If acceleration due to gravity on the surface of earth is $10 {\mathrm{ms}}^{–2}$ and let acceleration due to gravitational acceleration at surface of another planet of our solar system be $5 {\mathrm{ms}}^{–2}$. An astronaut weighing $50 \mathrm{kg}$ on earth goes to this planet in a spaceship with a constant velocity. The weight of the astronaut with time of flight is roughly given by,

A spherical hollow cavity is made in a lead sphere of radius $R$, such that its surface touches the outside surface of the lead sphere and passes through its centre. The mass of the sphere before hollowing was $M$. With what gravitational force will the hollowed-out lead sphere attract a small sphere of mass $m$, which lies at a distance d from the centre of the lead sphere on the straight line connecting the centres of the spheres and that of the hollow, if $d=2R$ 

A straight rod of length $l$ extends from $x=\alpha \text{ to} x=l+\alpha$ as shown in the figure. If the mass per unit length is $(a+b{x}^2).$ The gravitational force it exerts on a point mass $m$ placed at $x=0$ is given by, 

A uniform ring of mass $M$ is lying at a distance $\sqrt{3}R$ from the centre of a uniform sphere of mass $m$ just below the sphere as shown in the figure where $R$ is the radius of the ring as well as that of the sphere. Then gravitational force exerted by the ring on the sphere is 

Two small balls of mass $m$ each are suspended side by two equal threads of length $L$ as shown in the figure. If the distance between the upper ends of the threads be $a$, the angle $\mathrm{\theta }$ that the threads will make with the vertical due to attraction between the balls is 

A block of mass $m$ is lying at a distance $r$ from a spherical shell of mass $m$ and radius $r$as shown in the figure. Then