A half ring of mass $m$, radius $R$ is hanged at its one end in vertical plane and is free to oscillate in its plane. Find oscillations frequency of the half ring.

 

A rectangular rigid fixed block has a long horizontal edge. A solid homogeneous cylinder of radius $R$ is placed horizontally at rest with its length parallel to the edge such that the axis of the cylinder and the edge of the block are in the same vertical plane as shown in figure. There is sufficient friction present at the edge, so that a very small displacement causes the cylinder to roll of the edge without slipping. Determine :

(i) The angle ${\mathrm{\theta }}_{\mathrm{C}}$ through which the cylinder rotates before it leaves contact with the edge.

(ii) The speed of the centre of mass of the cylinder before leaving contact with the edge and

(iii) The ratio of the translational to rotational kinetic energies of the cylinder when its centre of mass is in horizontal line with the edge.

A uniform rod of length $4\ell$ and mass $m$ is free to rotate about a horizontal axis passing through a point distance, $\ell$ from its one end. When the rod is horizontal, its angular velocity is $\omega$ as shown in figure. Calculate

(i) reaction of axis at this instant,

(ii) acceleration of centre of mass of the rod at this instant,

(iii) reaction of axis and acceleration of centre of mass of the rod when rod becomes vertical for the first time.

(iv) minimum value of $\omega$ so that centre of rod can complete circular motion.

A semi circular track of radius $R=62.5\mathrm{cm}$ is cut in a block. Mass of block, having track, is $M=1\mathrm{kg}$ and rests over a smooth horizontal floor. A cylinder of radius $r=10 \mathrm{cm}$ and mass $m=0.5 \mathrm{kg}$ is hanging by thread such that axis of cylinder and track are in same level and surface of cylinder is in contact with the track as shown in figure. When the thread is burnt, cylinder starts to move down the track. Sufficient friction exists between surface of cylinder and track, so that cylinder does not slip. Calculate velocity of axis of cylinder and velocity of the block when it reaches bottom of the track. Also find force, applied by block on the floor at the moment. $\left(g=10 m{s}^{-2}\right)$

A thin rod is passing through the centre of a sphere. The rod is fixed to a vertical axis and the sphere is made to roll on a surface with friction. The radius of the sphere is $r,$ the mass is $m$ and the length of the rod is $l.$ The rod is rotating with an angular velocity ${\omega }_0.$ Find the energy of the sphere in terms of ${\omega }_0, m, l$ and $r.$ Assume the rod to be of negligible mass.

A man and a woman skate towards each other on smooth ice, but in parallel lines. The distance between the, lines is $l.$ The mass of the man is $M$ and that of the woman is $m.$ The velocity of the man is given by $V$ and that of the woman by $v.$ The woman holds a stick of length $l$ and negligible mass. The stick is directed normal to the direction of motion as shown in the figure. When the couple passes each other, the man grasps the stick and the, couple move together, each of them holding different ends of the stick.

(i) What is the angular velocity of the rod after the couple begin moving together ?

(ii) The couple start moving towards each other by pulling the stick until the distance between them is $l_0\left(l_0<l\right).$ What is the velocity of the centre of mass now ?

(iii) What is the angular velocity of the couple now ?

(iv) What is the work done by the couple as they move from $l$ to $l_0 ?$

A square frame is formed by four rods, each of length $l=60\mathrm{cm}.$ Mass of two rods $AB$ and $BC$ is $m=\frac{25}{18}\mathrm{kg}$ each while that of rods $AD$ and $CD$ is $2\mathrm{kg}$ each. The frame is free to rotate about a fixed horizontal axis passing through its geometric centre $O$ shown in figure. A spring is placed on the $AB$ at a distance $a=15\mathrm{cm}$ from $B.$ The spring is held vertical and a block is placed on upper end of the spring so that rod $AB$ is horizontal.

(i) Calculate mass $M$ of the block,

(ii) If the spring is initially compressed by connecting a thread between its ends and energy stored in it is $76.5$ joule, calculate velocity with which block bounces up when the thread is burnt.