Are Newton?s Laws valid for extended Rigid Bodies ?

Center of Mass 1

What is Center of Mass ? And how are Newton?s Laws applicable to the Center of Mass ?


A stick is thrown in the air as a projectile. 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

Center of Mass 2

How to find the position of Center of Mass of a system of particles ?


The centre of mass of a system of particles is at the origin. It follows that
(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.
The distance of the centre of mass of the system from the origin is


Find the CoM of the given system.


Find the CoM of the system where 5 particles of equal masses are placed at 5 vertices of a regular hexagon.

CoM Solids

How to find the position of Center of Mass of Continuous Mass Distributions or Rigid Bodies ? Does Center of Mass lie at the Symmetric center of the shape of Rigid Body ?


Find the CoM of a uniform wire frame in the shape of a isosceles right angled triangle.


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.

2nd Law & CoM

How are Newton?s Laws applicable to the Center of Mass ?


If the external forces acting on a system have zero resultant,
the centre of mass
(a) must not move
(b) must not accelerate
(c) may move
(d) may accelerate


Two blocks of masses 2kg and 5kg are lying on a frictionless surface. They are being pushed by a force of 10 N as shown.
Find the acceleration of the CoM 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)


Linear Momentum and Newton?s 2nd Law in terms of Linear Momentum.


Two blocks are connected by an ideal spring (as shown in the fig) and are free to slide along x axis on a frictionless horizontal surface. Initially the spring is compressed and the blocks are tied with a massless string.
When the string is cut, what is the ratio v1/v2 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.

(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 speed of 20 m/s bursts into two equals parts continuing in the same direction. If one of the parts moves at 30 m/s,
a) with what speed does the second part move and
b) what is the fractional change in the kinetic energy ?


Definition of Impulse and Relation between Impulse and Momentum.
Are Normal Reaction, Friction and Tension always Impulsive forces ? Is Gravity an Impulsive force ?


A ball collides with a vertical wall without any change in its speed.
Consider the change Dp in the balls liner momentum.
a) Is Dpx positive, negative, or zero?
b) Is Dpy positive, negative, or zero?
c) What is the direction of Dp ?

Collisions - 1

Analysis of Head-On Collision between two point objects. Will Linear Momentum be conserved in Collisions ? Will Mechanical Energy be conserved in Collisions ? Impulse of Deformation and Restoration. Coefficient of Restitution. Relative speed of Approach and Separation. Newton?s Law of Collisions. Completely In-elastic and Perfectly Elastic Collisions.


Two balls of masses 2 kg and 4 kg are moving towards each other with velocities 4 m/s and 2 m/s, respectively on a frictionless surface. After colliding, the 2 kg ball returns back with velocity 2 m/s. Then find
a) velocity of the 4 kg 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.

Collisions - 2

Final velocities of particles after a Head-On collision.

Collisions - 3

Analysis of Oblique Collision between 2 particles.


What will be the angle of reflection in case of an inelastic collision ? and
final velocity after rebound ?

Var Mass Sys - 1

Is Newton?s 2nd Law applicable to systems with varying or changing mass ?


A man is strapped on a trolley which is travelling at the speed vo on a smooth horizontal surface; he is carrying a rocket launcher. 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?

Var Mass Sys - 2

Variable Mass Systems, Thrust Force and Rocket Propulsion.


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 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 land ?


There are n identical spheres of mass m lying on a frictionless surface. If sphere 1 is given an initial velocity v1 as shown in the figure, 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 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.
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 of 9 m/s strikes an identical stationary ball such that after collision the direction of each ball make an angle of 300 with the original line of motion.


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 of length 2 l and is initially placed as shown in the figure. Find the velocity of the ball at the lowest point.


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. 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 moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly change from 60o to 30o at point B. The block is initially at rest at A. Assume that collisions between the block and the incline are totally inelastic. Find
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 other as shown and rest over a smooth horizontal table. Length of each string is l. 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 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. Find the
a) distance moved by wedge
b) speed of the wedge
when the block reaches ground.
( Assume all surfaces are frictionless )


A small body of mass m is placed over a larger mass M as shown in the figure. The smaller mass is given an initial velocity u and the system is left to itself. Assume that all the surface are frictionless.


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.


Find the center of mass of a
a) uniform semicircular wire
b) uniform semicircular plate


Find the CoM of
(a) full cone
(b) 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 CoM 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 the rate of ?.
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 the rate h. How much force is needed to keep the cart moving at the 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 ?


Sand falls from a hooper onto a box which is sliding with uniform velocity vo on a surface which has coefficient of friction m.The sand falls at the rate h. How much force is needed to keep the box moving at the speed v0, if
a) hooper moves with velocity v0
b) hooper 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.

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