The moment of inertia of semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is \(\frac{1}{ x } MR ^2\), where \(R\) is the radius and \(M\) is the mass of semicircular ring. The value of \(x\) will be \(...........\)

A square loop is carrying a steady current \(I\) and the magnitude of its magnetic dipole moment is \(m\). If this square loop is changed to a circular loop and it carries the same current, the magnitude of the magnetic dipole moment of circular loop will be

As shown below, bob \(A\) of a pendulum having massless string of length ' \(R\) ' is released from \(60^{\circ}\) to the vertical. It hits another bob \(B\) of half the mass that is at rest on a friction less table in the center. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take g as acceleration due to gravity.)

A disc is rolling without slipping on a surface. The radius of the disc is \(R\). At \(t=0\), the top most point on the disc is \(A\) as shown in figure. When the disc completes half of its rotation, the displacement of point \(A\) from its initial position is

A string is clamed at both the ends and it is vibrating in its \(4^{th}\) harmonic. The equation of the stationary wave is \(Y =0.3\,sin\,(0.157\,x) \,cos\,(200\pi t)\). The length of the string is ..... \(m\) (all quantities are in \(SI\) units)

A girl standing on road holds her umbrella at \(45^{\circ}\) with the vertical to keep the rain away. If she starts running without umbrella with a speed of \(15 \sqrt{2} \; kmh ^{-1}\), the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is ...........  \(kmh ^{-1}\)

A uniform heating wire of resistance \(36\, \Omega\) is connected across a potential difference of \(240\, {V}\) The wire is then cut into half and potential difference of \(240\, {V}\) is applied across each half separately. The ratio of power dissipation in first case to the total power dissipation in the second case would be \(1: {x}\), where \({x}\) is........... .

A long straight wire of circular cross-section (radius \(a\)) is carrying steady current \(I\). The current \(I\) is uniformly distributed across this cross-section. The magnetic field is

As shown in the figure, a combination of a thin plano concave lens and a thin plano convex lens is used to image an object placed at infinity. The radius of curvature of both the lenses is \(30\,cm\) and refraction index of the material for both the lenses is \(1.75\). Both the lenses are placed at distance of \(40\,cm\) from each other. Due to the combination, the image of the object is formed at distance \(x =..........cm\), from concave lens.

A test particle is moving in a circular orbit in the gravitational field produced by a mass density \(\rho \left( r \right) = \frac{K}{{{r^2}}}\). Identify the correct relation between the radius \(R\) of the particle’s orbit and its period \(T\)