Mercury is filled in a tube of radius \(2 \mathrm{~cm}\) up to a height of \(30 \mathrm{~cm}\). The force exerted by mercury on the bottom of the tube is _______ \(\mathrm{N}\). (Given, atmospheric pressure \(=10^5 \mathrm{Nm}^{-2}\), density of mercury \(=1.36 \times 10^4 \mathrm{~kg} \mathrm{~m}^3, \mathrm{~g}=10 \mathrm{~ms}^2\), \(\left.\pi=\frac{22}{7}\right)\)

What is the minimum energy required to launch a satellite of mass \(m\) from the surface of a planet of mass \(M\) and radius \(R\) in a circular orbit at an altitude of \(2R\)?

Water from a pipe is coming at a rate of \(100\, litres\) per minute. If the radius of the pipe is \(5\, cm\), the Reynolds number for the flow is of the order of : (density of water \(= 1000\, kg/m^3\), coefficient of viscosity of water \(= 1\, mPa\, s\))

For an electromagnetic wave propagating through vacuum, \(\vec{k}\), \(\vec{E}\) and \(\omega\) represent propagation vector, electric field and angular frequency, respectively. The magnetic field associated with this wave is represented by :

A thin smooth rod of length \(L\) and mass \(M\) is rotating freely with angular speed \({\omega _0}\) about an axis perpendicular to the rod and passing through its centre. Two beads of mass \(m\) and negligible size are at the centre of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod will be

A mass \(m\) is attached to two springs as shown in figure. The spring constants of two springs are \(K _1\) and \(K _2\). For the frictionless surface, the time period of oscillation of mass \(m\) is

A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical and the ratio of their kinetic energies is \(\frac{7}{\mathrm{x}}\) where \(\mathrm{x}\) is___________.

particle moves with simple harmonic motion in a straight line. In first \(\tau\ s\), after starting from rest it travels a distance \(a\), and in next \(\tau\ s\) it travels \(2a\), in same direction, then

Three particles \({P}, {Q}\) and \({R}\) are moving along the vectors \({A}=\hat{{i}}+\hat{{j}}, {B}=\hat{{j}}+\hat{{k}}\) and \({C}=-\hat{{i}}+\hat{{j}}\) respectively. They strike on a point and start to move in different directions. Now particle \(P\) is moving normal to the plane which contains vector \(\vec{A}\) and \(\vec{B} .\) Similarly particle \(Q\) is moving normal to the plane which contains vector \(\vec{A}\) and \(\vec{C} .\) The angle between the direction of motion of \(P\) and \(Q\) is \(\cos ^{-1}\left(\frac{1}{\sqrt{x}}\right)\). Then the value of \(x\) is ...... .

A point charge of \(+\,12 \,\mu C\) is at a distance \(6 \,cm\) vertically above the centre of a square of side \(12\, cm\) as shown in figure. The magnitude of the electric flux through the square will be ....... \(\times 10^{3} \,Nm ^{2} / C\)