$A$ van is moving with a speed of $108 \,km/hr$ on a level road where the coefficient of friction between the tyres and the road is $0.5$. For the safe driving of the van, the minimum radius of curvature of the road shall be (Acceleration due to gravity, $g = 10 \,m/s^2$) (in $\,m$)

$A$ stone of mass $1\,kg$ is tied to the end of a massless string of length $1\,m$. If the breaking tension of the string is $400\,N$,then the maximum linear velocity the stone can have without breaking the string,while rotating in a horizontal plane,is $.......\,ms^{-1}$.

$A$ 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 $\mu_{s},$ then the magnitude of negative lift $F_{L}$ acting downwards on the car is (Assume forces on the four tyres are identical and $g =$ acceleration due to gravity)

$A$ string just breaks under a load of $50 \ kg$. $A$ mass of $1 \ kg$ is attached to one end of this string $10 \ m$ long and is rotated in a horizontal circle. Calculate the greatest number of revolutions that the mass can make in one second without breaking the string. (Take $g = 10 \ m/s^2$)

Two cars are moving on a banked circular path of radius $8 \ m$ having an angle of banking $45^{\circ}$. If the coefficients of static friction between the road and the tyres of the two cars are $0.5$ and $0.4$ respectively,then the ratio of maximum permissible speeds of the cars to avoid slipping is

$A$ cyclist speeding at $18 \; km/h$ on a level road takes a sharp circular turn of radius $3 \; m$ without reducing the speed. The coefficient of static friction between the tyres and the road is $0.1$. Will the cyclist slip while taking the turn?