The centre of mass of a solid hemisphere of radius \(8\, cm\) is \(X \,cm\) from the centre of the flat surface. Then value of \(x\) is\(......\)

A block of mass \(1 \,kg\) attached to a spring is made to oscillate with an initial amplitude of \(12\, cm\). After \(2\, minutes\) the amplitude decreases to \(6\, cm\). Determine the value of the damping constant for this motion. (take In \(2=0.693\) )

A plano convex lens of refractive index \({\mu _1}\) and focal length \(f_1\) is kept in contact with another plano concave lens of refractive index \({\mu _2}\) and focal length \(f_2\). If the radius of curvature of their spherical faces is \(R\) each and \(f_1 = 2f_2\), then \({\mu _1}\) and \({\mu _2}\) are related as

A square loop of side \(2\) cm is placed in a time varying magnetic field with magnitude as \(B = 0.4\sin(300t)\) Tesla. The normal to the plane of loop makes an angle of \(60°\) with the field. The maximum induced emf produced in the loop is __________ mV.

A mass \(m = 1.0\,kg\) is put on a flat pan attached to a vertical spring fixed on the ground. The mass of the spring and the pan is negligible. When pressed slightly and released, the mass executes simple harmonic motion. The spring constant is \(500\,N/m.\) What is the amplitude \(A\) of the motion, so that the mass \(m\) tends to get detached from the pan ? (Take \(g = 10\,m/s^2\) ). The spring is stiff enough so that it does not get distorted during the motion.

As shown in the figure, a block of mass \(\sqrt{3}\, kg\) is kept on a horizontal rough surface of coefficient of friction \(\frac{1}{3 \sqrt{3}}\). The critical force to be applied on the vertical surface as shown at an angle \(60^{\circ}\) with horizontal such that it does not move, will be \(3 x\). The value of \(3x\) will be \(\left[ g =10 m / s ^{2} ; \sin 60^{\circ}=\frac{\sqrt{3}}{2} ; \cos 60^{\circ}=\frac{1}{2}\right]\)

A bullet of mass \(0.1\, kg\) is fired on a wooden block to pierce through it, but it stops after moving a distance of \(50\, cm\) into it. If the velocity of bullet before hitting the wood is \(10\) \(m / s\) and it slows down with uniform deceleration, then the magnitude of effective retarding force on the bullet is \('x' N.\) The value of \('x'\) to the nearest integer is................. \(N\)

A heavy box of mass \(50 \mathrm{~kg}\) is moving on a horizontal surface. If co-efficient of kinetic friction between the box and horizontal surface is \(0.3\) then force of kinetic friction is _______.

The threshold frequency of a metal with work function \(6.63\  \mathrm{eV}\) is:

A heating element has a resistance of  \(100\,\Omega \) at room temperature. When it is connected to a supply of  \(220\,V,\)  a steady current of \(2\,A\)  passes in it and temperature is \(500\,^oC\) more than room temperature. What is the temperature coefficient resistance of the heating element?