A thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega .$ Two objects, each of mass $m,$ are attached gently to the opposite ends of a diameter of the ring. The ring rotates now with an angular velocity
$\frac{{\omega M}}{{M + m}}$
$\frac{{\omega (M - 2m)}}{{M + 2m}}$
$\frac{{\omega M}}{{M + 2m}}$
$\frac{{\omega (M + m)}}{M}$
In the following figure, a body of mass $m$ is tied at one end of a light string and this string and this string is wrapped around the solid cylinder of mass $M$ and radius $R$. At the moment $t = 0$ the system starts moving. If the friction is negligible, angular velocity at time $t$ would be
In an experiment with a beam balance an unknown mass $m$ is balanced by two known masses of $16\,kg$ and $4 \,kg$ as shown in figure. The value of the unknown mass $m$ is ......... $kg.$
A smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size, each of mass $m$ , which can slide freely along the rod. Initially the two beads are at the centre of the rod and the system is rotating with angular velocity $\omega _0$ about its axis perpendicular to the rod and passing through its mid-point (see figure). There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is
A rod $P$ of length $1\ m$ is hinged at one end $A$ and there is a ring attached to the other end by a light inextensible thread . Another long rod $Q$ is hinged at $B$ and it passes through the ring. The rod $P$ is rotated about an axis which is perpendicular to plane in which both the rods are present and the variation between the angles $\theta $ and $\phi $ are plotted as shown. The distance between the hinges $A$ and $B$ is ........ $m.$
A uniform cube of side $a$ and mass $m$ rests on a rough horizontal table. A horizontal force $F$ is applied normal to one of the faces at a point that is directly above the centre of face, at a height $\frac {3a}{4}$ above the base. The minimum value of $F$ for which the cube begins to tilt about the edge is (Assume that the cube does not slide)