A small coin is kept at a distance \(r\) from the centre of a gramophone disc rotating at an angular speed \(\omega\). The minimum coefficient of friction for which a coin will not slip is:
1. \(\dfrac{rω^{2}}{g}\)
2. \(\dfrac{g}{r\omega^2}\)
3. \(\dfrac{r^2ω^{2}}{g}\)
4. \(\dfrac{rω}{g}\)

A body of mass \(m\) is moving on a concave bridge \(ABC\) of the radius of curvature \(R\) at a speed \(v.\) The normal reaction of the bridge on the body at the instant it is at the lowest point of the bridge is:

1. \(mg-\frac{mv^{2}}{R}\)
2. \(mg+\frac{mv^{2}}{R}\)
3. \(mg\)
4. \(\frac{mv^{2}}{R}\)

A roller coaster is designed such that riders experience "weightlessness" as they go round the top of a hill whose radius of curvature is \(20\) m. The speed of the car at the top of the hill is between:
1. \(14~\text{m/s}~\text{and}~15~\text{m/s}\)
2. \(15~\text{m/s}~\text{and}~16~\text{m/s}\)
3. \(16~\text{m/s}~\text{and}~17~\text{m/s}\)
4. \(13~\text{m/s}~\text{and}~14~\text{m/s}\)

Two stones of masses \(m\) and \(2m\) are whirled in horizontal circles, the heavier one in a radius \(\frac{r}{2}\) and the lighter one in the radius \(r.\) The tangential speed of lighter stone is \(n\) times that of heavier stone when they experience the same centripetal forces. The value of \(n\) is:

A block of mass \(10~\text{kg}\) is in contact with the inner wall of a hollow cylindrical drum of radius \(1~\text{m}.\) The coefficient of friction between the block and the inner wall of the cylinder is \(0.1.\) The minimum angular velocity needed for the cylinder, which is vertical and rotating about its axis, will be: 
\(\left(g= 10~\text{m/s}^2\right )\)
1. \(10~\pi~\text{rad/s}\)
2. \(\sqrt{10}~\pi~\text{rad/s}\)
3. \(\frac{10}{2\pi}~\text{rad/s}\)
4. \(10~\text{rad/s}\)