Calculate the acceleration of the block and trolly system shown in the figure. The coefficient of kinetic friction between the trolly and the surface is \(0.05.\) 
( \(g=10~\text{m/s}^2,\) the mass of the string is negligible and no other friction exists)

A body of mass \(m\) is kept on a rough horizontal surface (coefficient of friction = \(\mu).\) A horizontal force is applied to the body, but it does not move. The resultant of normal reaction and the frictional force acting on the object is given by \(\vec {F}\) where:
1. \(|{\vec {F}}| = mg+\mu mg\)
2. \(|\vec {F}| =\mu mg\)
3. \(|\vec {F}| \le mg\sqrt{1+\mu^2}\)
4. \(|\vec{F}| = mg\)

Which one of the following statements is incorrect?

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:
              

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:
1. \( \dfrac{\left({m}_2+\mu_{{k}}{m}_1\right) {g}}{\left({m}_1+{m}_2\right)}\)

2. \( \dfrac{\left({m}_2-\mu_{{k}} {m}_1\right) {g}}{\left({m}_1+{m}_2\right)}\)

3. \(\dfrac{{m}_1 {~m}_2\left(1-\mu_{{k}}\right) {g}}{\left({m}_1+{m}_2\right)}\)

4. \( \dfrac{{m}_1 {~m}_2\left(1+\mu_{{k}}\right)}{{m}_1+{m}_2} {g}\)

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 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. \(\mu=1/\tan \theta\)

A car of mass \(m\) is moving on a level circular track of radius \(R\). If \(\mu_s\) represents the static friction between the road and tyres of the car, the maximum speed of the car in circular motion is given by:
1. \(\sqrt{\frac{Rg}{\mu_s} }\)
2. \(\sqrt{\frac{mRg}{\mu_s}}\)
3. \(\sqrt{\mu_s Rg}\)
4. \(\sqrt{\mu_s m Rg}\)