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centre of mass Flashcards

1
Q

what is centre of mass

A

its basically the point which represents the average of the whole mass….it can be assumed that the whole mass of the body is concentrated at that point…rotation of the body is symmetric about that point… its situated near the heavier mass

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2
Q

what is centre of mass from a reference point

A

it is the distance from the reference point where the mass is concentrated

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3
Q

formula of centre of mass

A

x1m1+x2m2+x3m3……../m1+m2+m3…….

so basically the mass is getting cancelled

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4
Q

if there are two masses and the position of one of them is taken as the reference point then what is the ratio of the distance between m1 and com to distance between m2 and com

A

m2:m1

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5
Q

if masses are taken in a cartesian plane then where is centre of mass

A

m1r1+m2r2+m3r3…../m1+m2+m3….
where r1,r2,r3…. are the position vectors of the masses
r1 r2 r3… are basically the distances of the masses from the origin

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6
Q

how to find the x coordinate y coordinate and z coordinate of the centre of mass

A

m1x1+m2x2+m3x3……/m1+m2+m3…..

do the same thing for y and z coordinate

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7
Q

if the mass of a particle is uniformly distributed then what is the coordinated of the centre of mass

A

the normal formula of com is x1m1+x2m2+x3m3……../m1+m2+m3…….
here the mass is uniformly distributed so each small part of the particle has a mass of dm
so the x coordinate of com is integration of xdm/integration of dm( as the denominator represents the total mass of the whole object and the numerator is basically the summation of all the x coordinates of each segment of the particle)

DO THE SAME THING FOR Y AND Z COORDINATES

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8
Q

what is the trick while solving uniform distributed mass questions

A

you have to consider that a dm section of a mass has a length dx

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9
Q

whenever theres a question related to circumcircle of circle or semicircle or sphere then what is the trick to solve this

A

for this kinda shit u have to consider a small angle d(theta) and solve from there

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10
Q

what is the length of a small segment on circumference

A

Rd(theta)

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11
Q

whenever theres a question related to area of circle or semicircle or shit like that then what is the trick to solve this

A

then you have to take a thin section of the circle as the and assume it have an area dA. this section of area must be parallel to circumference

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12
Q

what is the centre of mass of a system of particles

A

x1M1+x2M2+x3M3……./M1+M2+M3…..
here M1 is the mass of system1 and x1 is the centre of mass of system1. M2 is the mass of system 2 and x2 is the centre of mass of system2 and so on…….

if the mass of the object is uniformly distributed then the centre of mass of the system is integration of xdm/integration of dm
here x is the centre of mass of each system of mass

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13
Q

what is meant by centre of mass of system of mass

A

it is the centre of mass of a collection of points

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14
Q

what is an example of system of mass

A

while finding the centre of mass of an uniformly distributed semicircle we have to take small sections of area parallel to the circumference. there will be infinite number of such areas so we consider each area as a system of mass and we use the above formula

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15
Q

what is the com of the circumference of a semicircle

A

2R/pi

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16
Q

what will be the com of a small section of a circle with angle d(theta)

A

the small section can be considered as a triangle

the answer is 2R/3

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17
Q

in any question related to 1d 2d and 3d how do we approach the problem

A

you have to take a small section one dimension less than the dimension given in the question
if it is 1d you have to convert it into something without dimensions (like a small section of length like dl)
if its 2d you have to convert it into 1d(like area of circle into circumference of circle)
if its 3d take a small section of volume(convert volume into area)

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18
Q

what is the shortcut while finding the com of 3d and 2d circular shapes

A

in a 3d circular shapes just divide it into concentric circles horizontally with a thickness of dy
in a 2d circular shapes just divide it into concentric circles parallel to circumference with a thickness of dx

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19
Q

what is the relation between radius of concentric circle and radius of sphere

A

if h is the height of the concentric circle and R is the radius of the big circle and r is the radius of the concentric circle then R^2=r^2+h^2

