A small ring is rolling without slipping on the circumference of a large bowl as shown below in the figure. The ring is moving down at $P_1$, comes down to the lower most point $P_2$ and is climbing up at $P_3$. Let ${\overrightarrow{v}}_{CM}$ denote the velocity of the centre of mass of the ring. Choose the correct statement regarding the frictional force on the ring.

The distance between the vertex and the centre, of mass of a uniform solid planar circular segment of angular size, $\theta$ and radius, $R$ is given by

The moments of inertia of a non-uniform circular disc (of mass $M$ and radius $R$) about four mutually perpendicular tangents, $AB, BC, CD, DA$ are $I_1, I_2, I_3$ and $I_4$, respectively (the square $ABCD$ circumscribes the circle). The distance of the centre of mass of the disc from its geometrical centre is given by

In the figure shown $ABC$ is a uniform wire. If the center of mass of the wire lies vertically below point $A$, then $\frac{BC}{AB}$ is close to

A hollow tilted cylindrical vessel of negligible mass rest on a horizontal plane as shown. The diameter of the base is a and the side of the cylinder makes an angle $\theta$ with the horizontal. Water is then slowly poured into the cylinder. The cylinder topples over when the water reaches a certain height h, given by

Two identical objects each of radii, $R$ and masses, $m_1$ and, $m_2$ are suspended using two strings of equal length, $L$ as shown in the figure, $(R<<L).$ The angle, $\theta$ which mass, $m_2$ makes with the vertical is approximately

Two uniform plates of the same thickness and area but of different materials, one shaped like an isosceles triangle and the other shaped like a rectangle are joined together to form a composite body as shown in the figure alongside. If the centre of mass of the composite body is located at the mid-point of their common side, then the ratio between masses of the triangle to that of the rectangle is, 

Four particles $\mathrm{A},\mathrm{ }\mathrm{B},\mathrm{ }\mathrm{C}$ and $\mathrm{D}$ with masses ${\mathrm{m}}_{\mathrm{A}}=\mathrm{m},{\mathrm{m}}_{\mathrm{B}}=2\mathrm{m},{\mathrm{m}}_{\mathrm{C}}=3\mathrm{m}$  and ${\mathrm{m}}_{\mathrm{D}}=4\mathrm{m}$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is:

Consider regular polygons with number of sides $n=\mathrm{3,4},5\dots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\mathrm{\Delta }$ . Then, $\mathrm{\Delta }$ depends on $n$ and $h$ as