A car of mass \(m\) is moving on a level circular track of radius \(R\). If \(\mu_s\) represent the static friction between the road and tyres of the car, then the maximum speed of the car in circular motion is given by:

A particle of mass \(m\) is suspended from a ceiling through a massless string. The particle moves in a horizontal circle as shown in the given figure. The tension in the string is:
 

1. \(mg\)

2. \(2mg\)

3. \(3mg\)

4. \(4mg\)

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. \(\frac{rω^{2}}{g}\)
2. \(\frac{g}{r\omega^2}\)
3. \(\frac{r^2ω^{2}}{g}\)
4. \(\frac{rω}{g}\)

A ball of mass \(0.1~\text{kg}\) is whirled in a horizontal circle of radius \(1\) m by means of a string at an initial speed of \(10~\text{rpm}\) . Keeping the radius constant, the tension in the string is reduced to one quarter of its initial value. The new speed is:

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. \(13~\text{m/s}~\text{and}~14~\text{m/s}\)

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. \(\frac{mv^{2}}{R}\)

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: