A motorcyclist moving with a velocity of \(72\ \text{km/h}\) on a flat road takes a turn on the road at a point where the radius of curvature of the road is \(20\ \text{m}\). The acceleration due to gravity is \(10\ \text{m/s}^2\). To avoid skidding, he must not bend with respect to the vertical plane by an angle greater than:

1. \(𝜃=\tan^{−1}6\)

2. \(𝜃=\tan^{−1}2\)$\mathrm{}$

3. \(𝜃=\tan^{−1}25.92\)$$

4. \(𝜃=\tan^{−1}4\)$$

A particle moves in a circular orbit under the action of a central attractive force inversely proportional to the distance \(r\). The speed of the particle is:

1. Proportional to \(r^2\)

2. Independent of \(r\)

3. Proportional to \(r\)

4. Proportional to \(\dfrac 1r\)

Two masses \(M\) and \(m\) are attached to a vertical axis by weightless threads of combined length \(l\). They are set in rotational motion in a horizontal plane about this axis with constant angular velocity \(ω\). If the tensions in the threads are the same during motion, the distance \(x\) of \(M\) from the axis is:

1. \(\dfrac{Ml}{M+m}\)

2. \(\dfrac{ml}{M+m}\)

3. \(\dfrac{M+m}{M}l\)

4. \(\dfrac{M+m}{m}l\)

A \(500\ \text{kg}\) car takes a round turn of radius \(50\ \text{m}\) with a velocity of \(36\ \text{km/h}\). The centripetal force is:

1. \(250\ \text{N}\)

2. \(750\ \text{N}\)

3. \(1000\ \text{N}\)

4. \(1200\ \text{N}\)

A ball of mass \(0.25\ \text{kg}\) attached to the end of a string of length \(1.96\ \text{m}\) is moving in a horizontal circle. The string will break if the tension is more than \(25\ \text{N}\). What is the maximum speed with which the ball can be moved:

1. \(14\ \text{m/s}\)

2. \(3\ \text{m/s}\)

3. \(3.92\ \text{m/s}\)

4. \(5\ \text{m/s}\)

A body of mass \(5\ \text{kg}\) is moving in a circle of radius \(1\ \text{m}\) with an angular velocity of \(2\ \text{rad/s}\). The centripetal force is:

1. \(10\ \text{N}\)

2. \(20\ \text{N}\)

3. \(30\ \text{N}\)

4. \(40\ \text{N}\)

A stone of mass \(16\ \text{kg}\) is attached to a string \(144\ \text{m}\) long and is whirled in a horizontal circle. The maximum tension the string can withstand is \(16\) Newton. The maximum velocity of revolution that can be given to the stone without breaking it will be:

1. \(20\ \text{ms}^{–1}\)
2. \(16\ \text{ms}^{–1}\)
3. \(14\ \text{ms}^{–1}\)
4. \(12\ \text{ms}^{–1}\)

The coefficient of friction between the road and tyre is \(0.2\). The maximum velocity for skidding of a car moving on a circular track of radius \(100\ \text{m}\) is:

1. \(0.14\ \text{m/s}\)

2. \(140\ \text{m/s}\)

3. \(1.4\ \text{m/s}\)

4. \(14\ \text{m/s}\)

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 cyclist riding a bicycle at a speed of \(14\sqrt3\ \text{ms}^{–1}\) takes a turn around a circular road of radius \( 20\sqrt3\ \text{m}\) without skidding. His inclination to the vertical is:

1. \(30^\circ\)

2. \(​​90^\circ\)

3. \(45^\circ\)

4. \(60^\circ\)