A particle executes S.H.M. of period $$\frac{2 \pi}{\sqrt{3}}$$ second along a straight line $$4 \mathrm{~cm}$$ long. The displacement of the particle at which the velocity is numerically equal to the acceleration is

Two consecutive harmonics of an air column in a pipe closed at one end are of frequencies 150 Hz and 250 Hz. The fundamental frequency of an air column is

Two masses '$$m_{\mathrm{a}}$$' and '$$\mathrm{m}_{\mathrm{b}}$$' moving with velocities '$$v_{\mathrm{a}}$$' and '$$v_{\mathrm{b}}$$' opposite directions collide elastically. Alter the collision '$$m_a$$' and '$$m_b$$' move with velocities and '$$v_{\mathrm{b}}$$' and '$$v_a$$' respectively, then the ratio $$\mathrm{m_a:m_b}$$ is

A sample of radioactive element contains $$8 \times 10^{16}$$ active nuclei. The halt-life of the element is 15 days. The number of nuclei decayed after 60 days is