**COM, Momentum & Collisions**

- This video introduces us to the strange motion of a particle constrained in its position when a force is applied.
- This video introduces us to the concept of centre of mass which is a point where all the mass of body seems to be concentrated at. The force acting on the body can be assumed to be acting at centre of mass only. This point of centre of mass can either be inside the body or outside.
- A stick is thrown in the air as a projectile. In which of the following case, the center of mass of the stick will move along a parabolic path. (a) only if the stick is uniform (b) only if stick does not have any rotational motion (c) only if center of mass of stick lies at some point on it and not outside it (d) in all cases
- This video illustrates algebraically the method to determine the centre of mass of a particle or system of particles.
- The centre of mass of a system of particles is at the origin. Which of the following is true? (a) The number of particles to the right of the origin is equal to the number of particles to the left (b) The total mass of the particles to the right of the origin is same as the total mass to the left of the origin (c) The number of particles on X-axis should be equal to the number of particles on Y-axis (d) If there is a particle on the positive X-axis, there must be at least one particle on the negative X-axis.
- All the particles of a system are situated at a distance R from the origin as shown in figure. The distance of the centre of mass of the system from the origin is….?
- Find the Centre of Mass of the system as shown in figure.
- Find the Centre of Mass of the system where 5 particles of equal masses are placed at 5 vertices of a regular hexagon.
- This video illustrates the method to determine the centre of mass of a non-uniform and uniform solid body with the help of integration in 3-D space.
- Find the Centre of Mass of a wire frame in the shape of an isosceles right angled triangle. Given the mass is uniformly distributed over the wire.
- There is a uniform metal plate of radius 2R from which a disk of radius R has been cut out. Locate the center of mass of the remaining portion of the plate.
- This video explains the validity and equation of the Newton's 2nd law of motion for a rigid body and the role of internal forces on the acceleration of the system.
- If the external forces acting on a system have zero resultant, then which of following is true for the centre of mass? Centre of mass (a) must not move(b) must not accelerate(c) may move (d) may accelerate
- Two blocks of known masses are lying on a frictionless surface and are being pushed by a force towards each other as shown in figure. Find the acceleration of the Centre of Mass of the blocks.
- A man of mass m1 is standing on one side of a board of mass m2 floating in water. He then walks to the other side of the board. Find the displacement of the board. (Ignore resistance from water)
- This video explains in detail, the concept of momentum for a particle and system of particles.
- Two blocks are connected by an ideal spring and are free to slide along x axis on a frictionless horizontal surface as shown in the figure. Initially the spring is compressed and the blocks are tied with a mass-less string. When the string is cut, what is the ratio of the velocity of block 1 to the velocity of block 2 as the separation between the block increases?
- Consider the following two statements: (A) Linear momentum of the system of particles is zero. (B) Kinetic energy of a system of particles is zero. Which of the following is true? (a) A implies B and B implies A. (b) A does not imply B and B does not imply A. (c) A implies B and B does not imply A. (d) B implies A but A does not imply B.
- A block moving horizontally on a smooth surface with a given speed bursts into two equals parts continuing in the same direction. If the speed of one of the parts is known, at what speed does the second part move? Also find the fractional change in the kinetic energy before and after the explosion.
- This video explains the concept of impulse when a mass changes its direction of momentum in a very small fraction of time with help of a graph.
- A ball collides with a vertical wall without any change in its speed as shown in figure. Considering the change Δp in the ball's liner momentum, answer the following. (a) Is Δpx positive, negative, or zero? (b) Is Δpy positive, negative, or zero? (c) What is the direction of Δp ?
- This video explains the features and parameters like restitution and impulse of deformation of 1-D head-on collision with equations. It also explains the Newton's law of collision for elastic and inelastic collision.
- Two balls of known masses are moving towards each other with given velocities on a frictionless surface. After colliding, first ball returns back with a known velocity. Then find the

a) velocity of the second ball after collision

b) coefficient of restitution e;

c) impulse of deformation JD;

d) Maximum potential energy of deformation.

(e) impulse of reformation JR. - A ball of mass m slides with velocity u on a frictionless surface towards a smooth inclined wall as shown in the figure. If the collision with the wall is perfectly elastic, find the speed with which the ball rebounds.
- This video illustrates the different equation related to the different types of head-on collision using the conservation of momentum, Newton's law of collision and kinetic energy consideration.
- This video illustrates the features and equations for collision in two dimensions.
- For the figure shown, what will be the angle of reflection and final velocity after rebound in case of an inelastic collision?
- This video explains the validity of Newton's second law of motion and conservation of momentum for a variable mass which is changing with time.
- A man carrying a rocket launcher is strapped on a trolley which is travelling at a speed v0 on a smooth horizontal surface. The combined mass of him, rocket launcher, trolley and the rocket is (M + m). The rocket has mass m. The muzzle velocity of the rocket is v. What will be the velocities of the Man and rocket just after firing?
- This video explains the change in momentum and thrust with equations of rocket whose mass is changing with time.
- A ball is dropped from height h on a surface. The coefficient of restitution is e. Find the height attained by the ball after the nth collision.
- Consider a block of mass m2 kept on a rough surface being hit by a particle of mass m1 moving with speed u1. Find the velocity of combined mass immediately after the particle sticks with the block.
- A projectile breaks into two parts in the mass ratio 1:3 at the highest point of trajectory. The smaller part lands at a distance of 3/4 R from the launching point. Where does the heavier piece lands?
- There are n identical spheres of mass m lying on a frictionless surface as shown in figure. If sphere 1 is given an initial velocity v1, find an expression for the velocity of the nth sphere immediately after being struck by the one adjacent to it. The coefficient of restitution for all the impacts is e.
- A heavy ball mass 2m moving with a velocity vo collides elastically head-on with a cradle of three identical balls each of mass m as shown in figure. Determine the velocity of each ball just after collision.
- A long plank of mass m2 is kept on a smooth horizontal surface and block of mass m1 is kept on it with initial velocity v1 as shown in figure. (a) Find the final velocity of the blocks. (b) Find total work done by friction between the plank and the block. (Assuming plank is sufficient long)
- A ball moving with a speed strikes an identical stationary ball such that after collision the direction of each ball make an angle with the original line of motion. Find the speed of both balls after the collision. Also tell in which direction (X or Y) the momentum will be conserved.
- As shown in figure, two equal spheres B and C, each of mass m, are in contact on a smooth horizontal table. A third sphere A of same size but mass m/2 impinges symmetrically on them with a velocity u and is itself brought to rest. Find the (a) velocity acquired by each of the spheres B and C after collision. (b) coefficient of restitution between the two spheres A and B.
- A ball of mass m is tied to an inextensible thread and is initially placed as shown in the figure. Find the velocity of the ball at the lowest point when released.
- Two small particles A and B with masses m and 2m are connected by a light, inextensible string of length 2 l, placed on a smooth horizontal plane, separated by a distance of l as shown in figure. Particle A is given a velocity v in a direction normal to AB. Find the velocity of A when the string just becomes taut.
- A small block of mass m starts from rest on a frictionless surface of an inclined plane. The angle of the incline suddenly changes at point B as shown in figure. Assume that collisions between the block and the incline are totally inelastic. Find the following.

