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 circular loop of radius \(r\) is carrying current I A. The ratio of magnetic field at the centre of circular loop and at a distance \(r\) from the center of the loop on its axis is:

In an experiment, electrons are accelerated, from rest, by applying, a voltage of \(500 \,V.\) Calculate the radius of the path if a magnetic field \(100\,mT\) is then applied. [Charge of the electron \(= 1.6 \times 10^{-19}\,C\) Mass of the electron \(= 9.1 \times 10^{-31}\,kg\) ]

Two cells of emf \(2\, E\) and \(E\) with internal resistance \(r _{1}\) and \(r _{2}\) respectively are connected in series to an external resistor \(R\) (see \(figure\)). The value of \(R ,\) at which the potential difference across the terminals of the first cell becomes zero is

A sinusoidal voltage \(V(t)=210\,\sin 3000 t\) volt is applied to a series \(LCR\) circuit in which \(L =10\) \(mH , C =25 \mu F\) and \(R =100 \Omega\). The phase difference \((\Phi)\) between the applied voltage and resultant current will be

The percentage increase in magnetic field (B) when space within a current carrying solenoid is filled with magnesium (magnetic susceptibility \(\left.\chi_{\mathrm{mg}}=1.2 \times 10^{-5}\right)\) is :

In the given figure switches \(S_{1}\) and \(S_{2}\) are in open condition. The resistance across \(a b\) when the switches \(S_{1}\) and \(S_{2}\) are closed is \(...\,\Omega\)

The position of a projectile launched from the origin at \(t = 0\) is given by \(\vec r = \left( {40\hat i + 50\hat j} \right)\,m\) at \(t = 2\,s\). If the projectile was launched at an angle \(\theta\) from the horizontal, then \(\theta\) is (take \(g = 10\, ms^{-2}\))

An inductor of \(0.5\,mH\), a capacitor of \(20\,\mu\,F\) and resistance of \(20\,\Omega\) are connected in series with a \(220\,V\) ac source. If the current is in phase with the emf, the amplitude of current of the circuit is \(\sqrt{x} A\). The value of \(x\) is -

A parallel plate capacitor is having separation between plates \(0.885\) mm. It has a capacitance of \(1\,\mu\)F when the space between the plates is filled with an insulating material of resistivity \(1\times 10^{13}\,\Omega\)m and resistance \(17.7\times 10^{14}\,\Omega\). Relative permittivity of the insulating material is \(\alpha\times 10^7\). The value of \(\alpha\) is ________. (Take permittivity of free space \(=8.85\times 10^{-12}\) F/m)

A bob of mass \('m'\) suspended by a thread of length \(l\) undergoes simple harmonic oscillations with time period \({T}\). If the bob is immersed in a liquid that has density \(\frac{1}{4}\) times that of the bob and the length of the thread is increased by \(1 / 3^{\text {rd }}\) of the original length, then the time period of the simple harmonic oscillations will be :-

An engine operates by taking a monatomic ideal gas through the cycle shown in the figure. The percentage efficiency of the engine is close to  \(.......\%\)