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20
Q

what is a common mistake made

A

the r of the concentric circle is different from R of main circle

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21
Q

in a cone how to relate dl dh r R h and h

A

by similarity and using the vertical angle

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22
Q

what is the com of the area of a semicircle

A

4R/3pi

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23
Q

what is the com of a uniformly distributed cone

A

H/4

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24
Q

what is the com of a hollow cone

A

2H/3

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25
what is the com of a uniformly distributed hemisphere
3R/8
26
what is the com of a hollow hemisphere
R/2
27
what is com of remaining portion when a certain part of the body has been removed
M_complete x_complete= m_removed x_removed+m_remainingx_remaining
28
what is com of a triangle
centroid
29
com of sphere
centre
30
what do you have to be careful about?
sign convention | remember take one side as positive and the other side as negative
31
what is the velocity of the centre of mass of multiple systems
v_cm=m1v1+m2v2+m3v3..../m1+m2+m3.....
32
what is the acceleration of the centre of mass of multiple systems
a_cm=m1a1+m2a2+m3a3..../m1+m2+m3......
33
what is the net force on multiple systems
total mass * acceleration of centre of mass
34
when you are taking a circular arc and you are integrating an angle which is on both sides of the axis what do you do
you have to integrate it by taking the initial limit as | positive and the final limit as negative as you are integrating on both sides of the axis
35
what do you do when there is something related to a quarter circle
remember the fact that two quarter circles make a semi circle and use the centre of mass of system formula
36
what so you have to keep in mind when you are solving problems related to acceleration and velocity of centre of mass
the acceleration and velocity does not depend on the size of the objects you also have to keep the sign conventions in mind while solving problems
37
how do you relate the displacements of objects which remain. in contact.... and in what direction can you use this equation
(m1+m2+m3....)*x_com=m1x1+m2x2+m3x3........... remember you can only use this in a particular direction for example the horizontal direction
38
what do you have to keep in mind while solving problems in which there is relative motion
in the above formulae the displacement must be wrt ground
39
Fnet on a system is equal to
F_external force | which is further equal to mass*acceleration of com
40
F_internal=?
0
41
if F_net=0 then
p1+p2+p3...... is constant | where p1,p2,p3 are linear momentums of the objects
42
what else if F_net=0
vcm is constant | acm=0
43
what if F_net=0 and initial velocity=0
then x_cm=0
44
before solving any question what do you have to see
if F_net=0
45
what do u have to remember while solving conservation of momentum questions
the velocity of com is not equal to the velocity of the objects
46
what is the other way of solving some problems
combine both conservation of linear momentum and work energy theorem
47
in these kinda questions what should you be careful about
you should keep a note on the initial horizontal and vertical velocities of each object and the final horizontal and vertical velocities of each object
48
relation between initial and final momentums
initial momentum=final momentum
49
what is very important in conservation of momentum
sign convention
50
if the directions of velocities are different in momentum questions
then use vectors for velocities
51
when the question. is find KE in the frame of cm then?
find the velocity of each particle relative to the v_cm
52
what is m1v-_A(wrt to cm)+m2v_B(wrt to cm)...... | what does this result conclude?
Mv_cm(wrt to cm) which is equal to 0 this concludes that the velocities in the conservation of momentum formula can be wrt any reference frame but the velocity of com will also be wrt the same frame
53
when theres any question related to finding final velocities of the objects then
you have to indirectly use work energy theorem and conservation of momentum
54
in which direction can you use conservation of linear momentum
you can use conservation of linear momentum in a particular direction... for example you can use conservation of linear momentum in the horizontal direction.....
55
the velocities in conservation of linear momentum have to be wrt which frame
it has to wrt a fixed frame... for example the ground
56
in order to find S_rel between 2 bodies what do you do
use kinematic equations with v_rel a_rel
57
what is a common silly mistakes made here
if the entire system is moving with a common velocity and an object is part of these system.... we HAVE TO INCLUDE THE VALUE OF VELOCITY OF SYSTEM WHILE CONSIDERING THE VELOCITY OF THE OBJECT
58
what is the first thing you have to see before solving any question
you have to see if the system is moving with a common velocity
59
whenever it is given that a carriage with a gun fires a shell with a velocity wrt carriage then
the velocity of the shell is wrt the final velocity of the carriage
60
what basic formula is crucial to solve these questions without silly mistakes
v(of mass wrt ground)=v(of mass wrt frame)+v(frame wrt ground)
61
what else can you do if the net force on the system is 0 | and what is the exception to this
final kinetic energy of the system=initial kinetic energy of the system however you cannot use this if the initial velocities of the system is 0 and the net external force is 0 as the work energy theorem wouldn't make sense
62
what is the com of an arc of angle theta
2Rsin(theta/2)/theta
63
when you are finding the com of any shape what do you have to keep in mind
if the shape is hollow and only the circumference has mass | or if the entire shape has mass
64
while finding the centre of mass of a shape what should you do
take the reference point where a x axis and y axis intersect....and locate all the other points according to this coordinate system
65
what do you do if there are multiple forces acting on a system(like monkeys hanging descending with different accelerations on a rope)
consider all the monkeys as 1 system by calculating the acceleration of com this is the acceleration of the system the mass of the system is the total mass of the monkeys now use force equations on this system
66
where else can you use the above principle
when you have to use force equations on an uniform rod
67
what happens if you apply a force on the centre of mass
then the object would move with the same acceleration as would a point mass of the same mass
68
what do you mean by the com of a system of many particles
