What is the radius of curvature of the parabola, traced out by the projectile in the previous problem, at a point where the particle velocity makes an angle $\theta /2$ with the horizontal?

A block of mass $m$ moves on a horizontal circle, against the wall of a cylindrical room of radius $R$. The floor of the room, on which the block moves, is smooth but the friction coefficient between the wall and the block is $\mu$. The block is given an initial speed $v_0.$ As a function of the speed $v$, write

(a) the normal force by the wall on the block,

(b) the frictional force by the wall and

(c) the tangential acceleration of the block.

(d) Integrate the tangential acceleration $\left(\frac{dv}{dt}=v\frac{dv}{ds}\right)$ to obtain the speed of the block after one revolution.

A table with a smooth horizontal surface is fixed in a cabin that rotates with a uniform angular velocity $\omega$ in a circular path of radius $R$, as shown in the figure. A smooth groove $AB$ of length $L(<<R)$ is made on the surface of the table. The groove makes an angle $\mathit{\text{θ}}$ with the radius $OA$ of the circle in which the cabin rotates. A small particle is kept at the point $A$ in the groove and is released to move along $AB$. Find the time taken by the particle to reach the point $B$. 

A car moving at a speed of $36 \mathrm{km} {\mathrm{h}}^{-1}$ is taking a turn on a circular road of radius $50 \mathrm{m}.$ A small wooden plate is kept on the seat with its plane perpendicular to the radius of the circular road, as shown in the figure. A small block of mass $100 \mathrm{g}$ is kept on the seat which rests against the plate. The friction coefficient between the block and the plate is $\mu =0·58$. 
(a) Find the normal contact force exerted by the plate on the block.

(b) The plate is slowly turned so that the angle between the normal to the plate and the radius of the road slowly increases. Find the angle at which the block will just start sliding on the plate.

A table with a smooth horizontal surface is placed in a cabin which moves in a circle of a large radius $R$, as shown in the figure. A smooth pulley of a small radius $r$ is fastened to the table. Two masses $m$ and $2m$, placed on the table, are connected through a string going over the pulley. Initially, the masses are held by a person with the strings along the outward radius and then the system is released from rest (with respect to the cabin). Find the magnitude of the initial acceleration of the masses as seen from the cabin and the tension in the string., ( Assume $r<<R$, also length of string is very less as compared to radius of rotating circle.)