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 slides \(4.0\) m down the plank in \(4.0\) 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:
1. \(3\)
2. \(4\)
3. \(1\)
4. \(2\)

Three blocks A, B, and C of masses 4 kg, 2 kg, and 1 kg respectively, are in contact on a frictionless surface, as shown. If a force of 14 N is applied to the 4kg block, then the contact force between A and B is: 
       

1. 2 N

2. 6 N

3. 8 N

4. 18 N

A block A of mass ${\mathrm{m}}_1$ 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 B of mass m₂ is suspended. The coefficient of kinetic friction between block A and the table is ${\mathrm{\mu }}_{\mathrm{k}}$. When block A is sliding on the table, the tension in the string is:

A system consists of three masses m₁, m_2, and m₃ connected by a string passing over a pulley P. The mass m₁ hangs freely, and m₂ and m₃ are on a rough horizontal table (the coefficient of friction = μ). The pulley is frictionless and of negligible mass. The downward acceleration of mass m₁ is: (Assume m₁ = m₂ = m₃ = m and g is the acceleration due to gravity.) 
          

1. $\frac{g(1-g\mu )}{9}$

2. $\frac{2g\mu }{3}$

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:
      

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 \(a\)?
1. \( \frac{2 m a}{g+a} \)
2. \( \frac{2 m a}{g-a} \)
3. \( \frac{m a}{g+a} \)
4. \( \frac{m a}{g-a}\)

A body of mass (4m) is lying in the x-y plane at rest. It suddenly explodes into three pieces. Two pieces, each of mass (m) move perpendicular to each other with equal speeds (u). The total kinetic energy generated due to explosion is:

1. $m{u}^2$

2. 1.5$m{u}^2$

3. 2$m{u}^2$

4. 3$m{u}^2$

The upper half of an inclined plane of inclination θ 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. μ = 2/tanθ
2. μ = 2tanθ
3. μ = tanθ
4. μ = 1/tanθ

Three blocks with masses \(m\), \(2m\), and \(3m\) are connected by strings as shown in the figure. After an upward force \(F\) is applied on block \(m\), the masses move upward at constant speed \(v\). What is the net force on the block of mass \(2m\)? (\(g\) is the acceleration due to gravity)

           

1. \(2~mg\)
2. \(3~mg\)
3. \(6~mg\)
4. zero