$A$ particle of mass $m$ is suspended from a ceiling through a string of length $L$. The particle moves in a horizontal circle of radius $r$ such that $r = \frac{L}{\sqrt{2}}$. The speed of the particle will be:

$A$ person with his hands in his pockets is skating on ice at a velocity of $10 \, m/s$ and describes a circle of radius $50 \, m$. What is his inclination with the vertical?

$A$ car is moving with velocity $v$ at the top of a semi-circular hill of radius $40 \,m$ such that the normal force on it is zero. Find the velocity $(v)$ of the car. [Use $g=10 \,ms^{-2}$] (in $\,ms^{-1}$)

$A$ vehicle of mass $200\,kg$ is moving along a levelled curved road of radius $70\,m$ with an angular velocity of $0.2\,rad/s$. The centripetal force acting on the vehicle is $.........\,N$.

$A$ string of length $2 \ m$ can withstand a maximum tension of $16000 \ N$. $A$ particle of mass $2 \ kg$ is tied to one end of the string and rotated in a horizontal circle. What is the maximum number of rotations per second the particle can make so that the string does not break? (Take $\pi^2 = g = 10 \ m/s^2$)

What should be the speed of a car in $m/s$ to cross a bridge with a radius of curvature of $20 \; m$ without losing contact with the bridge? $(g = 9.8 \; m/s^2)$