$m_1=20\mathrm{kg}, m_2=30\mathrm{kg}. m_2$ is on smooth surface. Surface between $m_1$ and $m_2$ has ${\mu }_s=0.5$ and ${\mu }_k=0.3$. Find the acceleration of $m_1$ and $m_2$ for the followings figures $\left(a\right)$ and $\left(b\right)$. When Take $F=160 \mathrm{N}$.

 

Three identical rigid cylinders $A,B$ and $C$ are arranged on smooth inclined surfaces as shown in figure. Find the least value of $\theta$ that prevent the arrangement from collapse.

Block $C$ descends vertically at $1 {\text{m s}}^{-1}$. Find the velocity of $A$ relative to $B$.

A bead of mass $m$ is attached to one end of a spring of natural length $\sqrt{3}\mathrm{R}$ and spring constant $k=\frac{\left(\sqrt{3}+1\right)\mathrm{mg}}{R}$. The other end of the spring is fixed at point $A$ on a smooth fixed vertical ring of radius $R$ as shown in the figure. What is the normal reaction at $B$ just after the bead is released?

The blocks are of mass $\text{2 kg}$ shown is in equilibrium. At $t=0$ right spring in figure(i) and right string in figure (ii) breaks. Find the ratio of instantaneous acceleration of blocks?

Fig. shows a bead of mass $m$ moving with uniform speed $v$ through a U-shaped smooth wire the wire has a semicircular bending between $A$ and $B$. Calculate the average force exerted by the bead on the part $AB$ of the wire.

A monkey of mass, $40 \mathrm{kg}$ climbs on a rope that can stand a maximum tension of $600 \mathrm{N}$. Calculate the tension in rope in the following cases. In which case will the rope break:

(i) The monkey climbs up with an acceleration of $6 \mathrm{m} {\mathrm{s}}^{-2}$

(ii) The monkey climbs down with an acceleration of $4 \mathrm{m} {\mathrm{s}}^{-2}$

(iii) The monkey climbs up with a uniform speed of, $5 \mathrm{m} {\mathrm{s}}^{-1}$. Neglect the mass of string. 

Consider a conical pendulum having a bob of mass, $m$ is suspended from a ceiling through a string of length, $L$. The bob moves in a horizontal circle of radius, $r$. Find (a) the angular speed of the bob and (b) the tension in the string.