Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle \(\theta\) with each other. When suspended in water the angle remains the same. If density of the material of the sphere is \(1.5 \mathrm{~g} / \mathrm{cc}\), the dielectric constant of water will be _______. (Take density of water \(=1 \mathrm{~g} / \mathrm{cc}\) )

In a Screw Gauge, fifth division of the circular scale coincides with the reference line when the ratchet is closed. There are \(50\) divisions on the circular scale, and the main scale moves by \(0.5 \,{mm}\) on a complete rotation. For a particular observation the reading on the main scale is \(5\, {mm}\) and the \(20^{\text {th }}\) division of the circular scale coincides with reference line. Calculate the true reading. (in \({mm}\))

A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample is _______.

For the given circuit the current \(i\) through the battery when the key in closed and the steady state has been reached is .....\(A\)

In an experiment, a set of reading are obtained \(-1.24 mm, 1.25 mm, 1.23 mm, 1.21 mm\). The expected least count of the instrument used in recording these readings is ___________ mm.

A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical and the ratio of their kinetic energies is \(\frac{7}{\mathrm{x}}\) where \(\mathrm{x}\) is___________.

List - I	List - II
(A) Heat capacity of body	\(\text{(I) } \mathrm{J} \mathrm{kg}^{-1}\)
(B) Specific heat capacity of body	\(\text{(II) } \mathrm{JK}^{-1}\)
(C) Latent heat	\(\text{(III) } \mathrm{J} \mathrm{kg}^{-1} \mathrm{~K}^{-1}\)
(D) Thermal conductivity	\(\text{(IV) } \mathrm{Jm}^{-1} \mathrm{~K}^{-1} \mathrm{~s}^{-1}\)

Choose the correct answer from the options given below:

Two blocks of masses \(m\) and \(M,(M \gt m)\), are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released then
(\(\mu=\) coefficient of friction between the two blocks)

(A) The time period of small oscillation of the two blocks is \(\mathrm{T}=2 \pi \sqrt{\frac{(\mathrm{~m}+\mathrm{M})}{\mathrm{k}}}\)
(B) The acceleration of the blocks is \(\mathrm{a}=\frac{\mathrm{kx}}{\mathrm{M}+\mathrm{m}}\) (\(\mathrm{x}=\) displacement of the blocks from the mean position)
(C) The magnitude of the frictional force on the upper block is \(\frac{m \mu|x|}{M+m}\)
(D) The maximum amplitude of the upper block, if it does not slip, is \(\frac{\mu(M+m) g}{k}\)
(E) Maximum frictional force can be \(\mu(\mathrm{M}+\mathrm{m}) \mathrm{g}\).
Choose the correct answer from the options given below:

Paramagnetic substances: \(A\). align themselves along the directions of external magnetic field. \(B\). attract strongly towards external magnetic field. \(C\). has susceptibility little more than zero. \(D\). move from a region of strong magnetic field to weak magnetic field. Choose the most appropriate answer from the options given below:

A body is released from a height equal to the radius ( \(R\) ) of the earth. The velocity of the body when it strikes the surface of the earth will be : (Given \(g\) = acceleration due to gravity on the earth.)

The velocity of sound in air is doubled when the temperature is raised from \(0^{\circ} C\) to \(\alpha^{\circ} C\). The value of \(\alpha\) is ________.

A Carnot engine whose heat \(\operatorname{sinks}\) at \(27\,^{\circ} C\), has an efficiency of \(25 \%\). By how many degrees should the temperature of the source be changed to increase the efficiency by \(100 \%\) of the original efficiency \(?\)

When photons of wavelength \(\lambda _1\)  are incident on an isolated sphere, the corresponding stopping potential is found to be \(V.\)  When photons of wavelength \(\lambda _2\) are used, the corresponding stopping potential was thrice that of the above value . If light of wavelength \(\lambda _3\) is used then find the stopping potential for this case