Conservation of Linear Momentum and Mechanical Energy

A small sphere is attached to a cord and rotates in a vertical circle about a point $O$. If the range speed of the sphere is increased, the cord is most likely to break at the orientation when the mass is at:

A stone tied to a string is rotated in a vertical circle. The minimum speed of the stone during a complete vertical circular motion 

A smooth semicircular wire track of radius $R$ is fixed in a vertical plane One end of a massless spring of natural length  $\frac{\mathrm{}\mathrm{}\mathrm{}3R}{4}$  is attached to the lowest point $O$of the wire track. A small ring of mass $m$ which can slide on the track is attached to the other end of the spring. The ring is held stationary at point $P$ such that the spring makes an angle of  $60°$ with the vertical. The spring constant $K=\frac{mg}{R}$ . Consider the instant when the ring is released. Find the normal reaction on the ring by the track.

A small block is shot into each of the four smooth tracks as shown below. Each of the tracks rises to the same height. The speed with which the block enters the track is the same in all cases. At the highest point of the track, the normal reaction is maximum in

 Two bodies$A$ and B of masses $5.00 \mathrm{kg}$ and $10.00 \mathrm{kg}$ moving in free space in opposite directions with velocities $4.00 \mathrm{m} {\mathrm{s}}^{-1}$ and $0.50 \mathrm{m} {\mathrm{s}}^{-1}$ respectively undergo a head-on collision. The force $F$ of their mutual interaction varies with time $t$ according to the given graph. What can you conclude from the given information?

A rubber ball is dropped from a height of $5 \mathrm{m}$ on a plane, where the acceleration due to gravity is not shown. On bouncing, it rises to $1.8 \mathrm{m}$. The ball loses its velocity on bouncing by a factor of $\frac{\alpha }{5}$. Write value of $\alpha$.

A chain of mass $m$ and length $L$ is initially held in vertical position as shown in the figure. It is released from rest, then which of the following graph(s) is/are correct. Here $N$ represents normal force by the ground on the chain and $v$ represents velocity of free end

A bob of mass $m$ connected to the end of an inextensible string of length $l$, is released from position shown in figure. If impacts of bob with smooth floor is perfectly inelastic. Choose the correct option(s):

A point mass $\mathrm{M}$ is hanging by a string of length $\mathrm{l}.$ The velocity $v$ which must be imparted to it in order for it to just barely reach the top is

A mass of $10\mathrm{ }\mathrm{kg}$ is shot out of a cannon at an initial speed of $1 \mathrm{m}\mathrm{ }{\mathrm{s}}^{-1}$at $45°$ upwards from a height $1.5 \mathrm{m}$above the ground. What is the kinetic energy (in $\mathrm{J}$) of the cannonball when it reaches the ground? [Consider negligible air resistance and take $g=10\mathrm{ }\mathrm{m}\mathrm{ }{\mathrm{s}}^{-2}$]

A pendulum string of length $𝓁$ moves up to a horizontal position as shown in figure and released.




What should be the minimum strength of the string to withstand the tension as the pendulum passes through the position of equilibrium? The mass of the pendulum is $m$.

A small ball is pushed from a height $h$ along a smooth hemispherical bowl. With what speed should the ball be pushed so that it just reaches the top of opposite end of the bowl? The height of the top of the bowl is $R$.

There is a parabolic bridge across a river of width $10 \mathrm{m}$. The highest point of the bridge is $4 \mathrm{m}$ above the level of the bank. A car of mass $72 \mathrm{kg}$ is crossing the bridge at a constant speed of $5 \mathrm{m} {\mathrm{s}}^{-1}$. Find the force (in Newton) exerted on the bridge by the car when it is at the highest point $P$ of the bridge in newton. (Take, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.)

A stationary bomb explodes into three pieces. One piece of $2 \mathrm{kg}$ mass moves with a velocity of $8 \mathrm{m} {\mathrm{s}}^{-1}$ at right angles to the other piece of mass $1 \mathrm{kg}$ moving with a velocity of $12 \mathrm{m} {\mathrm{s}}^{-1}$. If the mass of the third piece is $0.5 \mathrm{kg}$, then its velocity is:

A particle is tied to a string describes a vertical circular motion of radius $r$ continually. If it has a velocity $\sqrt{3gr}$ at the highest point, then the ratio of the respective tensions in the string holding it at the highest and lowest points is

A bucket, full of water is revolved in a vertical circle of radius $1.5 \mathrm{m}.$ The maximum time period of revolution so that the water doesn't fall out of the bucket is $\left(g\approx {\pi }^2\right)$

Consider a point $P$ on the circumference of a disc rolling along a horizontal surface. If radius of the disc is $1 \mathrm{m}$ then the distance (in meter) travelled by $P$ in one full rotation of the disc is

A bomb of mass $9 \mathrm{kg}$ initially at rest explodes into two piece of masses $3 \mathrm{kg}$ and $6 \mathrm{kg}$. The kinetic energy of the $3 \mathrm{kg}$ mass is $216 \mathrm{J}$. The kinetic energy of the $6 \mathrm{kg}$ mass will be