A square shaped hole of side \(l=\frac{a}{2}\) is carved out at a distance \(d =\frac{ a }{2}\) from the centre \('O'\) of a uniform circular disk of radius \(a\). If the distance of the centre of mass of the remaining portion from \(O\) is \(-\frac{a}{X},\) value of \(X\) (to the nearest integer) is.......

A metallic sphere cools from \(50^{\circ} C\) to \(40^{\circ} C\) in \(300 \,s.\) If atmospheric temperature around is \(20^{\circ} C ,\) then the sphere's temperature after the next \(5\) minutes will be close to\(.....C\)

Three masses 200 kg, 300 kg and 400 kg are placed at the vertices of an equilateral triangle with sides 20 m . They are rearranged on the vertices of a bigger triangle of side 25 m and with the same centre. The work done in this process _________ J.

Two identical strings \(X\) and \(Z\) made of same material have tension \(T _{ x }\) and \(T _{ z }\) in them. If their fundamental frequencies are \(450\, Hz\) and \(300\, Hz ,\) respectively, then the ratio \(T _{ x } / T _{ z }\) is\(.....\)

Identify the pair of physical quantities which have different dimensions 

Assuming the earth to be a sphere of uniform mass density, a body weighed \(300 \mathrm{~N}\) on the surface of earth. How much it would weigh at \(R / 4\) depth under surface of earth?

In a hydrogen like ion, the energy difference between the \(2^{\text {nd }}\) excitation energy state and ground is 108.8 eV. The atomic number of the ion is

The figure represents the momentum time \((p-t)\) curve for a particle moving along an axis under the influence of the force. Identify the regions on the graph where the magnitude of the force is maximum and minimum respectively ? If \(\left(t_3-t_2\right) < t_1\)

An amount of ice of mass \(10^{-3} \mathrm{~kg}\) and temperature \(-10^{\circ} \mathrm{C}\) is transformed to vapour of temperature \(110^{\circ} \mathrm{C}\) by applying heat. The total amount of work required for this conversion is, (Take, specific heat of ice \(=2100 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}\), specific heat of water \(=4180 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}\), specific heat of steam \(=1920 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}\), Latent heat of ice \(=3.35 \times 10^5 \mathrm{Jkg}^{-1}\) and Latent heat of steam \(=2.25 \times 10^6\) \(\mathrm{Jkg}^{-1}\) )

Consider a water tank as shown in the figure. It's cross-sectional area is \(0.4\, m ^{2}\). The tank has an opening \(B\) near the bottom whose cross-section area is \(1\, cm ^{2}\). A load of \(24\, kg\) is applied on the water at the top when the height of the water level is \(40\, cm\) above the bottom, the velocity of water coming out the opening \(B\) is \(v\, ms ^{-1}\). The value of \(v\), to the nearest integer, is ......\(m/s\). [Take value of \(g\) to be \(10 \,ms ^{-2}\) ]

For the four sets of three measured physical quantities as given below. Which of the following options is correct ? \((i)\) \(\mathrm{A}_{1}=24.36, \mathrm{B}_{1}=0.0724, \mathrm{C}_{1}=256.2\) \((ii)\) \(\mathrm{A}_{2}=24.44, \mathrm{B}_{2}=16.082, \mathrm{C}_{2}=240.2\) \((iii)\) \(\mathrm{A}_{3}=25.2, \mathrm{B}_{3}=19.2812, \mathrm{C}_{3}=236.183\) \((iv)\) \(\mathrm{A}_{4}=25, \mathrm{B}_{4}=236.191, \mathrm{C}_{4}=19.5\)

Two particles \(A\) and \(B\) of equal mass \(M\) are moving with the same speed \(v\) as shown in the figure . They collide completely inelastically and move as a single particle \(C\). The angle \(\theta \) that the path of \(C\) makes with the \(X-\) axis is given by