A uniform sphere of mass \(500\; g\) rolls without slipping on a plane horizontal surface with its centre moving at a speed of \(5.00\; \mathrm{cm} / \mathrm{s}\). Its kinetic energy is

Given below are two statements:
Statement I: Pressure of fluid is exerted only on a solid surface in contact as the fluid-pressure does not exist everywhere in a still fluid.
Statement II: Excess potential energy of the molecules on the surface of a liquid, when compared to interior, results in surface tension.
In the light of the above statements, choose the correct answer from the options given below.

A galaxy is moving away from the earth at a speed of \(286\, kms ^{-1}\). The shift in the wavelength of a red line at \(630\, nm\) is \(x \times 10^{-10}\, m\). The value of \(x ,\) to the nearest integer, is........ [Take the value of speed of light \(c,\) as \(3 \times 10^{8}\) \(\left. ms ^{-1}\right]\)

Four bulbs \(B_1 , B_2, B_3\) and \(B_4\) of \(100\, W\) each are connected to \(220\, V\) main as shown in the figure. The reading in an ideal ammeter will be ............... \(A\)

The explosive in a Hydrogen bomb is a mixture of \({ }_1 \mathrm{H}^2,{ }_1 \mathrm{H}^3\) and \({ }_3 \mathrm{Li}^6\) in some condensed form. The chain reaction is given by \( { }_3 \mathrm{Li}^6+{ }_0 \mathrm{n}^1 \rightarrow{ }_2 \mathrm{He}^4+{ }_1 \mathrm{H}^3 \) \( { }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^3 \rightarrow{ }_2 \mathrm{He}^4+{ }_0 \mathrm{n}^1\) During the explosion the energy released is approximately [Given : \(\mathrm{M}(\mathrm{Li})=6.01690\  \mathrm{amu} . \mathrm{M}\left({ }_1 \mathrm{H}^2\right)=2.01471 \  amu.\) \(\mathrm{M}\left({ }_2 \mathrm{He}^4\right)=4.00388\  \mathrm{amu}\),\( and\ \)\(1 \ \mathrm{amu}=931.5\) \(\mathrm{MeV}]\)

A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle \((\theta)\) of thread deflection in the extreme position will be :

An electron, moving along the \(x-\) axis with an initial energy of \(100\, eV\), enters a region of magnetic field \(\vec B = (1.5\times10^{-3}T)\hat k\) at \(S\) (See figure). The field extends between \(x = 0\) and \(x = 2\, cm\). The electron is detected at the point \(Q\) on a screen placed \(8\, cm\) away from the point \(S\). The distance \(d\) between \(P\) and \(Q\) (on the screen) is :......\(cm\) (electron's charge \(= 1.6\times10^{-19}\, C\), mass of electron \(= 9.1\times10^{-31}\, kg\))

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A :\) An electron microscope can achieve better resolving power than an optical microscope. Reason \(R\) : The de Broglie's wavelength of the electrons emitted from an electron gun is much less than wavelength of visible light. In the light of the above statements, choose the correct answer from the options given below

An object of mass \(0.2 \mathrm{~kg}\) executes simple harmonic motion along \(\mathrm{x}\) axis with frequency of \(\left(\frac{25}{\pi}\right) \mathrm{Hz}\). At the position \(\mathrm{x}=0.04 \mathrm{~m}\) the object has kinetic energy \(0.5 \mathrm{~J}\) and potential energy \(0.4 \mathrm{~J}\) The amplitude of oscillation is _______ cm.

In the given figure, each diode has a forward bias resistance of \(30\, \Omega\) and infinite resistance in reverse bias. The current \({I}_{1}\) will be \(....\,A\)

For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric \(\left( U _{ e }\right)\) and magnetic \(\left( U _{ m }\right)\) fields is

Two metal spheres of radius \(R\) and \(3 R\) have same surface charge density \(\sigma\). If they are brought in contact and then separated, the surface charge density on smaller and bigger sphere becomes \(\sigma_1\) and \(\sigma_2\), respectively. The ratio \(\frac{\sigma_1}{\sigma_2}\) is ________.

\(ABC\) is a plane lamina of the shape of an equilateral triagnle. \(D,\) \(E\) are mid points of \(AB\), \(AC\) and \(G\) is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through \(G\) and perpendicular to the plane \(ABC\) is \(I _{0} .\) If part \(ADE\) is removed, the moment of inertia of the remaining part about the same axis is \(\frac{ NI _{0}}{16}\) where \(N\) is an integer. Value of \(N\) is