. If m1 and m2 are the masses constituting the rigid body, bound by some internal forcers so that the
distance between the masses remain constant then
(1) Define centre of mass
(II) Derive expression for position vector of centre of mass.
Answers
Answer:
Let two bodies of masses m
1
and m
2
moving with velocities u
1
and u
2
along the same straight line.
And consider the two bodies collide and after collision v
1
and v
2
be the velocities of two masses.
Before collision
Momentum of mass m
1
=m
1
u
1
Momentum of mass m
2
=m
2
u
2
Total momentum before collision
p
1
=m
1
u
1
+m
2
u
2
Kinetic energy of mass m
1
=
2
1
m
1
u
1
2
Kinetic energy of mass m
2
=
2
1
m
2
u
2
2
Thus,
Total kinetic energy before collision is
K.E=
2
1
m
1
u
1
2
+
2
1
m
2
u
2
2
After collision
Momentum of mass m
1
=m
1
v
1
Momentum of mass m
2
=m
2
v
2
Total momentum before collision
P
f
=m
1
v
1
+m
2
v
2
Kinetic energy of mass m
1
=
2
1
m
1
v
1
2
Kinetic energy of mass m
2
=
2
1
m
2
v
2
2
Total kinetic energy after collision
K
f
=
2
1
m
1
v
1
2
+
2
1
m
2
v
2
2
So, according to the law of conservation of momentum
m
1
u
1
+m
2
u
2
=m
1
v
1
+m
2
v
2
⇒m
1
(u
1
−v
1
)=m
2
(v
2
−u
2
)
............(1)
And according to the law of conservation of kinetic energy
2
1
m
1
u
1
2
+
2
1
m
2
u
2
2
=
2
1
m
1
v
1
2
+
2
1
m
2
v
2
2
⇒m
1
(u
1
2
−v
1
2
)=m
2
(v
2
2
−v
1
2
)
.............(2)
Now dividing the equation
(u
1
+v
1
)=(v
2
+u
2
)
⇒u
1
−u
2
=v
2
−v
1
Therefore, this is, relative velocity of approach is equal to relative velocity of separation.
(1) A particle system's centre of mass is a point that represents the mean position of all the particles in the system.
- It is the point at which the entire mass of the system can be assumed to be concentrated, and around which the particles move as if they were a single particle.
- The center of mass is an important concept in physics, as it allows us to simplify the analysis of the motion of a system of particles.
(II) The formula for the position vector of the centre of mass is:
R = (m1r1 + m2r2)/(m1 + m2)
- where R is the position vector of the center of mass, m1 and m2 are the masses of the particles, and r1 and r2 are the position vectors of the particles with respect to some origin.
- This formula can be derived by considering the definition of the center of mass as the point at which the net torque on the system is zero.
- By setting the net torque to zero and solving for the position vector R, we arrive at the above formula.
In summary, the center of mass is a point that represents the average position of all the particles in a system, and the position vector of the center of mass is given by a formula that takes into account the masses and position vectors of the individual particles.
To learn more about mass from the given link.
https://brainly.in/question/24482524
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