Rotational Motion — JEE Main PYQs

Two blocks of masses 2 kg and 1 kg respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in 2 s is ________ m. (Take $g = 10 \text{ m/s}^2$)

[Figure: A pulley fixed to the ceiling with a string over it; a 2 kg block hangs on the left side and a 1 kg block hangs on the right side at the same horizontal level.]

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 value of $\alpha$ is ________.

[Figure: A vertical axis AB; a horizontal rod of length 3 m and mass $M = 40$ kg with one end at the axis; below it, a solid sphere of radius $R$ whose center is at a distance 3 m from the axis, with its own rotation axis parallel to AB.]

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 having radius $r$ about an axis through its centre and $I_2$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $I_2/I_1 =$ _____.

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first $2 \text{ s}$ it rotates through an angle $\theta_1$ and in the next $2 \text{ s}$ it rotates through an angle $\theta_2$. The ratio $\dfrac{\theta_2}{\theta_1}$ 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 $\dfrac{7 v_0^2}{10\, g}$ ($g$ = acceleration due to gravity), the object is a __________.

[Figure: an object at the bottom of a concave curved ramp moving with initial velocity $v_0$ toward the ramp, which curves upward to a vertical rise]

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 comes to rest are ______ and ________ respectively.

The position of center of mass of three masses 2 kg, 3 kg and 15 kg placed with respect to mid point ($p$) of normal bisector, as shown in the figure is ________.

[Figure: An isosceles triangle arrangement: 2 kg at the left base corner, 3 kg at the right base corner, and 15 kg at the apex; the two slant sides are 10 m each with $120°$ angle at the apex; point $p$ is the midpoint of the normal bisector from the apex to the base]

A solid sphere ($A$) of mass $5m$ and a 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 $a_A$ and $a_B$, respectively. The ratio of $a_A$ and $a_B$ is __________.

Two identical bodies A and B of equal masses have initial velocities $\vec{v}_1 = 4\hat{i}$ m/s and $\vec{v}_2 = 4\hat{j}$ m/s respectively. The body A has acceleration $\vec{a}_1 = 6\hat{i} + 6\hat{j}$ m/s$^2$ while the acceleration of the other body B is zero. The centre of mass of the two bodies moves in __________ path.

A solid cylinder having radius $R$ and length $L$ is slipping on a rough horizontal plane. At time $t = 0$ the cylinder has a translational velocity $v_0 = 49\,\text{m/s}$, perpendicular to its axis and a rotational velocity $v_0/4R$ about the centre. The time taken by the cylinder to start rolling is ________ seconds. (coefficient of kinetic friction $\mu_K = 0.25$ and $g = 9.8\,\text{m/s}^2$)

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m, would be :

[Figure: a wheel of radius R with a cord hanging from its rim, pulled downward with force F = 20 N]

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 :

[Figure: square OABC of side 10 cm with O at bottom-left, A at bottom-right, B at top-right, C at top-left; forces of 10 N act upward at C and B, rightward at B and A, downward at A, and force F acts leftward at C]

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 axis passing through its center and normal to plane containing cross is :

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 :

[Figure: a wheel (pulley) mounted at a fixed support with a string wrapped over its rim; the string hangs down and is pulled by a force of 10 N]

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 $2~\mathrm{rad/s^2}$. Moment of intertia of the wheel is _________ $\mathrm{kg~m^2}$.

(Acceleration due to gravity $= 10~\mathrm{m/s^2}$)

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

[Figure: a solid sphere of radius r resting on a rough horizontal surface with a horizontal force F = 49 N applied tangentially at its topmost point]

Which of the following are correct expression for torque acting on a body?

A. $\vec{\tau} = \vec{r} \times \vec{L}$

B. $\vec{\tau} = \frac{d}{dt}\left(\vec{r} \times \vec{p}\right)$

C. $\vec{\tau} = \vec{r} \times \frac{d\vec{p}}{dt}$

D. $\vec{\tau} = I\,\vec{\alpha}$

E. $\vec{\tau} = \vec{r} \times \vec{F}$

($\vec{r}$ = position vector; $\vec{p}$ = linear momentum; $\vec{L}$ = angular momentum; $\vec{\alpha}$ = angular acceleration; $I$ = moment of inertia; $\vec{F}$ = force; $t$ = time)

Choose the correct answer from the options given below:

If $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass '$m$' having position vector as $\vec{r} = a\left(\hat{i}\,\cos\omega t + \hat{j}\,\sin\omega t\right)$. The direction of force is

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

A solid sphere with uniform density and radius R is rotating initially with constant angular velocity ($\omega_1$) about its diameter. After some time during the rotation its starts loosing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become R/2 is $x\omega_1$. The value of $x$ is ________.

A rod of length 5 L is bent right angle keeping one side length as 2 L.

[Figure: an L-shaped bent rod on x–y axes; vertical arm of length 3L along the y-axis, horizontal arm of length 2L along the x-axis]

The position of the centre of mass of the system:

(Consider L = 10 cm)

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.

[Figure: three circles of equal radius touching each other — A (disc) on top, B (solid sphere) at bottom left and C (spherical shell) at bottom right; a vertical axis PQ passes through the centre of A and between B and C]

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

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 $\dfrac{4}{x}\,MR^2$. The value of $x$ is __________.

[Figure: a disc of radius R centred at O with vertical axis AB; a small disc of radius R/3 centred at O' is removed near the left edge, with O' on the horizontal diameter]

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 $25\pi$ Nm for 40s, the speed increases to 2100 rpm. The diameter of the disk is __________ m.

