EXERCISE

A particle being acted on by constant forces $\left(4 \hat{i}+\hat{j}-3 \hat{k}\right)$ and $\left(3 \hat{i}+\hat{j}-\hat{k}\right)$ is displaced from the point $\left(\hat{i}+2 \hat{j}-3 \hat{k}\right)$ to the point $\left(5 \hat{i}+4 \hat{j}-\hat{k}\right)$. Find the total work done by the forces.

A $0.5 \mathrm{kg}$ body slides from the point $A$, figure along a horizontal track with an initial speed of $3 {\mathrm{ms}}^{-1}$ towards a weightless horizontal spring of length $1 \mathrm{m}$ and force constant $2 {\mathrm{Nm}}^{-1}$. The part $AB$ of the track is frictionless and the part $BC$ has the coefficient of static and dynamic friction $0.22$ and $0.2$ respectively. Find the total distance through which the block moves before it comes to rest completely $\left(g=10 \mathrm{m}/{\mathrm{s}}^2\right)$, if $AB=2 \mathrm{m}$ and $BD=2.14 \mathrm{m}$.

A ball with a speed of $9 {\mathrm{ms}}^{-1}$ strikes another identical ball such that after collision the direction of each ball makes an angle $30°$ with the original line of motion. Find the speeds of the two balls after the collision. Is the kinetic energy conserved in this collision process?

Two plates of masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ are connected by a spring of force constant $k$, figure below. What force should be applied to the upper plate for it to raise the lower one after the force is removed? Disregard the mass of the spring.

A simple pendulum is suspended from a peg on a vertical wall. The pendulum is pulled away from the wall to a horizontal position, see figure below and released. The ball hits the wall, the coefficient of restitution being $\frac{2}{\sqrt{5}}$. What is the minimum number of collisions after which the amplitude of oscillations becomes less than $60°$?

A $20 \mathrm{gm}$ bullet pierces through a plate of mass $M_1=1 \mathrm{kg}$ and then come to rest inside a second plate of mass $M_2=2.98 \mathrm{kg}$ as shown in figure below. It is found that two plates initially at rest, now move with equal velocities. Find the percentage loss in initial velocity of the bullet when it is between $M_1$ and $M_2$. Neglect any loss of material of the plates due to action of the bullet.

In the below figure (a) and (b) $AC, DG$ and $GF$ are fixed inclined planes, $BC=EF=x$ and $AB=DE=y$. A small block of mass $M$ is released from the point $A$. It slides down $AC$ and reaches $C$ with a speed $v_c$. The same block is released from rest from the point $D$. It slides down $DGF$ and reaches the point $F$ with speed $v_F$. The coefficient of friction between the block and both surfaces $AC$ and $DGF$ are $\mu$. Calculate $v_c$ and $v_F$.

A string, with one end fixed on a rigid wall, passing over a fixed frictionless pulley at a distance of $2 \mathrm{m}$ from the wall, figure below, has a point mass $M=2 \mathrm{kg}$ attached to it at a distance of $1 \mathrm{m}$ from the wall. A mass $m=0.5 \mathrm{kg}$ attached at the free end is held at rest so that the string is horizontal between the wall and the pulley and vertical beyond the pulley. What will be the speed with which the mass $M$ will hit the wall when the mass $m$ is released? (Take $g=9.8 \mathrm{m}/{\mathrm{s}}^2$).