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
$${\omega _M} = {v \over {2L}}\sqrt 3 $$
2
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 $$\sqrt {{{3g} \over {2L}}} $$.
3
JEE Advanced 2018 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
The potential energy of a particle of mass $$m$$ at a distance $$r$$ from a fixed point $$O$$ is given by $$V\left( r \right) = k{r^2}/2,$$ where $$k$$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $$R$$ about the point $$O$$. If $$v$$ is the speed of the particle and $$L$$ is the magnitude of its angular momentum about $$O,$$ which of the following statements is (are) true?
A
$$v = \sqrt {{k \over {2m}}} R$$
B
$$v = \sqrt {{k \over m}} R$$
C
$$L = \sqrt {mk} {R^2}$$
D
$$L = \sqrt {{{mk} \over 2}} {R^2}$$
4
JEE Advanced 2018 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Consider a body of mass $$1.0$$ $$kg$$ at rest at the origin at time $$t=0.$$ A force $$\overrightarrow F = \left( {\alpha t \widehat i + \beta \widehat j} \right)$$ is applied on the body, where $$\alpha = 1.0N{s^{ - 1}}$$ and $$\beta = 1.0\,N.$$ The torque acting on the body about the origin at time $$t=1.0s$$ is $$\overrightarrow \tau .$$ Which of the following statements is (are) true?
