A flexible chain of mass \(m\) hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is \(30^{\circ}\). Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is _________ .

Substance \(A\) has atomic mass number \(16\) and half life of \(1\) day. Another substance \(B\) has atomic mass number \(32\) and half life of \(\frac{1}{2}\) day. If both \(A\) and \(B\) simultaneously start undergo radio activity at the same time with initial mass \(320\,g\) each, how many total atoms of \(A\) and \(B\) combined would be left after \(2\) days \(.........\times 10^{24}\)

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m , would be :

Unpolarized light of intensity \(I_0\) incident on surface of a block of glass at Brewster's angle. In that case, which one of ihe following statements is true ?

A force \(\overrightarrow{ F }=(\hat{ i }+2 \hat{ j }+3 \hat{ k }) N\) acts at a point \((4 \hat{ i }+3 \hat{ j }-\hat{ k }) m \cdot\) Then the magnitude of torque about the point \((\hat{i}+2 \hat{j}+\hat{k}) m\) will be \(\sqrt{ x } N - m .\) The value of \(x\) is\(........\)

The threshold wavelength for photoelectric emission from a material is \(5500\,\mathring A\). Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a \(A.\) \(75\,W\) infra-red lamp \(B.\) \(10\,W\) infra-red lamp \(C.\) \(75\,W\) ultra-violet lamp \(D.\) \(10\,W\) ultra-violet lamp Choose the correct answer from the options given below :

A bob of mass \('m'\) suspended by a thread of length \(l\) undergoes simple harmonic oscillations with time period \({T}\). If the bob is immersed in a liquid that has density \(\frac{1}{4}\) times that of the bob and the length of the thread is increased by \(1 / 3^{\text {rd }}\) of the original length, then the time period of the simple harmonic oscillations will be :-

There are \(100\) divisions on the circular scale of a screw gauge of pitch \(1 \mathrm{~mm}\). With no measuring quantity in between the jaws, the zero of the circular scale lies \(5\) divisions below the reference line. The diameter of a wire is then measured using this screw gauge. It is found the \(4\) linear scale divisions are clearly visible while \(60\) divisions on circular scale coincide with the reference line. The diameter of the wire is _______.

Using the given \(\mathrm{P}-\mathrm{V}\) diagram, the work done by an ideal gas along the path ABCD is :

The acceleration due to gravity at height \(h\) above the earth if \(h \ll R\) (radius of earth) is given by