A very long (length \(L\) ) cylindrical galaxy is made of uniformly distributed mass and has radius \(R(R < < L).\) A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of star is \(T\) and its distance from the galaxy's axis is \(r,\) then

An engine is attached to a wagon through a shock absorber of length \(1.5\, {m}\). The system with a total mass of \(40,000\, {kg}\) is moving with a speed of \(72\, {kmh}^{-1}\) when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by \(1.0\, {m}\). If \(90\, \%\) of energy of the wagon is lost due to friction, the spring constant is \(....\, \times 10^{5}\, {N} / {m}\)

\(P-T\) diagram of an ideal gas having three different densities \(\rho_1, \rho_2, \rho_3\) (in three different cases) is shown in the figure. Which of the following is correct?

Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the area of cross-sections at \(A\) and \(B\) are \(6 cm^2\) and \(3 cm^2\) respectively.
The rate of flow will be ___________ \(cm ^3 / s\). (take \(g =10 m / s ^2\) )

A uniform rectangular thin sheet \(ABCD\) of mass \(M\) has length \(a\) and breadth \(b\), as shown in the figure. If the shaded portion \(HBGO\) is cut off, the coordinates of the centre of mass of the remaining portion will be

Three solid spheres each of mass \(\mathrm{m}\) and diameter \(d\) are stuck together such that the lines connecting the centres form an equilateral triangle of side of length \(d\). The ratio \(\mathrm{I}_{0} / \mathrm{I}_{\mathrm{A}}\) of moment of inertia \(\mathrm{I}_{0}\) of the system about an axis passing the centroid and about center of any of the spheres \(\mathrm{I}_{\mathrm{A}}\) and perpendicular to the plane of the triangle is

A physical quantity \(y\) is represented by the formula \(y=m^{2}\, r^{-4}\, g^{x}\,l^{-\frac{3}{2}}\). If the percentage error found in \(y, m, r, l\) and \(g\) are \(18,1,0.5,4\) and \(p\) respectively, then find the value of \(x\) and \(p\).

The time period of a satellite, revolving above earth's surface at a height equal to \(R\) will be (Given \(g =\pi^2 m / s ^2, R =\) radius of earth)

The de-Broglie wavelength associated with an electron and a proton were calculated by accelerating them through same potential of \(100\, V\). What should nearly be the ratio of their wavelengths ? \(\left( m _{ P }=1.00727\, u , m _{ e }=0.00055 \,u \right)\)

A parallel plate capacitor with plate area \('A'\) and distance of separation \('d'\) is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as : \(\varepsilon(x)=\varepsilon_{0}+k x, \text { for }\left(0\,<\,x \leq \frac{d}{2}\right)\) \(\varepsilon(x)=\varepsilon_{0}+k(d-x)\), for \(\left(\frac{d}{2} \leq x \leq d\right)\)

A hydrogen atom makes a transition from \(n = 2\) to  \(n = 1\) and emits a photon. This photon strikes a doubly ionized lithium atom \((z = 3)\)  in excited state and completely removes the orbiting electron. The least quantum number for the excited state of the ion for the process is

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : The Bohr model is applicable to hydrogen and hydrogen-like atoms only.
Reason R : The formulation of Bohr model does not include repulsive force between electrons.
In the light of the above statements, choose the correct answer from the options given below :

In a metre-bridge when a resistance in the left gap is \(2\ \Omega\) and unknown resistance in the right gap, the balance length is found to be \(40\ \mathrm{cm}\). On shunting the unknown resistance with \(2\ \Omega\), the balance length changes by _______.