Rotational Motion

Match List-I with List-II

Choose the correct answer from the options given below :

The position vector of 1 kg object is $\vec r=\left(3\hat i-\hat j\right)$ m and its velocity $\vec v=\left(3\hat j+\hat k\right)$ ms$^{-1}$. The magnitude of its angular momentum is $\sqrt x$ Nm where $x$ is ______.

A ball is spun with angular acceleration $\alpha=6t^2-2t$ where $t$ is in second and $\alpha$ is in rads$^{-2}$. At $t=0$, the ball has angular velocity of 10 rads$^{-1}$ and angular position of 4 rad. The most appropriate expression for the angular position of the ball is :

A $\sqrt{34}$ m long ladder weighing 10 kg leans on a frictionless wall. Its feet rest on the floor 3 m away from the wall as shown in the figure. If $F_f$ and $F_w$ are the reaction forces of the floor and the wall, then ratio of $F_w/F_f$ will be :

(Use $g=10$ m/s$^2$.)

A uniform disc with mass $M=4$ kg and radius $R=10$ cm is mounted on a fixed horizontal axle as shown in figure. A block with mass $m=2$ kg hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is ______ N.

(Take $g=10$ ms$^{-2}$)

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 $\dfrac a3R^2\omega$. The value of a will be :

The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is $I_1$. The same rod is bent into a ring and its moment of inertia about a diameter is $I_2$. If $\dfrac{I_1}{I_2}$ is $\dfrac{x\pi^2}3$, then the value of $x$ will be ______.

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 plane of the square, is ______ kg m$^2$.

A solid cylinder and a solid sphere, having same mass M and radius R, roll down the same inclined plane from top without slipping. They start from rest. The ratio of velocity of the solid cylinder to that of the solid sphere, with which they reach the ground, will be :

A disc of mass 1 kg and radius R is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, its angular speed will be $4\sqrt{\dfrac x{3R}}$ rad s$^{-1}$ where $x=$ ______.

($g=10$ ms$^{-2}$)

The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the center will be ______ m.

Given, the length of the rod is $10\sqrt3$ m.

A pulley of radius 1.5 m is rotated about its axis by a force $F=\left(12t-3t^2\right)$ N applied tangentially (while t is measured in seconds). If moment of inertia of the pulley about its axis of rotation is 4.5 kg m$^2$, the number of rotations made by the pulley before its direction of motion is reversed, will be $\dfrac K\pi$. The value of K is ______.

A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should by unbinding the strings to achieve a speed of 4 ms$^{-1}$, is ______ cm. (take g = 10 ms$^{-2}$)

Four identical discs each of mass 'M' and diameter 'a' are arranged in a small plane as shown in figure. If the moment of inertia of the system about OO' is $\dfrac x4Ma^2$. Then, the value of $x$ will be ______.

The torque of a force $5\hat i+3\hat j-7\hat k$ about the origin is τ. If the force acts on a particle whose position vector is $2\hat i+2\hat j+\hat k$, then the value of τ will be

A fly wheel of mass 10 kg and radius 50 cm is rotating at a rate of 360 rpm. Assuming mass to be concentrated at the rim, the constant retarding torque required to stop the rotation of wheel in 6 rotations will be $x\pi$ N m. The value of $x$ is _______

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 mass of the object has an acceleration $a$. The acceleration due to gravity is $g$. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

A particle of mass $M = 0.2$ kg is initially at rest in the $xy$-plane at a point $(x = -l, y = -h)$, where $l = 10$ m and $h = 1$ m. The particle is accelerated at time $t = 0$ with a constant acceleration $a = 10\ \text{m/s}^2$ along the positive $x$-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\vec L$ and $\vec\tau$, respectively. $\hat i$, $\hat j$ and $\hat k$ are unit vectors along the positive $x$, $y$ and $z$-directions, respectively. If $\hat k = \hat i \times \hat j$ then which of the following statement(s) is(are) correct?

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 free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point O is $\left(\dfrac{Ma^2\Omega}{48}\right)n$.

The value of $n$ is ____.

One end of a horizontal uniform beam of weight $W$ and length $L$ is hinged on a vertical wall at point $O$ and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point $Q$, at a height $L$ above the hinge at point $O$. A block of weight $\alpha W$ is attached at the point $P$ of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of $(2\sqrt2)W$. Which of the following statement(s) is(are) correct?