha }_{3}t\hat{\mathrm{k}}\right)\mathrm{m}$
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 circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that $\theta (t)=5t^2-8t$, where $\theta (t)$ is the angular position of the rotating disc as a function of time $t$.
How much p …

The coordinates of a particle with respect to origin in a given reference frame is $(1,1,1)$ meters. If a force of $\vec{F}=\hat{i}-\hat{j}+\hat{k}$ acts on the particle, then the magnitude of torque (with respect to origin) in z -direction is $\_\_\_\_$ .

A uniform rod of mass 250 g having length 100 cm is balanced on a sharp edge at 40 cm mark. A mass of 400 g is suspended at 10 cm mark. To maintain the balance of the rod, the mass to be suspended at 90 cm mark, is

The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is $n$ times higher than the …

Two iron solid discs of negligible thickness have radii $R_1$ and $R_2$ and moment of intertia $I_1$ and $I_2$, respectively. For $R_2=2R_1$, the ratio of $I_1$ and $I_2$ would be $1/\mathrm{x}$, where $\mathrm{x}=$ $\_\_\_\_$

A solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is :

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be $t_1$ and $t_2$, respectively, then

An object of mass 'm' is projected from origin in a vertical xy plane at an angle $45^{\circ }$ with the x-axis with an initial velocity ${\mathrm{v}}_0$. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum heigh …

A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination $45^{\circ }$. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be :-

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is :

A wheel of radius 0.2 m rotates freely about its center when a string that is wrapped over its rim is pulled by force of 10 N as shown in figure. The established torque produces an angular acceleration of $2\mathrm{rad}/{\mathrm{s}}^2$. Moment of inertia of the wheel is ______ …

A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg , kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is

Which of the following are correct expression for torque acting on a body?
A. $\vec{\tau }=\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{L}}$
B. $\vec{\tau }=\frac{d}{dt}(\vec{r}\times \vec{p})$
C. …

A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{x}{5}}$ where $x=$ ______.

$M$ and $R$ be the mass and radius of a disc. A small disc of radius $R/3$ is removed from the bigger disc as shown in figure. The moment of inertia of remaining part of bigger disc about an axis AB passing through the centre O and perpendicular to the plane of disc is …

A, B and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about PQ is $\frac{x}{15}I$, where $I$ is the moment of inertia of the disc about its diameter. Th …

A rod of linear mass density '$\lambda$' and length '$L$' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is :

A thin solid disk of 1 kg is rotating along its diameter axis at the speed of 1800 rpm. By applying an external torque of $25\pi \mathrm{Nm}$ for 40 s, the speed increases to 2100 rpm. The diameter of the disk is ______ m.

A wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is $8\mathrm{m}/\mathrm{s}$. The speed of the particle on the rim of the wheel at the same level as the centre of wheel, will be :

A square Lamina OABC of length 10 cm is pivoted at 'O'. Forces act at Lamina as shown in figure. If Lamina remains stationary, then the magnitude of $F$ is:

Moment of inertia of a rod of mass 'M' and length 'L' about an axis passing through its center and normal to its length is '$\alpha$'. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axi …

Three point masses of 2 kg, 4 kg and 6 kg are placed at the corners of an equilateral triangle of side 2 m. The moment of inertia of the system about an axis passing through the centroid and perpendicular to the plane of the triangle is ______ ${\mathrm{kgm}}^2$.

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 ${\mathrm{h}}_1$ and ${\mathrm{h}}_2$, respectively, above the initial level. The ratio …