A block of mass $$M$$ has a circular cut with a frictionless surface as shown. The block resets on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at $$x=0,$$ in a co-ordinate system fixed to the table. A point mass $$m$$ is released from rest at the topmost point of the path as shown and it slides down.
When the mass loses contact with the block, its position is $$x$$ and the velocity is $$v.$$ At that instant, which of the following options is/are correct?
A
The position of the point mass $$m$$ is :
$$x = - \sqrt 2 {{mR} \over {M + m}}$$
B
The velocity of the point mass $$m$$ is :
$$v = \sqrt {{{2gR} \over {1 + {m \over M}}}} $$
C
The $$x$$ component of displacement of the center
of mass of the block $$M$$ is: $$ - {{mR} \over {M + m}}$$
D
The velocity of the block $$M$$ is:
$$V = - {m \over M}\sqrt {2gR} $$
2
JEE Advanced 2016 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a
massless, rigid rod of length $$l = \sqrt {24} a$$ through their centers. This assembly is laid on a firm and flat
surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $$\omega $$. The angular momentum of the entire assembly about the point ‘O’ is $$\overrightarrow L $$ (see the figure). Which of the following statement(s) is(are) true?
A
The center of mass of the assembly rotates about the z-axis with an angular speed of $${\omega \over 5}$$
B
The magnitude of angular momentum of center of mass of the assembly about the point O is $$81\,m{a^2}\omega $$
C
The magnitude of angular momentum of the assembly about its center of mass is $${{17m{a^2}\omega } \over 2}$$
D
The magnitude of the z-component of $$\overrightarrow L $$
is $$55m{a^2}\omega $$
3
JEE Advanced 2016 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
The position vector $$\overrightarrow r $$ of a particle of mass m is given by the following equation
$$$\overrightarrow r \left( t \right) = \alpha {t^3}\widehat i + \beta {t^2}\widehat j,$$$where $$\alpha = {{10} \over 3}m{s^{ - 3}}$$, $$\beta = 5\,m{s^{ - 2}}$$ and m = 0.1 kg. At t = 1 s, which of the following
statement(s) is(are) true about the particle?
A
The velocity $$\overrightarrow v $$ is given by $$\overrightarrow v = \left( {10\widehat i + 10\widehat j} \right)$$ ms-1
B
The angular momentum $$\overrightarrow L $$
with respect to the origin is given by $$\overrightarrow L = - \left( {{5 \over 3}} \right)\widehat k\,N\,m\,s$$
C
The force $$\overrightarrow F $$
is given by $$\overrightarrow F = \left( {\widehat i + 2\widehat j} \right)N$$
D
The torque $$\overrightarrow \tau $$ with respect to the origin is given by $$\overrightarrow \tau = - \left( {{{20} \over 3}} \right)\widehat k\,N\,m$$
4
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
A ring of mass M and radius R is rotating with angular speed $$\omega$$ about a fixed vertical axis passing through its centre O with two point masses each of mass $${M \over 8}$$ at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant, the angular speed of the system is $${8 \over 9}$$$$\omega$$ and one of the masses is at a distance of $${3 \over 5}$$R from O. At this instant, the distance of the other mass from O is
