Constrained Motion

A light string goes over a frictionless polley. At its one end hangs a mass of $2 \mathrm{kg}$ and at the other end hangs a mass of $6 \mathrm{kg}$. Both the masses are supported by hands to keep them at rest. When the masses are released, they being to move and the string gets taut. (Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$) The tension in the string during the motion of the masses is :

Two smooth rings $A$ and $B$ are connected by a string. Another string connects Ring  $A$ and block, if strings are tight then at the given moment :

Acceleration (with respect to ground) of pulleys and blocks are as shown in the figure. All pulleys and strings are massless and  frictionless than the magnitude of $a_A$ and $a_B$ are:

A block of mass $m$ lies on wedge of mass $M,$ which lies on fixed horizontal surface. The wedge is free to move on the horizontal surface. A horizontal force of magnitude $F$ is applied on block as shown, neglecting friction at all surfaces, the value of force $F$ such that block has no relative motion w.r.t. wedge will be : (where $\mathrm{g}$ is acceleration due to gravity)

In the given figure, inclined surface and pulleys are smooth. Strings and pulleys are massless. Acceleration of mass $m$ is

In the figure shown acceleration of monkey relative to the rope if it exerts a force of $80 \mathrm{N}$ on string will be

Figure shows a $5 \mathrm{kg}$ ladder hanging from a string that is connected with a ceiling and is having a spring balance connected in between. A boy of mass $25 \mathrm{kg}$ is climbing up the ladder at acceleration $1 \mathrm{m} {\mathrm{s}}^{-2}$. Assuming the spring balance and the string to be massless and the spring to show a constant reading, the reading of the spring balance is (take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ )

Same spring is attached with $2 \mathrm{kg},3 \mathrm{kg}$ and $1 \mathrm{kg}$ blocks in three different cases as shown in figure. If $x_1,x_2$ and $x_3$ be the extensions in the spring in these cases then (Assume all the blocks to move with uniform acceleration)

Mass $m$ shown in figure is in equilibrium. If it is displaced further by $x$ and released find its acceleration just after it is released. Take pulleys to be light and smooth and strings light.