A alternating current at any instant is given by \(i=\left[6+\sqrt{56} \sin \left(100 \pi t+\frac{\pi}{3}\right)\right] A\). The rms value of the current is _________ A.

 Consider a water jar of radius \(R\) that has water filled up to height \(H\) and is kept on a stand of height \(h\) (see figure) . Through a hole of radius \(r\)  \((r << R)\) at its bottom, the water leaks out and the stream of water coming down towards the ground has a shape like a funnel as shown in the figure. If the radius of the cross-section of water stream when it hits the ground is \(x.\) Then

In an ac generator, a rectangular coil of \(100\) turns each having area \(14 \times 10^{-2}\,m ^2\) is rotated at \(360\,rev / min\) about an axis perpendicular to a uniform magnetic field of magnitude \(3.0 T\). The maximum value of the emf produced will be \(............V\). \(\left(\right.\) Take \(\left.\pi=\frac{22}{7}\right)\)

Calculate the time (in \(minutes\)) interval between \(33 \,\%\) decay and \(67\, \%\) decay if half-life of a substance is \(20\, minutes.\)

The Young's modulus of steel wire of radius \(r\) and length \(L\) is \(Y\). If the radius \(r\) and length \(L\) of the wire are doubled then the value of \(Y\)

A radioactive nucleus can decay by two different processes. Half-life for the first process is \(3.0\, hours\) while it is \(4.5\, hours\) for the second process. The effective half- life of the nucleus will be \(.........\,hours.\)

A particle of mass \(500 \,gm\) is moving in a straight line with velocity \(v=b x^{5 / 2}\). The work done by the net force during its displacement from \(x=0\) to \(x =4 \,m\) is ...................\(J\) (Take \(b =0.25 \,m ^{-3 / 2} s ^{-1}\) )

If the initial velocity in horizontal direction of a projectile is unit vector \(\hat{i}\) and the equation of trajectory is \(y =5 x (1- x )\). The \(y\) component vector of the initial velocity is. (Take \(g=10\,m / s ^{2}\) )