A solid cylinder of mass \(m\) is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is: [The coefficient of static friction, \(\mu_{ s },\) is \(\left.0.4\right]\)

A solid sphere of mass \(M\) and radius \(R\) is divided into two unequal parts. The smaller part having mass \(M/8\) is converted into a sphere of radius \(r\) and the larger part is converted into a circular disc of thickness \(t\) and radius \(2R\). If \(I_1\) is moment of inertia of a sphere having radius \(r\) about an axis through its centre and \(I_2\) is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia \(I_2/I_1 = \) _____.

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of \(\theta ,\) where \(\theta \) is the angle by which it has rotated, is given as \(k\theta ^2.\) If its moment of inertia is \(I\) then the angular acceleration of the disc is

Consider the equation \(H = \dfrac{x^p \epsilon^q E^r}{t^s}\), where \(H=\) magnetic field; \(E=\) electric field, \(\epsilon=\) permittivity, \(x=\) distance, \(t=\) time. The values of \(p, q, r\) and \(s\) respectively are:

The space between the plates of a parallel plate capacitor is filled with a 'dielectric' whose 'dielectric constant' varies with distance as per the relation: \(K(x) = K_0 + \lambda x\) ( \(\lambda  =\) constant) The capacitance \(C,\) of the capacitor, would be related to its vacuum capacitance \(C_0\) for the relation

Two ideal electric dipoles \(A\) and \(B\) having their dipole moment \(p_{1}\) and \(p_{2}\) respectively are placed on a plane with their centres at \(O\) as shown in the figure. At point \(C\) on the axis of dipole \(A\), the resultant electric field is making an angle of \(37^{\circ}\) with the axis. The ratio of the dipole moment of \(A\) and \(B, \frac{P_{1}}{P_{2}}\) is \(....\) (take \(\left.\sin 37^{\circ}=\frac{3}{5}\right)\)

A simple pendulum, made of a string of length \(l\) and a bob of mass \(m\) , is released from a small angle \(\theta_0\). It strikes a block of mass \(M\), kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle \(\theta_1\). Then \(M\) is given by

A particle of mass \(m\) moves on a straight line with its velocity increasing with distance according to the equation \(\mathrm{v}=\alpha \sqrt{\mathrm{x}}\), where \(\alpha\) is a constant. The total work done by all the forces applied on the particle during its displacement from \(\mathrm{x}=0\) to \(\mathrm{x}=\mathrm{d}\), will be _______.

The variation of stopping potential \(\left(V_0\right)\) as a function of the frequency \(\nu \left(\times 10^{14} Hz\right).\) of the incident light for a metal is shown in figure. The work function of the surface is \(...........\,eV\)

Width of one of the two slits in a Young's double slit interference experiment is half of the other slit. The ratio of the maximum to the minimum intensity in the interference pattern is _______.

Two batteries with e.m.f \(12\ V\) and \(13\ V\) are connected in parallel across a load resistor of \(10\,\Omega\) . The internal resistances of the two batteries are \(1\,\Omega\) and \(2\,\Omega\) respectively. The voltage across the load lies between

Find out the surface charge density at the intersection of point \(x =3\, m\) plane and \(x\) -axis, in the region of uniform line charge of \(8\, nC / m\) lying along the \(z\) -axis in free space.

Consider two uniform discs of the same thickness and different radii \(R _{1}= R\) and \(R _{2}=\alpha R\) made of the same material. If the ratio of their moments of inertia \(I_{1}\) and \(I_{2}\), respectively, about their axes is \(I _{1}: I _{2}=1: 16\) then the value of \(\alpha\) is