A radioactive material decays by simultaneous emissions of two particles with half lives of \(1400\, years\) and \(700\, years\) respectively. What will be the time after which one third of the material remains? (Take In \(3=1.1\) ) (In \(years\))

An electron of mass \(m _{ e }\) and a proton of mass \(m _{ p }=1836 m _{ e }\) are moving with the same speed. The ratio of their de Broglie wavelength \(\frac{\lambda_{\text {electron }}}{\lambda_{\text {proton }}}\) will be ....... .

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)

A string is wound around a hollow cylinder of mass \(5\, kg\) and radius \(0.5\,m\). If the string is now pulled with a horizontal force of \(40\, N\), and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be ......... \(rad/s^2\) .(Neglect the mass and thickness of the string)

By what percentage will the illumination of the lamp decrease if the current drops by \(20 \%\) ?

Let \([ {\varepsilon _0} ]\) denote the dimensional formula of the permittivity of vacuum. If \(M =\) mass, \(L=\) length, \(T =\) time and \(A=\) electric current, then:

Statement \(I:\) If three forces \(\vec{F}_{1}, \vec{F}_{2}\) and \(\vec{F}_{3}\) are represented by three sides of a triangle and \(\overrightarrow{{F}}_{1}+\overrightarrow{{F}}_{2}=-\overrightarrow{{F}}_{3}\), then these three forces are concurrent forces and satisfy the condition for equilibrium. Statement \(II:\) A triangle made up of three forces \(\overrightarrow{{F}}_{1}, \overrightarrow{{F}}_{2}\) and \(\overrightarrow{{F}}_{3}\) as its sides taken in the same order, satisfy the condition for translatory equilibrium. In the light of the above statements, choose the most appropriate answer from the options given below:

A capacitor P with capacitance \( 10\times10^{-6} F \) is fully charged with a potential difference of 6.0 V and disconnected from the battery. The charged capacitor P is connected across another capacitor Q with capacitance \( 20\times10^{-6} F \). The charge on capacitor Q when equilibrium is established will be \( \alpha\times10^{-5} C \).(assume capacitor \(Q\) does not have any charge initially),The value of \( \alpha \) is __________ .

Two identical electric point dipoles have dipole moments \(\overrightarrow{p}_{1}= p\hat i\) and \(\overrightarrow{p}_{2}=- p\hat i\) and are held on the \(x-\)axis at distance \('a'\) from each other. When released, they move along the \(x-\)axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is \('m',\) their speed when they arc infinitely far apart is

Find the electric field at point \(P\) (as shown in figure) on the perpendicular bisector of a uniformly charged thin wire of length \(L\) carrying a charge \(Q.\) The distance of the point \(P\) from the centre of the rod is \(a=\frac{\sqrt{3}}{2} L\).

The angular frequency of the damped oscillator is given by, \(\omega  = \sqrt {\left( {\frac{k}{m} - \frac{{{r^2}}}{{4{m^2}}}} \right)} \) where \(k\) is the spring constant, \(m\) is the mass of the oscillator and \(r\) is the damping constant. If the ratio \(\frac{{{r^2}}}{{mk}}\) is \(8\%\), the change in time period compared to the undamped oscillator is approximately as follows

Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is \(\sqrt{2}\) times the speed of sound, then the value of \(n\) will be