A thin uniform disc of mass $M$ and radius $R$ is rotating with angular velocity $\omega$ about an axis passing through its center and perpendicular to its plane. A thin ring of mass $m$ and radius $r$ is gently placed on the disc concentrically. The final angular velocity of the combined system is:

If velocity (v), time (T) and mass (M) are taken as fundamental quantities, then dimensional formula for gravitational potential would be.

Universal time is based on

Two quantities A and B are related by the relation $\frac{A}{B}$=m, where m is linear mass den sity and A is force. The dimensions of B will be

A lens has a focal length 'f'. Derive the lens maker's formula, relating f to the refractive index 'n' of the lens material and the radii of curvature $R_1$ and $R_2$ of its surfaces.

What is dimensional formula of thermal conductivity?

If P represents radiations pressure, c represents speed of light and Q represents radiation energy striking a unit area per second, the non-zero integers x,y and z such that ${P^x}{Q^y}{c^z}$ is dimensionless, are

An important milestone in the evolution of the universe just after the Big Bang is the Planck time ${t_P}$, the value of which depends on three fundamental constants-speed c of light in vacuum, gravitational constant G and Planck’s constant h. Then, ${t_P} \propto$

A quantity X is given by ${\varepsilon _0}L\frac{{\Delta V}}{{\Delta t}},\;$, where ${\varepsilon _0}$ is the permittivity of free space, L is a length, ${\Delta V}$ is a potential difference and ${\Delta t}$ is a time interval. The dimensional formula for X is the same as that of

The equation for power ($P = \frac{W}{t}$) is derived from the relationship between work and which quantity?

A particle of mass $m$ moves under the influence of a central force given by $F(r) = -k/r^3$. If the particle's orbit is circular with radius $r_0$, its time period of revolution is: