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. \(\frac{g(1-2 \mu)}{2}\)

Two masses, \(A\) and \(B\), of mass \(4\) kg and \(1\) kg, respectively, are connected with the help of a massless inextensible string. Mass \(A\) is placed on a rough horizontal table, and mass \(B\) is suspended with the help of a string passing through a smooth hole at the centre of the table. For the system to be in equilibrium, what should be the minimum value of the coefficient of friction?
1. \(0.5\)
2. \(0.25\)
3. \(2.5\)
4. \(0.125\)

A block of mass \(m\) is in contact with the cart \((C)\) as shown in the figure. 
                    
The coefficient of static friction between the block and the cart is \(\mu.\) The acceleration \(a\) of the cart that will prevent the block from falling satisfies:
1. \(a > \frac{mg}{\mu}\)
2. \(a > \frac{g}{\mu m}\)
3. \(a \ge \frac{g}{\mu}\)
4. \(a < \frac{g}{\mu}\)

A block \(B\) is pushed momentarily along a horizontal surface with an initial velocity \(v.\) If \(\mu\) is the coefficient of sliding friction between \(B\) and the surface, the block \(B\) will come to rest after a time: 
 
1. \(v \over g \mu\)
2. \(g \mu \over v\)
3. \(g \over v\)
4. \(v \over g\)

A block of mass \(m\) lying on a rough horizontal plane is acted upon by a horizontal force \(P\) and another force \(Q\) inclined at an angle \(\theta\) to the vertical. The block will remain in equilibrium if the coefficient of friction between it and the surface is: 

                        
1. \(\dfrac{(P+Q\sin\theta)}{(mg+Q\cos\theta)}\)

2. \(\dfrac{(P\cos\theta+Q)}{(mg-Q\sin\theta)}\)

3. \(\dfrac{(P+Q\cos\theta)}{(mg+Q\sin\theta)}\)

4. \(\dfrac{(P\sin\theta-Q)}{(mg-Q\cos\theta)}\)

A uniform chain of length \(L\) hangs partly from a table which is kept in equilibrium by friction. If the maximum length that can be supported without slipping is \(l,\) then the coefficient of friction between the table and the chain is: 
1. \(\frac{l}{L}\)
2. \(\frac{l}{L+l}\)
3. \(\frac{l}{L-l}\)
4. \(\frac{L}{L+l}\)

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

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 \(\overrightarrow F\) where:
1. \(|{\overrightarrow F}| = mg+\mu mg\)
2. \(|\overrightarrow F| =\mu mg\)
3. \(|\overrightarrow F| \le mg\sqrt{1+\mu^2}\)
4. \(|\overrightarrow F| = mg\)

Block \(\mathrm{A}\) has a mass of \(10\) kg. Between block \(\mathrm{A}\) and the table, the coefficient of static friction is \(0.2\), and the coefficient of kinetic friction is also \(0.2\). The required mass of \(\mathrm{B}\) to start the motion will be:

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. \(\mu=1/\tan \theta\)