A block of mass \(m\) is placed on a smooth inclined wedge \(ABC\) of inclination \(\theta\) as shown in the figure. The wedge is given an acceleration '\(a\)' towards the right. The relation between \(a\) and \(\theta\) for the block to remain stationary on the wedge is:
          
1. \(a = \dfrac{g}{\mathrm{cosec }~ \theta}\)
2. \(a = \dfrac{g}{\sin\theta}\)
3. \(a = g\cos\theta\)
4. \(a = g\tan\theta\)

A car is negotiating a curved road of radius \(R\). The road is banked at an angle \(\theta\). The coefficient of friction between the tyre of the car and the road is \(\mu_s\). The maximum safe velocity on this road is:
1. \(\sqrt{\operatorname{gR}\left(\dfrac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\mathrm{s}} \tan \theta}\right)}\)
2. \(\sqrt{\frac{\mathrm{g}}{\mathrm{R}}\left(\dfrac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\mathrm{s}} \tan \theta}\right)}\)
3. \(\sqrt{\frac{\mathrm{g}}{\mathrm{R}^2}\left(\dfrac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\operatorname{s}} \tan \theta}\right)}\)
4. \(\sqrt{\mathrm{gR}^2\left(\dfrac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\mathrm{s}} \tan \theta}\right)}\)

A plank with a box on it at one end is gradually raised about the other end. As the angle of inclination with the horizontal reaches \(30^\circ,\) the box starts to slip and slide \(4.0~\text m\) down the plank in \(4.0~\text s.\) The coefficients of static and kinetic friction between the box and the plank will be, respectively:
              

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 a radius \(r\). The tangential speed of lighter stone is \(n\) times that of the value of heavier stone when they experience the same centripetal forces. The value of \(n\) is:

Three blocks \(\mathrm{A}\), \(\mathrm{B}\), and \(\mathrm{C}\) of masses \(4~\text{kg}\), \(2~\text{kg}\), and \(1~\text{kg}\) respectively, are in contact on a frictionless surface, as shown. If a force of \(14~\text{N}\) is applied to the \(4~\text{kg}\) block, then the contact force between \(\mathrm{A}\) and \(\mathrm{B}\) is: 
          
1. \(2~\text{N}\)
2. \(6~\text{N}\)
3. \(8~\text{N}\)
4. \(18~\text{N}\)

A block \(\mathrm{A}\) of mass \(m_1\) rests on a horizontal table. A light string connected to it passes over a frictionless pulley at the edge of the table and from its other end, another block \(\mathrm{B}\) of mass \(m_2\) is suspended. The coefficient of kinetic friction between block \(\mathrm{A}\) and the table is \(\mu_k\). When block \(\mathrm{A}\) is sliding on the table, the tension in the string is:

A system consists of three masses \(m_1,\) \(m_2,\) and \(m_3\) connected by a string passing over a pulley \(\mathrm{P}.\) The mass \(m_1\) hangs freely, and \(m_2\) and \(m_3\) are on a rough horizontal table (the coefficient of friction \(=\mu.\)) The pulley is frictionless and of negligible mass. The downward acceleration of mass \(m_1\) is:
(Assume \(m_1=m_2=m_3=m\) and \(g\) is the acceleration due to gravity.) 
                
1. \(\frac{g(1-g \mu)}{9}\)
2. \(\frac{2 g \mu}{3}\)
3. \( \frac{g(1-2 \mu)}{3}\)
4. \(\frac{g(1-2 \mu)}{2}\)

The force \(F\) acting on a particle of mass \(m\) is indicated by the force-time graph shown below. The change in momentum of the particle over the time interval from \(0\) to \(8\) s is:

1. \(24~\text{N-s}\)
2. \(20~\text{N-s}\)
3. \(12~\text{N-s}\)
4. \(6~\text{N-s}\)

A balloon with mass \(m\) is descending down with an acceleration \(a\) (where \(a<g\)). How much mass should be removed from it so that it starts moving up with an acceleration \(a\)?

The upper half of an inclined plane of inclination \(\theta\) is perfectly smooth while the lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom if the coefficient of friction between the block and the lower half of the plane is given by:
1. \(\mu=2/\tan \theta\)
2. \(\mu=2\tan \theta\)
3. \(\mu=\tan \theta\)
4. \(\mu=1/\tan \theta\)