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. \(4mg\)

Choose the incorrect alternative:

A block slides down on a \(45^{\circ}\) rough incline in thrice the time it takes to slide down on a frictionless \(45^{\circ}\) incline of the same length. The coefficient of friction between the block and the rough incline is:

The kinetic energy 'K' of a particle moving in a circular path varies with the distance covered S as K = a${\mathrm{S}}^2$, where a is constant. The angle between the tangential force and the net force acting on the particle is: (R is the radius of the circular path)

1.  ${\tan }^{-1}\left(\frac{\mathrm{S}}{\mathrm{R}}\right)$

2.  ${\tan }^{-1}\left(\frac{\mathrm{R}}{\mathrm{S}}\right)$

3.  ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{R}}\right)$

4.  ${\tan }^{-1}\left(\frac{\mathrm{R}}{\mathrm{a}}\right)$

The tension in the string connecting blocks, \(B\) and \(C\), placed on a smooth horizontal surface in the following diagram is:

If two forces \(\left(6 \hat{i}+8\hat j\right)\) and \(\left(4 \hat{i}+4\hat j \right)\)N are acting on a body of mass \(2\) kg, then the acceleration produced in the body (in \(\text{m/s}^{2}\)) will be:
1. \(\left(5 \hat{i}+6\hat j\right)\)
2. \(\left(10 \hat{i}+12\hat j\right)\)
3. \(\left(6 \hat{i}+12\hat j\right)\)
4. \(\left(2 \hat{i}+3\hat j\right)\)

The friction between the front foot and the back foot when walking on a horizontal surface is, respectively:

1. Forward, forward

2. Backward, backward

3. Forward, backward

4. Backward, forward

A simple pendulum hangs from the roof of a train moving on horizontal rails. If the string is inclined towards the front of the train, then the train is:

A block of mass \(M\) is pulled by a force \(F\), making an angle \(\theta\) with the horizontal on a smooth horizontal surface as shown. If \(a\) is the acceleration of block on the surface, then the contact force between the block and the surface will be:
        
1. \(Mg+ Ma\cos\theta\)
2. \(Mg- Ma\cos\theta\)
3. \(Mg+ Ma\tan\theta\)
4. \(Mg- Ma\tan\theta\)

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. \(\frac{mv^{2}}{R}\)