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 $J_0 \hat{x}$ 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
For $\mu \neq 0$ and $a \rightarrow 0, h_m=b / 2$.
B
For $\mu \neq 0$ and $a \rightarrow b, h_m=b$.
C
For $h=h_m$, the initial angular velocity does not depend on the inner radius $a$.
D
For $\mu=0$ and $h=0$, the wheel always slides without rolling.
2
JEE Advanced 2021 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
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
For the same F, the value of a does not depend on whether the cylinder is solid or hollow
B
For a solid cylinder, the maximum possible value of a is 2$$\mu$$g
C
The magnitude of the frictional force on the object due to the ground is always $$\mu$$mg
D
For a thin-walled hollow cylinder, $$a = {F \over {2m}}$$
3
JEE Advanced 2020 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
A rod of mass m and length L, pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed v strikes the rod horizontally at a distance x from its pivoted end and gets embedded in it. The combined system now rotates with angular speed $$\omega$$ about the pivot. The maximum angular speed $$\omega$$M is achieved for x = xM. Then
A
$$\omega = {{3vx} \over {{L^2} + 3{x^2}}}$$
B
$$\omega = {{12vx} \over {{L^2} + 12{x^2}}}$$
C
$${x_M} = {L \over {\sqrt 3 }}$$
D
$${\omega _M} = {v \over {2L}}\sqrt 3 $$
4
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
A thin and uniform rod of mass M and length L is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle 60$$^\circ $$ with vertical? [g is the acceleration due to gravity]
A
The angular acceleration of the rod will be $${{2g} \over L}$$.
B
The normal reaction force from the floor on the rod will be $${{Mg} \over 16}$$.
C
The radial acceleration of the rod's center of mass will be $${{3g} \over 4}$$.
D
The angular speed of the rod will be $$\sqrt {{{3g} \over {2L}}} $$.
