A soap bubble of radius \(3\, {cm}\) is formed inside the another soap bubble of radius \(6\, {cm}\). The radius of an equivalent soap bubble which has the same excess pressure as inside the smaller bubble with respect to the atmospheric pressure is .......... \({cm}\) 

Given below are two statements: one is labelled as Assertion \(\mathrm{A}\) and the other is labelled as Reason \(\mathrm{R}\) Assertion A : The kinetic energy needed to project a body of mass m from earth surface to infinity is \(\frac{1}{2} \mathrm{mgR}\), where R is the radius of earth.
Reason \(\mathrm{R}\) : The maximum potential energy of a body is zero when it is projected to infinity from earth surface.
In the light of the above statements, choose the correct answer from the option given below

An experiment takes \(10\, minutes\) to raise the temperature of water in a container from \(0\,^oC\) to \(100\,^oC\) and another \(55\, minutes\) to convert it totally into steam by a heater supplying heat at a uniform rate . Neglecting the specific heat of the container and taking specific heat of water to be \(1\, cal / g\,^oC\), the heat of vapourization according to this experiment will come out to be ........ \(cal/g\)

A particle is moving \(5\) times as fast as an electron. The ratio of the de-Broglie wavelength of the particle to that of the electron is \(1.878 \times 10^{-4}\). The mass of the particle is close to

Voltage rating of a parallel plate capacitor is \(500\,V\). Its dielectric can withstand a maximum electric field of \({10^6}\,\frac{V}{m}\). The plate area is \(10^{-4}\, m^2\) . What is the dielectric constant if the capacitance is \(15\, pF\) ? (given \({ \in _0} = 8.86 \times {10^{ - 12}}\,{C^2}\,/N{m^2}\))

Consider two ideal diatomic gases \(\mathrm{A}\) and \(\mathrm{B}\) at some temperature \(T\). Molecules of the gas \(A\) are rigid, and have a mass \(m\). Molecules of the gas \(\mathrm{B}\) have an additional vibrational mode, and have a mass \(\frac{\mathrm{m}}{4} .\) The ratio of the specific heats \((\mathrm{C}_{\mathrm{v}}^{\mathrm{A}}\) and \(\mathrm{C}_{\mathrm{v}}^{\mathrm{B}})\) of gas \(\mathrm{A}\) and \(\mathrm{B}\), respectively is

A force acts on a \(2\,kg\) object so that its position is given as a function of time as \(x= 3t^2 + 5.\) What is the work done by this force in first \(5\,seconds\) ? ................ \(\mathrm{J}\)

Experimentally it is found that \(12.8\,eV\) energy is required to separate a hydrogen atom into a proton and an electron. So the orbital radius of the electron in a hydrogen atom is \(\frac{9}{ x } \times 10^{-10}\,m\). The value of the \(x\) is \(\left(1\,eV =1.6 \times 10^{-19}\,J , \frac{1}{4 \pi \epsilon_0}=9 \times 10^9\,Nm ^2 / C ^2\right.\) and electronic charge \(=1.6 \times 10^{-19}\,J\,C\) )

A uniform metal chain of mass \(m\) and length ' \(L\) ' passes over a massless and frictionless pulley. It is released from rest with a part of its length ' \(l\) ' is hanging on one side and rest of its length ' \(L -l\) ' is hanging on the other side of the pulley. At a certain point of time, when \(l=\frac{L}{x}\), the acceleration of the chain is \(\frac{g}{2}\). The value of \(x\) is ........

Consider two blocks A and B of masses \(m_1=10 \mathrm{~kg}\) and \(m_2=5 \mathrm{~kg}\) that are placed on a frictionless table. The block A moves with a constant speed \(v=3 \mathrm{~m} / \mathrm{s}\) towards the block B kept at rest. A spring with spring constant \(\mathrm{k}=3000 \mathrm{~N} / \mathrm{m}\) is attached with the block B as shown in the figure. After the collision, suppose that the blocks A and B, along with the spring in constant compression state, move together, then the compression in the spring is _______. (Neglect the mass of the spring)

Two coherent sources of light interfere. The intensity ratio of two sources is \(1: 4\). For this interference pattern if the value of \(\frac{I_{\max }+I_{\min }}{I_{\max _{\max }}-I_{\min }}\) is equal to \(\frac{2 \alpha+1}{\beta+3}\), then \(\frac{\alpha}{\beta}\) will be.

The time period of a satellite in a circular orbit of radius \(R\) is \(T\). The period of another satellite in a circular orbit of radius \(9 R\) is............ \(T\)