A coin is placed at the edge of a horizontal disc rotating about a vertical axis through its axis with a uniform angular speed $2 \mathrm{rad} {\mathrm{s}}^{-1}$. The radius of the disc is $50 \mathrm{cm}$. Find the minimum coefficient of friction between disc and coin so that the coin does not slip $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$.

A man revolves a stone of mass $m$ tied to the end of a string in a vertical circle of radius $R$. The net forces at the lowest and highest points of the circle directed vertical downwards are: [Here $T_1, T_2$ (and $v_1, v_2$) denote the tension in the string (and the speed of the stone) at the lowest and highest points, respectively.]

A small ring $P$ is threaded on a smooth wire bent in the form of a circle of radius $a$ and center $O$. The wire is rotating with constant angular speed $\omega$ about a vertical diameter $XY$, while the ring remains at rest relative to the wire at a distance $\frac{a}{2}$ from $XY$. Then ${\omega }^2$ is equal to

When the string of a conical pendulum makes an angle of $45°$ with the vertical, its time period is $T_1$. When the string makes an angle of $60°$ with the vertical, its time period is $T_2$. Then $T_1^2/T_2^2$ is

Three identical particles are joined together by a thread as shown in the figure. All the three particles are moving in a horizontal plane. If the velocity of the outermost particle is $v_0$, then the ratio of tensions in the three sections of the string ($T_1:T_2:T_3=$?) is?

Two identical particles are attached at the ends of a light string which passes through a hole at the centre of a table. One of the particles is made to move in a circle on the table with angular velocity ${\omega }_1$ and the other is made to move in a horizontal circle as a contact pendulum with angular velocity ${\omega }_2$. If $l_1$ and $l_2$ are the length of the string over and under the table, then in order that particle under the table neither moves down nor moves up, the ratio $l_1/l_2$ is

A block of mass $m$ is revolving in a smooth horizontal plane with a constant speed $v$. If the radius of the circular path is $R$, find the total contact force received by the block.

A particle is projected with a speed $v_0=\sqrt{gR}$, as shown in figure. The coefficient of friction between the particle and the hemispherical plane is $\mu =0.5$. Then, the acceleration (at $t=0$) of the particle is

A car travels with constant speed on a circular road on level ground. In figure, $F_{\mathrm{air}}$ is the force of air resistance on the car. Air resistance acts opposite to the motion of car. Which of the other forces shown best represents the horizontal force of the road on the car’s tires?