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 R/2 is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

[Figure: disc of radius R with a removed circular part of radius R/2 whose diameter spans from the centre of the disc to its rim]

The position vectors of two 1 kg particles, (A) and (B), are given by $\vec{r}_A = \left(\alpha_1 t^2 \hat{i} + \alpha_2 t \hat{j} + \alpha_3 t \hat{k}\right)$ m and $\vec{r}_B = \left(\beta_1 t \hat{i} + \beta_2 t^2 \hat{j} + \beta_3 t \hat{k}\right)$ m, respectively; $(\alpha_1 = 1\ \mathrm{m/s^2}$, $\alpha_2 = 3n\ \mathrm{m/s}$, $\alpha_3 = 2\ \mathrm{m/s}$, $\beta_1 = 2\ \mathrm{m/s}$, $\beta_2 = -1\ \mathrm{m/s^2}$, $\beta_3 = 4p\ \mathrm{m/s})$, where t is time, n and p are constants. At $t = 1$ s, $\left|\vec{V}_A\right| = \left|\vec{V}_B\right|$ and velocities $\vec{V}_A$ and $\vec{V}_B$ of the particles are orthogonal to each other. At $t = 1$ s, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{L}\ \mathrm{kg\,m^2 s^{-1}}$. The value of L is ________.

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

A tube of length 1m is filled completely with an ideal liquid of mass 2M, and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is F then angular velocity of the tube is $\sqrt{\dfrac{F}{\alpha M}}$ in SI unit. The value of $\alpha$ is __________.

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 power is delivered by the applied torque, when $t = 2$ s ?

An object of mass 'm' is projected from origin in a vertical xy plane at an angle $45°$ with the x-axis with an initial velocity $v_0$. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [g is acceleration due to gravity]

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

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

The center of mass of a thin rectangular plate (fig - x) with sides of length $a$ and $b$, whose mass per unit area ($\sigma$) varies as $\sigma = \frac{\sigma_0 x}{ab}$ (where $\sigma_0$ is a constant), would be _____

[Figure: fig-x — a rectangular plate in the x-y plane with one corner at origin O, side $a$ along the x-axis and side $b$ along the y-axis]

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 $1/x$, where $x$ = _____.

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 moment of inertia of the given ring. Here, n = _____. Consider all the bodies have equal masses.

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

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 $\mathrm{h_1} : \mathrm{h_2}$ is $\frac{n}{10}$. The value of n is ________.

In a system two particles of masses $m_1 = 3~kg$ and $m_2 = 2~kg$ are placed at certain distance from each other. The particle of mass $m_1$ is moved towards the center of mass of the system through a distance 2 cm. In order to keep the center of mass of the system at the original position, the particle of mass $m_2$ should move towards the center of mass by the distance ______ $cm$.

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 AB as shown in figure is $\sqrt{\frac{8}{x}}$. The value of $x$ is :

[Figure: a hollow sphere of mass M and radius R with diameter axis AB, and a solid cylinder of diameter 2R and length 4R with axis AB through its centre perpendicular to its length]

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 ________.

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _________ hours 30 minutes.

Three balls of masses 2kg, 4kg and 6kg respectively are arranged at centre of the edges of an equilateral triangle of side 2 m. The moment of intertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be ________ $\mathrm{kg\,m^2}$.

A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of x and y coordinates of center of mass of plate in $\frac{n}{9}$. The value of $n$ is ______.

[Figure: plate occupying the region from (0,0) to (3,2) with a rectangular notch cut from the top between (1,1), (1,2), (2,2) and (2,1)]

A thin circular disc of mass M and radius 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 dimensions but of mass $\mathrm{M}/2$ is placed gently on the first disc co-axially, then the new angular velocity of the system is :

A heavy iron bar, of weight W is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $\theta$ with the horizontal. The weight experienced by the person is :

A string is wrapped around the rim of a wheel of moment of inertia $0.40\ \mathrm{kgm^2}$ and radius 10 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 40 N. The angular velocity of the wheel after 10 s is $x$ rad/s, where $x$ is ________.

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

The identical spheres each of mass 2M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 4 m each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is $\dfrac{4\sqrt{2}}{x}$, where the value of $x$ is ________

A disc of radius R and mass 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 :

[Figure: a disc rolling horizontally with speed v towards an inclined plane; maximum height reached on the incline marked h]

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

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

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

A particle of mass m is projected with a velocity 'u' making an angle of $30°$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height h is :

Consider a Disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 rad/s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is ________ J.

[Figure: a disc of mass 5 kg and radius 2 m rotating at $\omega = 10~\mathrm{rad/s}$ about a vertical axis through its centre]

Two discs of moment of inertia $I_1 = 4\ kg\ m^2$ and $I_2 = 2\ kg\ m^2$, about their central axes & normal to their planes, rotating with angular speeds $10\ rad/s$ & $4\ rad/s$ respectively are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is ________ J.

A solid circular disc of mass 50 kg rolls along a horizontal floor so that its center of mass has a speed of 0.4 m/s. The absolute value of work done on the disc to stop it is __________ J.

A body of mass '$m$' is projected with a speed '$u$' making an angle of $45°$ with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as $\frac{\sqrt{2}\,mu^3}{Xg}$. The value of '$X$' is ________.

Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is $\frac{x}{20}\ kg\ m^2$, where the value of $x$ is ________.