A particle is suspended vertically from a point $O$ by an inextensible massless string of length $L$. A vertical line $AB$ is at a distance $L/8$ from $O$ as shown in Figure. The object is given a horizontal velocity $u$. At some point, its motion ceases to be circular and eventually the object passes through the line $AB$. At the instant of crossing $AB$, its velocity is horizontal. Find $u$.

 

A block of mass $0.18 \mathrm{kg}$ is attached to a spring of force constant $2 \raisebox{1ex}{$\mathrm{N}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{m}$}\right.$. The coefficient of friction between the block and the floor is $0.1$. Initially, the block is at rest and the spring is unstretched. An impulse is given to the block as shown in the Figure. The block slides a distance of $0.06 \mathrm{m}$ and comes to rest for the first time. The initial velocity of the block is $v=\frac{N}{10}$. Find $N$.

A particle of mass $m=0.2 \mathrm{kg}$ is moving in one dimension under a force that delivers constant power $0.5 \mathrm{W}$ to the particle. If the initial speed of the particle is zero, find its speed after $5 \mathrm{s}$.

Consider an elliptically shaped rail $PQ$ in the vertical plane with $OP=3 \mathrm{m}$ and $OQ=4 \mathrm{m}$. A block of mass $1 \mathrm{kg}$ is pulled along the rail from $P$ to $Q$ with a force of $18 \mathrm{N}$, which is always parallel to line $PQ$ (see Fig.). Assuming no frictional losses, the kinetic energy of the block when it reaches $Q$ is $\left(\mathrm{n}\times 10\right) \mathrm{J}$ Joules. Find the value of $n$ (take acceleration due to gravity$=10 \mathrm{m}/{\mathrm{s}}^2$).

A light Inextensible string that goes over a smooth fixed pulley as shown In Fig. connects two blocks of masses $0.36 \mathrm{kg}$ and $0.72 \mathrm{kg}$. Taking $g=10 \raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{s}}^2$}\right.$, find the work done (in Joule) by string on the block of mass $0.36 \mathrm{kg}$ during the first second after the system is released from rest.