A ring of mass \(M\) and radius \(R\) is rotating about its axis with angular velocity \(\omega \). Two identical bodies each of mass \(m\) are now gently attached at the two ends of a diameter of the ring. Because of this, the kinetic energy loss will be

A cube having a side of 10 cm with unknown mass and 200 gm mass were hung at two ends of an uniform rigid rod of 27 cm long. The rod along with masses was placed on a wedge keeping the distance between wedge point and 200 gm weight as 25 cm. Initially the masses were not at balance. A beaker is placed beneath the unknown mass and water is added slowly to it. At given point the masses were in balance and half volume of the unknown mass was inside the water.
(Take the density of unknown mass is more than that of the water, the mass did not absorb water and water density is \(1 \mathrm{gm} / \mathrm{cm}^3\).) The unknown mass is ________ kg.

A wire of length \(L\, (=20\, cm)\), is bent into a semicircular arc. If the two equal halves of the arc were each to be uniformly charged with charges \( \pm Q\,,\,\left[ {\left| Q \right| = {{10}^3}{\varepsilon _0}} \right]\) Coulomb where \(\varepsilon _0\) is the permittivity (in \(SI\, units\)) of free space] the net electric field at the centre \(O\) of the semicircular arc would be

A concave mirror of focal length \(10\) cm forms an image which is double the size of object when the object is placed at two different positions. The distance between the two positions of the object is __________ cm.

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Net dipole moment of a polar linear isotropic dielectric substance is not zero even in the absence of an external electric field.
Reason (R) : In absence of an external electric field, the different permanent dipoles of a polar dielectric substance are oriented in random directions.
In the light of the above statements, choose the most appropriate answer from the options given below :

Two solids \(A\) and \(B\) of mass \(1\, kg\) and \(2\, kg\) respectively are moving with equal linear momentum. The ratio of their kinetic energies \((K.E.)_{ A }:( K.E. )_{ B }\) will be \(\frac{ A }{1},\) so the value of \(A\) will be ..... .

A metal sample carrying a current along \(X-\) axis with density \(J_x\) is subjected to a magnetic field \(B_z\) ( along \(z-\) axis ). The electric field \(E_y\) developed along \(Y-\) axis is directly proportional io \(J_x\) as well as \(B_z\) . The constant of proportionality has \(SI\, unit\).

The mean and standard deviation of \(15\) observations were found to be \(12\) and \(3\) respectively. On rechecking it was found that an observation was read as \(10\) in place of \(12\) . If \(\mu\) and \(\sigma^2\) denote the mean and variance of the correct observations respectively, then \(15\left(\mu+\mu^2+\sigma^2\right)\) is equal to ...........

A mass of \(10\,kg\) is suspended vertically by a rope from the roof. When a horizontal force is applied on the rope at some point, the rope deviated at an angle of \(45^o\) at the roof point. If the suspended mass is at equilibrium, the magnitude of the force applied is ..........  \(N\) \((g = 10\,ms^{-2})\) 

The value of \(\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{j=1}^{n} \frac{(2 j-1)+8 n}{(2 j-1)+4 n}\) is equal to:

Let \(A = \{1, 2, 3, 4, 5, 6\}\). The number of one-one functions \(f: A \rightarrow A\) such that \(f(1) \geq 3\), \(f(3) \leq 4\) and \(f(2) + f(3) = 5\), is __________.

As shown in the figure. a configuration of two equal point charges \(\left( q _0=+2 \mu C \right)\) is placed on an inclined plane. Mass of each point charge is \(20\,g\). Assume that there is no friction between charge and plane. For the system of two point charges to be in equilibrium (at rest) the height \(h = x \times 10^{-3}\,m\) The value of \(x\) is \(..........\).(Take \(\left.\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9\,Nm ^2 C ^{-2}, g=10\,ms ^{-1}\right)\)

A point on the straight line, \(3x + 5y = 15\) which is equidistant from the coordinate, axes will lie only in