A weightless thread can withstand tension upto 30 N. A stone of mass 0.5 kg is tied to it and is revolved in a circular path of radius 2m in a vertical plane. If g = 10 ms^-2, then the,maximum angular velocity of the stone can be :

A modern grand-prix racing car of mass $m$ is travelling on a flat track in a circular arc of radius $R$ with a speed $v.$ If the coefficient of static friction between the tyres and the track is ${\mu }_{\mathrm{s}},$ then the magnitude of negative lift $F_L$ acting downwards on the car is:

Statement I: A cyclist is moving on an unbanked road with a speed of $7 \mathrm{km} {\mathrm{h}}^{-1}$ and takes a sharp circular turn along a path of the radius of $2 \mathrm{m}$ without reducing the speed. The static friction coefficient is $0.2$. The cyclist will not slip and pass the curve $\left(g=9.8 \mathrm{m} {\mathrm{s}}^{-2}\right)$

Statement II : If the road is banked at an angle of $45°$, cyclist can cross the curve of $2 \mathrm{m}$ radius with the speed of $18.5 \mathrm{km} {\mathrm{h}}^{-1}$ without slipping. In the light of the above statements, choose the correct answer from the options given below.

The Earth (of mass $6\times {10}^{24} \mathrm{kg}$ ) revolves round the sun with an angular velocity of $2\times {10}^{-7} \mathrm{rad} {\mathrm{s}}^{-1}$ in a circular orbit of radius $1.5\times {10}^{11} \mathrm{km}.$ The force exerted by the sun on the earth is $\underline{ }$
 

Two stones of masses $m$ and $2m$ are whirled in horizontal circles, the heavier one in radius $\frac{r}{2}$ and the lighter one in radius $r$. The tangential speed of lighter stone is $n$ times that of the tangential speed of the heavier stone. They are reported to experience the same centripetal force. The value of $n$ is,