Rotational Motion

Consider a star of mass $m_2$ kg revolving in a circular orbit around another star of mass $m_1$ kg with $m_1 \gg m_2$. The heavier star slowly acquires mass from the lighter star at a constant rate of $\gamma$ kg/s. In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is $r$, then its relative rate of change $\dfrac{1}{r}\dfrac{dr}{dt}$ (in s$^{-1}$) is given by:

A conducting solid sphere of radius $R$ and mass $M$ carries a charge $Q$. The sphere is rotating about an axis passing through its center with a uniform angular speed $\omega$. The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as $\alpha\dfrac{Q}{2M}$. The value of $\alpha$ is _____.

A disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R/2$ is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed $\omega$. If the angular speed at which the large disc rotates is $\omega/n$, then the value of $n$ is _____.

A thin uniform rod of length $L$ and certain mass is kept on a frictionless horizontal table with a massless string of length $L$ fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O. If a horizontal impulse $P$ is imparted to the rod at a distance $x = L/n$ from the mid-point of the rod (see figure), then the rod and string revolve together around the point O, with the rod remaining aligned with the string. In such a case, the value of $n$ is _____.

A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass $m$ and radius $r$ and it is in a uniform vertical magnetic field $B_0$, as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity $g$, on two conducting supports at P and Q. When a current $I$ is passed through the loop, the loop turns about the line PQ by an angle $\theta$ given by

A bar of mass $M = 1.00$ kg and length $L = 0.20$ m is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m = 0.10$ kg is moving on the same horizontal surface with $5.00\ \mathrm{m\,s^{-1}}$ speed on a path perpendicular to the bar. It hits the bar at a distance $L/2$ from the pivoted end and returns back on the same path with speed $v$. After this elastic collision, the bar rotates with an angular velocity $\omega$. Which of the following statement is correct?

Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2\times10^{-8}\ \mathrm{N\,m\,rad^{-1}}$. The angular frequency of the oscillations in $n\times10^{-3}\ \mathrm{rad\,s^{-1}}$. The value of $n$ is _____.

An annular disk of mass $M$, inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$, as shown in the figure. At some time, an impulse $\mathcal J_0\hat x$ is applied at a height $h$ above the center of the disk. If $h = h_m$ then the disk rolls without slipping along the $x$-axis. Which of the following statement(s) is(are) correct?

A thin circular coin of mass 5 gm and radius $4/3$ cm is initially in a horizontal $xy$-plane. The coin is tossed vertically up ($+z$ direction) by applying an impulse of $\sqrt{\dfrac\pi2}\times10^{-2}$ N-s at a distance $2/3$ cm from its center. The coin spins about its diameter and moves along the $+z$ direction. By the time the coin reaches back to its initial position, it completes $n$ rotations. The value of $n$ is _____.

[Given: The acceleration due to gravity $g = 10\ \mathrm{m\,s^{-2}}$]

At time $t = 0$, a disk of radius 1 m starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha = \dfrac23\ \mathrm{rad\,s^{-2}}$. A small stone is stuck to the disk. At $t = 0$, it is at the contact point of the disk and the plane. Later, at time $t = \sqrt\pi$ s, the stone detaches itself and flies off tangentially from the disk. The maximum height (in m) reached by the stone measured from the plane is $\dfrac12 + \dfrac{x}{10}$. The value of $x$ is _____. [Take $g = 10\ \mathrm{m\,s^{-2}}$.]

A solid sphere of mass 1 kg and radius 1 m rolls without slipping on a fixed inclined plane with an angle of inclination $\theta = 30^\circ$ from the horizontal. Two forces of magnitude 1 N each, parallel to the incline, act on the sphere, both at distance $r = 0.5$ m from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is _____ $\mathrm{m\,s^{-2}}$. (Take $g = 10\ \mathrm{m\,s^{-2}}$.)

A particle of mass 1 kg is subjected to a force which depends on the position as $\vec F = -k(x\hat i + y\hat j)\ \mathrm{kg\,m\,s^{-2}}$ with $k = 1\ \mathrm{kg\,s^{-2}}$. At time $t = 0$, the particle's position $\vec r = \left(\frac{1}{\sqrt2}\hat i + \sqrt2\hat j\right)$ m and its velocity $\vec v = \left(-\sqrt2\hat i + \sqrt2\hat j + \frac2\pi\hat k\right)\ \mathrm{m\,s^{-1}}$. Let $v_x$ and $v_y$ denote the $x$ and the $y$ components of the particle's velocity, respectively. Ignore gravity. When $z = 0.5$ m, the value of $(x v_y - y v_x)$ is _____ $\mathrm{m^2s^{-1}}$.

A flat surface of a thin uniform disk $A$ of radius $R$ is glued to a horizontal table. Another thin uniform disk $B$ of mass $M$ and with the same radius $R$ rolls without slipping on the circumference of $A$, as shown in the figure. A flat surface of $B$ also lies on the plane of the table. The center of mass of $B$ has fixed angular speed $\omega$ about the vertical axis passing through the center of $A$. The angular momentum of $B$ is $nM\omega R^2$ with respect to the center of $A$. Which of the following is the value of $n$?

A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be $x\times10^{-2}$ kg. The value of $x$ is ______.

A fly wheel is accelerated uniformly from rest and rotates through 5 rad in the first second. The angle rotated by the fly wheel in the next second, will be :

If force $\vec F=3\hat i+4\hat j-2\hat k$ acts on a particle having position vector $2\hat i+\hat j+2\hat k$ then, the torque about the origin will be :

Moment of Inertia (M.I.) of four bodies having same mass '$M$' and radius '$2R$' are as follows :

$I_1=$ M.I. of solid sphere about its diameter

$I_2=$ M.I. of solid cylinder about its axis

$I_3=$ M.I. of solid circular disc about its diameter

$I_4=$ M.I. of thin circular ring about its diameter

If $2(I_2+I_3)+I_4=x\cdot I_1$ then the value of $x$ will be ______.

A thin circular ring of mass M and radius R is rotating with a constant angular velocity 2 rad s$^{-1}$ in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass m be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in rad s$^{-1}$).

A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is -

A rolling wheel of 12 kg is on an inclined plane at position P and connected to a mass of 3 kg through a string of fixed length and pulley as shown in figure. Consider PR as friction free surface.

The velocity of centre of mass of the wheel when it reaches at the bottom Q of the inclined plane PQ will be $\dfrac12\sqrt{xgh}$ m/s. The value of $x$ is ______.