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. \(|\vec{F}| = mg\)

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. \(180~\text N\) 
2. \(216~\text N\) 
3. \(0\)
4. none of these

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:
              

Which one of the following statements is incorrect?

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

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:

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. \(\mu=1/\tan \theta\)