A block of mass \(m\) is placed on a surface with a vertical cross section given by \(y=\frac{x^3}{6}\). If the coefficient of friction is \(0.5\), then the maximum height above the ground at which the block can be placed without slipping is:
1. \( \frac{2}{3}~\text{m} \)
2. \( \frac{1}{3} ~\text{m} \)
3. \( \frac{1}{2} ~\text{m} \)
4. \( \frac{1}{6} ~\text{m}\)

Given in the figure are two blocks \(A\) and \(B\) of weight \(20~\text{N}\) and \(100~\text{N}\), respectively. These are being pressed against a wall by a force \(F\) as shown. If the coefficient of friction between the blocks is \(0.1\) and between block \(B\) and the wall is \(0.15\), the frictional force applied by the wall on block \(B\) is:
       
1. \(100~\text{N}\)
2. \(80~\text{N}\)
3. \(120~\text{N}\)
4. \(150~\text{N}\)

Two masses \(m_1=5~\text{kg}\) and \(m_2=10~\text{kg}\), connected by an inextensible string over a frictionless pulley, are moving as shown in the figure. The coefficient of friction of the horizontal surface is \(0.15\). The minimum weight \(m\) that should be put on top of \(m_2\) to stop the motion is:

1. \(18.3~\text{kg}\)
2. \(23.3~\text{kg}\)
3. \(43.3~\text{kg}\)
4. \(10.3~\text{kg}\)

A particle is moving with a uniform speed in a circular orbit of radius \(R\) in a central force inversely proportional to the \(n^{\text{th}}\) power of \(R\). If the period of rotation of the particle is \(T\), then:
1. \(T \propto R^{3 / 2} ~\text{for any } n\)
2. \(T \propto R^{\frac{{n}}{2}+1} \)
3. \({T} \propto {R}^{({n}+1) / 2} \)
4. \( T \propto R^{n / 2} \)

The mass of a hydrogen molecule is \(3.32 \times 10^{-27}~\text{kg}.\) If \(10^{23}\) hydrogen molecules strike, per second, a fixed wall of area \(2~\text{cm}^2\) at an angle of \(45^\circ\) to the normal, and rebound elastically with a speed of \(10^3~\text{m/s},\) then the pressure on the wall is nearly:
1. \( 2.35 \times 10^3 ~\text{N/m}^2 \)
2. \(4.70 \times 10^3 ~\text{N/m}^2 \)
3. \(2.35 \times 10^2 ~\text{N/m}^2 \)
4. \(4.70 \times 10^2 ~\text{N/m}^2 \)