The time taken by simple pendulum for one oscillation is T on earth's surface. Its time period becomes xT when taken to a height R (equal to earth's radius) above the earth's surface. The value of x is

The masses and radii of the earth and moon are $M_1$, $R_1$ and $M_2$, $R_2$ and respectively, Their centres are at a distance 'd' apart. The minimum speed with which body of mass 'm' should be projected from a distance $2d/3$ from the centre of $M_1$ so as to escape to infinity is

The distance of the two planets A and B from the sun are '$r_A$' and '$r_B$' respectively such that $r_B = 100\, r_A$. The ratio of the speed of planet A to that of planet B is (both the planets are revolving around the sun)

What should be the angular velocity of earth due to rotation about its own axis so that the weight at equator becomes $\left(\dfrac{3}{5}\right)^{th}$ of initial value? ($g =$ acceleration due to gravity, R = radius of earth)