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. \(\frac{g(1-2 \mu)}{2}\)

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

2. $\mathrm{\alpha }>\frac{\mathrm{g}}{\mathrm{\mu m}}$

3. $\mathrm{\alpha }\ge \frac{\mathrm{g}}{\mathrm{\mu }}$

4. $\mathrm{\alpha }<\frac{\mathrm{g}}{\mathrm{\mu }}$

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 θ to the vertical. The block will remain in equilibrium if the coefficient of friction between it and the surface is: 

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

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

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

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

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:
    

If \(\mu\) between block \(A\) and inclined plane is \(0.5\) and that between block \(B\) and the inclined plane is \(0.8\), then the normal reaction between blocks \(A\) and \(B\) will be:
                     
1. \(180\) N
2. \(216\) N
3. \(0\)
4. None of these

Two blocks of masses \(2\) kg and \(3\) kg placed on a horizontal surface are connected by a massless string. If \(3\) kg is pulled by \(10\) N as shown in the figure, then the force of friction acting on the \(2\) kg block will be: [Take \(g=10~\text{m/s}^2\)] $\mathrm{}$