(a) the speed of the block at point B immediately after it strikes the second incline

(b) the speed of the block at point C, immediately before it leaves the second incline

(c) if the collision between the block and the incline is completely elastic, find the vertical(upward) component of the velocity of the block at point B, immediately after it strikes the second incline - A ball is suspended by an inextensible thread of length l. Another identical ball is thrown vertically downward such that its surface remains just in contact with thread during downward motion and collides elastically with the suspended ball. If the suspended ball just completes vertical circle after collision, calculate the velocity of the falling ball just before collision.
- Three identical balls are connected by light inextensible string with each and rest over a smooth horizontal table as shown in figure. At moment t = 0, ball B is imparted a velocity vo perpendicular to the strings and then the system is left on its own. (a) Calculate the velocity of B just before A collides with ball C. (b) Calculate the velocity of A at the above given instant. (c) If collision between the balls is completely inelastic. Find the loss in kinetic energy of the system.
- Three particles of masses m1, m2 and m3 lie on a smooth horizontal surface, and are fastened to two light inextensible strings. The particle 1 is imparted an impulse J at an angle with string as shown in the figure. Find the initial speed of each particle.
- A block of mass m slides down the wedge of mass M as shown in figure. Find the (a) distance moved by wedge (b) speed of the wedge when the block reaches ground. (Assume all surfaces are frictionless)
- A small block of mass m is placed over a larger block of mass M as shown in the figure. The smaller mass is given an initial velocity u and the system is left to itself. Assuming that all surfaces are frictionless, find the following. (a) speed of larger block when smaller block is sliding on the vertical part. (b) speed of the smaller block when it breaks off the larger block at height h. (c) maximum height to which smaller block ascends (d) distance covered by larger block during the time of smaller block's flight under gravity.
- A pan of mass m1 and a block of mass m2 are connected with each other by an ideal, inextensible string, passing over an ideal pulley. Initially the block is resting over a horizontal floor as shown in figure. At t = 0, an inelastic ball of mass m collides with the pan with velocity u. Calculate the maximum height up to which the block rises.
- An ideal string passing over a pulley has a ladder of mass M and a small robot of mass m on the ladder at one end. On the other end, a counterweight of mass M+m is attached. Initially everything is at rest. The robot climbs upward by distance l on the ladder and then stops. Ignoring the masses of the pulley as well as the friction, find the work done by the robot?
- A ball of mass m makes head on elastic collision with a ball of mass nm which is initially at rest. Show that the fractional transfer of energy by the first ball is 4n/(1 + n)^2. Deduce the value of n for which the transfer is maximum.
- Locate the center of mass of a

(a) uniform semicircular wire

(b) Uniform semicircular plate. - Find the Centre of Mass of a full cone and a half cone of height H and radius R.
- Determine the position of center of mass of thin hemispherical shell of mass M and radius R, assuming uniform mass distribution.
- Find the Centre of Mass of a uniform solid hemisphere of radius R.
- A cart full of sand is moving with the velocity of v0. Sand is leaking from the cart at a given rate. What is the external force required on the cart so that it keeps moving with the constant velocity.
- Sand falls from a stationary hopper onto a cart which is moving with uniform velocity vo. The sand falls at a given rate. How much force is needed to keep the cart moving at the constant speed vo?
- Find the external force required on the cart so that it moves with constant velocity in the situation as shown in the figure. Rate at which sand drops on the cart is given.
- Sand falls from a hopper onto a box which is sliding with uniform velocity vo on a surface which has known coefficient of friction .The sand falls at a given rate. How much force is needed to keep the box moving at the speed v0, if

a) hopper moves with velocity v0

b) hopper is stationary. - A uniform chain of mass m and length l touches the surface of a movable table by its lower end, as shown in the figure. Find the normal force exerted by the table on the chain of length x has fallen on the table. (Surface of the table is moving so that the fallen part does not form heap. Ignore the horizontal forces between the table and the chain)
- A uniform chain of length l and having mass l per unit length is hanging from ceiling by two light, inextensible threads of equal length as shown in figure. Distance between the two ends of the chain is very small. Thread on the right is burnt at t = 0. Calculate the tension in thread on the left as a function of time.