A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should by unbinding the strings to achieve a speed of \(4\,ms ^{-1}\), is\(........cm\). \(\left(\right.\) take \(\left.g=10\,ms ^{-2}\right)\)

Moment of inertia of a dise of mass \(M\) and radius ' \(R\) ' about any of its diameter is \(\frac{ MR ^2}{4}\). The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, \(\frac{ x }{2} MR ^2\). The value of \(x\) is \(..........\).

A flexible chain of mass \(m\) hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is \(30^{\circ}\). Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is _________ .

The electric field in an electromagnetic wave is given by \(E =56.5 \sin \omega( t - x / c ) \;NC ^{-1}\). Find the intensity of the wave if it is propagating along \(x-\)axis in the free space. (Given \(\left.\varepsilon_{0}=8.85 \times 10^{-12} \;C ^{2} N ^{-1} m ^{-2}\right)\)

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason \(\mathrm{R}\) Assertion A : Work done in moving a test charge between two points inside a uniformly charged spherical shell is zero, no matter which path is chosen.
Reason \(\mathrm{R}\) : Electrostatic potential inside a uniformly charged spherical shell is constant and is same as that on the surface of the shell. In the light of the above statements, choose the correct answer from the options given below

Identify the correct statements :
A. Effective capacitance of a series combination of capacitors is always smaller than the smallest capacitance of the capacitor in the combination.
B. When a dielectric medium is placed between the charged plates of a capacitor, displacement of charges cannot occur due to insulation property of dielectric.
C. Increasing of area of capacitor plate or decreasing of thickness of dielectric is an alternate method to increase the capacitance.
D. For a point charge, concentric spherical shells centered at the location of the charge are equipotential surfaces.
Choose the correct answer from the options given below.

Two light waves of wavelengths \(800\) and \(600\,nm\) are used in Young's double slit experiment to obtain interference fringes on a screen placed \(7\,m\) away from plane of slits. If the two slits are separated by \(0.35\,mm\), then shortest distance from the central bright maximum to the point where the bright fringes of the two wavelength coincide will be \(.............mm\).

If an air bubble of diameter \(2\) mm rises steadily through a liquid of density \(2000\) kg/m\(^3\) at a rate of \(0.5\) cm/s, then the coefficient of viscosity of liquid is _______ Poise. (Take \(g = 10\) m/s\(^2\))

When an object is kept at a distance of \(30\, cm\) from a concave mirror, the image is formed at a distance of \(10\, cm\) from the mirror. If the object is moved with a speed of \(9\, cms ^{-1}\), the speed (in \(cms ^{-1}\) ) with which image moves at that instant is\(....m/s\)

Given below are two statements : one is labeled as Assertion \(A\) and the other is labeled as Reason \(R\) Assertion \(A:\) Photodiodes are used in forward bias usually for measuring the light intensity. Reason \(R:\) For a \(p-n\) junction diode, at applied voltage \(V\) the current in the forward bias is more than the current in the reverse bias for \(\left|V_z\right| > \pm V \geq\left|V_0\right|\) where \(V_0\) is the threshold voltage and \(V _{ z }\) is the breakdown voltage. In the light of the above statements, choose the correct answer from the options given below

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion-(A) : If Young's double slit experiment is performed in an optically denser medium than air, then the consecutive fringes come closer.
Reason-(R) : The speed of light reduces in an optically denser medium than air while its frequency does not change.
In the light of the above statements, choose the most appropriate answer from the options given below :

A capacitor is made of a tlat plate of area \(A\) and a second plate having a stair-like structure as shown in figure. If the area of each stair is \(\frac{A}{3}\) and the height is \(\mathrm{d}\), the capacitance of the arrangement is _______.

The equation of a transverse wave travelling along a string is \(y(x, t)=4.0 \sin \left[20 \times 10^{-3} x+600 t\right] \mathrm{mm}\), where \(x\) is in mm and \(t\) is in second. The velocity of the wave is :