Page 159 - Class-11-Physics-Part-1_Neat

SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 145

particles. The centre of mass of the system is m (y + y + y ) y + y + y
that point C which is at a distance X from O, Y = 1 2 3 = 1 2 3
3m 3
where X is given by
Thus, for three particles of equal mass, the
m x + m x
X = 1 1 2 2 centre of mass coincides with the centroid of the
m + m (7.1)
1 2 triangle formed by the particles.
In Eq. (7.1), X can be regarded as the mass- Results of Eqs. (7.3a) and (7.3b) are
weighted mean of x and x . If the two particles generalised easily to a system of n particles, not
1 2
have the same mass m = m = m then necessarily lying in a plane, but distributed in
1 2 ,
mx + mx x + x space. The centre of mass of such a system is
X = 1 2 = 1 2 at (X, Y, Z ), where
2m 2
Thus, for two particles of equal mass the X = ∑ m x i (7.4a)
i
centre of mass lies exactly midway between M
them. ∑
i
If we have n particles of masses m , m , Y = m y i
1 2 (7.4b)
...m respectively, along a straight line taken as M
n
the x- axis, then by definition the position of the ∑
i
centre of the mass of the system of particles is and Z = m z i (7.4c)
given by. M
n Here M = ∑ m is the total mass of the
∑ m x i
m x + m x + ... + m x i i ∑ m x i
i
X = 1 1 2 2 n n = i= 1 = (7.2) system. The index i runs from 1 to n; m is the
i
n
m + m +... + m n ∑ m ∑ m i mass of the i particle and the position of the
th
1
2
th
i= 1 i i particle is given by (x , y , z ).
i i i
where x , x ,...x are the distances of the Eqs. (7.4a), (7.4b) and (7.4c) can be
1 2 n
particles from the origin; X is also measured from combined into one equation using the notation
the same origin. The symbol ∑ (the Greek letter of position vectors. Let r be the position vector
i
sigma) denotes summation, in this case over n of the i particle and R be the position vector of
th
particles. The sum the centre of mass:
∑ m = M r i = x i + y j + z k

i
i
i
i
is the total mass of the system. and R = X + Y + k

Z
j
i
Suppose that we have three particles, not
lying in a straight line. We may define x– and y– ∑ m r
axes in the plane in which the particles lie and Then R = i i (7.4d)
represent the positions of the three particles by M
coordinates (x ,y ), (x ,y ) and (x ,y ) respectively. The sum on the right hand side is a vector
1 1 2 2 3 3
Let the masses of the three particles be m , m sum.
1 2
and m respectively. The centre of mass C of Note the economy of expressions we achieve
3
the system of the three particles is defined and by use of vectors. If the origin of the frame of
located by the coordinates (X, Y) given by reference (the coordinate system) is chosen to
be the centre of mass then ∑ m r = 0 for the
m x + m x + m x i i
X = 1 1 2 2 3 3
m + m + m (7.3a) given system of particles.
3
1
2
A rigid body, such as a metre stick or a
m y + m y + m y flywheel, is a system of closely packed particles;
Y = 1 1 2 2 3 3
m + m + m (7.3b) Eqs. (7.4a), (7.4b), (7.4c) and (7.4d) are
3
2
1
therefore, applicable to a rigid body. The number
For the particles of equal mass m = m = m
1 2 of particles (atoms or molecules) in such a body
= m ,
3 is so large that it is impossible to carry out the
m (x + x + x ) x + x + x summations over individual particles in these
X = 1 2 3 = 1 2 3
3m 3 equations. Since the spacing of the particles is
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