To mop-clean a floor, a cleaning machine presses a circular mop of radius \(R\) vertically down with a total force \(F\) and rotates it with a constant angular speed about its axis. If the force \(F\) is distributed uniformly over the mop and the floor is \(\mu \), the torque, applied by the machine on the mop is

A parallel plate capacitor with plate separation \(5\) \(\mathrm{mm}\) is charged up by a battery. It is found that on introducing a dielectric sheet of thickness \(2 \mathrm{~mm}\), while keeping the battery connections intact, the capacitor draws \(25 \%\) more charge from the battery than before. The dielectric constant of the sheet is _______.

What is the relative decrease in focal length of a lens for an increase in optical power by 0.1 D from 2.5D ? ['D' stands for dioptre]

One of the two identical conducing wires of length \(L\) is bent in the form of a circular loop and the other one into a circular coil of \(N\) identical turns. If the same current is passed in both, the ratio of the magnetic field at the central of the loop \((B_L)\) to that at the centre of the coil \((B_C),\); \(.\,\frac {B_L}{B_C}\) will be

A wedge of mass \(M = 4\,m\) lies on a frictionless plane. A particle of mass \(m\) approaches the wedge with speed \(v\). There is no friction between the particle and the plane or between the particle and the wedge. The maximum height climbed by the particle on the wedge is given by

The orbital angular momentum of a satellite is \(L\), when it is revolving in a circular orbit at height \(h\) from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will be \(............\,L\)

Given below are two statements: one is labelled as Assertion\((A)\) and the other is labelled as Reason \((R).\) \(Assertion\) \((A)\) : In Vernier calliper if positive zero error exists, then while taking measurements, the reading taken will be more than the actual reading. \(Reason\) \((R)\) : The zero error in Vernier Calliper might have happened due to manufacturing defect or due to rough handling. In the light of the above statements, choose the correct answer from the options given below :

The displacement time graph of a particle executing \(S.H.M.\) is given in figure: (sketch is schematic and not to scale) Which of the following statements is are true for this motion? \((A)\) The force is zero \(t=\frac{3 T}{4}\) \((B)\) The acceleration is maximum at \(t=T\) \((C)\) The speed is maximum at \(t =\frac{ T }{4}\) \((D)\) The \(P.E.\) is equal to \(K.E.\) of the oscillation at \(t=\frac{T}{2}\)

A concave mirror for face viewing has focal length of \(0.4\,m.\) The distance at which you hold the mirror from your face in order to see your image upright with a magnification of \(5\) is......\(m\)

Motion of a particle in \(x - y\) plane is described by a set of following equations \(x=4 \sin \left(\frac{\pi}{2}-\omega t\right) m\) and \(y=4 \sin (\omega t) m\). The path of particle will be 

Light of wavelength \(550\, nm\) falls normally on a slit of width \(22.0 \times 10^{-5}\,cm\). The angular position of the second minima from the central maximum will be (in radians)