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.

In figure two identical springs, each with a relaxed length of $50 \mathrm{cm}$ and a spring constant of $500 {\mathrm{Nm}}^{-1}$, are connected by a short cord of length $10 \mathrm{cm}$. The upper string is attached to the ceiling, a box that weighs $100\mathrm{N}$ hangs from the lower spring. Two additional cords, each $85 \mathrm{cm}$ long, are also tied to the assembly; they are limp (i.e,slack).

(i) If the short cord is cut, so that the box then hangs from the springs and the two longer cords, does the box move up or down?

(ii) How far does the box move before coming to rest again?

Two particles $A$ and $B$ of mass $2m$ and $m$ respectively are attached to the ends of a light inextensible string of length $4a$ which passes over a small smooth peg at a height $3a$ from an inelastic table. The system is released from rest with each particle at a height a from the table.

Find- (i) The speed of $B$ when $A$ strikes the table.

Two particles $A$ and $B$ of mass $2m$ and $m$ respectively are attached to the ends of a light inextensible string of length $4a$ which passes over a small smooth peg at a height $3a$ from an inelastic table. The system is released from rest with each particle at a height a from the table.

Find- (ii) The time that elapses before $A$ first hits the table.

Two particles $A$ and $B$ of mass $2m$ and $m$ respectively are attached to the ends of a light inextensible string of length $4a$ which passes over a small smooth peg at a height $3a$ from an inelastic table. The system is released from rest with each particle at a height a from the table.

Find- (iii) The time for which $A$ is resting on the table after the first collision & before it is first jerked off.

Block $A$ of mass $\frac{m}{2}$ is connected to one end of light rope which passes over a pulley as shown in the figure. Man of mass $m$ climbs the other end of rope with a relative acceleration of $\frac{g}{6}$ with respect to rope find acceleration of block $A$ and tension in the rope.