this indicates the com of many particles... or it is the point where the averAge mass OF ALL particles... it'll be more towards heavier particles.... it need not be in any of the particles.....for example if there are 3 particles of same mass which form a triangle then the com of the system will be the centroid of the masses forming the triangle this point is not in any of the masses
69
where is the com of pulley block system
if there are 2 blocks of equal mass hanging from a pulley...then the com of the system will be the midpoint of the air between the blocks
70
describe the motion of centre of mass
lets say there are two systems in having mass m1 and m2 and there are moving in with different velocities or accelerations then the com will move in such a way that it always remains on the line connecting the com of both the systems and always divides that line in the ratio of m2 is to m1
71
if there is no net external force on a system of objects and the system has an initial velocity then
the com will keep moving with that constant initial velocity
72
when ever a object is undergoing projectile motion and the object suddenly breaks into pieces then
the com will continue to move in the projectile motion
73
what is the meaning of retracing path
following the same path it initially followed
74
if a particle resting on a platform is thrown out of the platform with a velocity of u wrt platform then what is the velocity of the particle wrt ground
u-v where v is the velocity of platform after the particle is thrown
75
what are isolated systems
systems in which no external force acts on it
76
where does conservation of momentum work
it only works in isolated systems
77
how does conservation of momentum work in a rocket
the rocket releases some amount of exhaust gases which were initially part of the rocket.... so basically some parts which were initially part of the rocket are getting thrown away...so there is no external force acting on the rocket....
78
in a question if it says that something is thrown from the platform perpendicular to it.... then what do you do
if velocity of train is a i cap and velocity of abject wrt train is b j cap then velocity of the object wrt ground is ai+bj
79
how to solve a question where it is given that there are two trains travelling in opposite directions and 2 men on the trains exchange their positions
in this kinda questions you have to use conservation of momentum where you the equation of one train will be (initial velocity of train)(M+m)+m(initial velocity of 2nd train)=(M+m)(final velocity of train)+m(initial velovity of 1st train) where m is the masses of the men and M is the masses of the trains [THIS QUESTION IS Q12 OF EX1.3 IN CENGAGE]
80
it they are n men on a train... if they jump off 1 by 1 or if they jump together which case will impart a greater velocity to the cart
jumping 1 by 1
81
what is the potential energy of a chain or a system of particles
Mgy_cm | y_cm is the com in the y direction
82
how to find the find the final velocity of a chain when it has descended through a certain height
change in potential energy of com of chin is equal to change in KE of chain
83
total kinetic energy of. a system of n particles is equal to?
sum of kinetic energy of all particles relative to ground
84
kinetic energy of system of particles is equal to
kinetic energy of the system relative to com plus KE of com
85
linear momentum of a system is equal to
linear momentum of com | this can be derived from conservation of momentum
86
what is ke of particles relative to com
internal kinetic energy
87
what is reduced mass
mu(reduced mass)=m1m2/m1+m2)
88
what do you have to remember when you are finding kinetic energy in the frame of com
you have to consider both the components of velocity of com when you are subtracting the velocities of the particles with the velocity of com
89
total work done by pseudo force in centroidal system
0
90
in a system what is the work done by internal forces
-(change in potential energy of internal particles)
91
what is work done by external force
(change in ke of centre of mass)+(internal energy)
92
what is meant by internal energy
it is basically the work done by internal forces elastic potential energy, gravitational potential energy, electromagnetic energy, chemical energy, nuclear energy, thermal energy and so on
93
when is work done by external force 0
when no external force acts on the system
94
change in ke in the above formula means?
``` it means (final ke of com-initial ke) this initial ke could be the ke of any individual object in the system ```
95
what is work energy theorem relative to centre of mass | and what shuld you be careful about here
``` total work(internal work+external work)= 1/2*u((v_initial relative speed of blocks)^2-(v_initial relative speed of blocks)^2)) here u is reduced mass the work done is wrt centre of mass ```
96
sum of mass moments in centroidal frame
it is 0 i.e. m1r1(wrt com)+m2r2(wrt com)........=0 here r1,r2.... are position vectors wrt com
97
total linear momentum of the system in centroidal frame
its 0 | miv1(wrt com)+m2v2(wrt com)..........=0
98
product of mass and acceleration of the particles=
0
99
in a 2 particle system momenta of the particles
are equal and opposite relative to com
100
what is the kinetic energy of two particle system relative to the centre of mass frame
1/2*u*(v_relative of 1 block to another)^2 | here u is m1m2/m1+m2
101
what is pseudo force in experienced due to acceleration of com
it is equal to (mass of object)*acceleration of com | this pseudo force acts in the opposite direction to the acceleration of the object
102
explain how pseudo forces act in a system where there are two blocks connected by a spring
so when a force acts towards right on the block m2 the acceleration of centre of mass= F/m1+m2 so the block m1 experiences a pseudo force towards left of magnitude m1F/m1+m2 the block m2 experiences a pseudo force towards left of magnitude m2F/m1+m2
103
explain how the forces act during the state of maximum elongation
when the spring starts elongating spring force starts acting on the blocks as well. during maximum elongation the relative velocity of the blocks become 0. at this stage the spring force balances the pseudo force in each block
104
how to solve com questions with blocks hanging from a table
remember that com represents the object | so in order for the system of blocks to remain on the table... the COM MUST REMAIN ON THE TABLE
105
during maximum compression or elongation
relative velocity of the blocks is 0
106
when you are finding kinetic energy in the frame of com
just use the formula directly and dont waste time in finding relative velocity of blocks with com and all that
107
suppose there is a rod inclined against the wall at some angle then if the bottom part of the rod is moving with some velocity then how to find the velocity of each point on the rod
first find the position vector of the point | then dr/dt=v

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