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In kinematics, only one body travels along a linear line. However, every system comprises more than one body. A two-body system or numerous bodies can be observed depending on the arrangement. The motion equations become more complex while considering a system of particles. Specifically, in such cases, we should examine the centre of mass. Therefore, in this case, we will consider the centre of mass and different aspects of rotational motion, along with focusing on their relations for understanding the entire concept of system of particles and rotational motion class 11.
Types of systems of particles
Two different types of systems of particles, namely continuous and discrete systems, are considered in classical mechanics.
Discrete system
In the discrete system, all bodies are placed at a distance from each other so that no point of contact occurs. A typical example of this would be an auditorium where people sit in a scattered pattern.
Continuous system
In the continuous system, all bodies present are arranged compactly with little space between them. In this case, the centre of motion plays a crucial role in deciding the type of work done and the point of force application.
Centre of mass in a system of particles
Each particle will have its own mass in a system involving different particles. While considering the individual body masses, we cannot generalise the work energy function or the rigid body system equations. In this case, we can consider the mass concept. The mass concept is a point at which all masses of the body are concentrated. At the centre of mass, only one magnitude is considered and can be used further in evaluating the system of particles and rotational motion. According to Newton’s second law, the force applied to a body is directly proportional to uniform acceleration.
Therefore, the equation can be written as F = ma where F is the applied force, m is the mass of the body, and a is the acceleration of the concerned body in motion.
Two particle system
In the two particle system, we will consider two bodies A and B with mass m1 and m2 , respectively. In this case, the instantaneous position vectors of both bodies from the origin can be represented as r1 and r2.
In this system, the centre of the mass can be described by the following equation:
rcm = (m1r1 + m2r2)/(m1 + m2) ……. (i)
As vectors are considered, all three coordinates of the Cartesian system need to be examined. Therefore,
xcm = (m1x1 + m2x2)/(m1 + m2) ……. (ii)
ycm = (m1y1 + m2y2)/(m1 + m2) ……. (iii)
zcm = (m1z1 + m2z2)/(m1 + m2) ……. (iv)
Discrete system of particles
Let us consider n number of particles in a discrete system of particles, with mass values as m1, m2, m3, and m4, and position vectors as r1, r2, r3, and r4. In this case, the centre of mass equation can be written as:
rcm = (m1r1 + m2r2 + m3r3 + ….. + mnrn) / (m1 +m2 +m3 + ……. + mn) ……. (i)
Conservation law for momentum
Momentum is defined as the product of mass and velocity. It depends on the force applied to the system of particles. Momentum can be stated as:
For two or more bodies in an isolated system acting upon each other, the total momentum remains constant unless and until an external force is applied. Thus, momentum cannot be created or destroyed.
Mathematically, if we consider two particles A and B, then momentum is written as:
A = m1 (vf1 – vi1) …. (i)
B = m2 (vf2 – vi2) …. (i)
Combining these equations and comparing them with Newton’s second law of motion, the conservation law of momentum can be written as:
m1u1 + m2u2 = m1v1 + m2v2 …… (iii)
Angular momentum of a system of particles
Based on the theories of rotational motion, we can determine the angular momentum of a body by L and can measure it from a fixed point where the vector momentums of the individual particles are concentrated.
Thus, L = L1 + L2 +L3 +L4 + ……. + Ln …. (iv)
Kinetic and potential energy of a system of particles
When a body is at rest, it possesses potential energy denoted by U. In the state of motion, the body possesses kinetic energy denoted by K. The mathematical expressions are written as:
U = mgh
K = ½ (mv2)
where m is the mass, g is the acceleration due to gravity, h is the height from sea level, and v is the velocity with which the body is moving.
According to the law of conservation of energy, new energy cannot be created or destroyed. When the body starts gaining velocity, the potential energy is converted to kinetic energy. This energy increases till the body attains the rest position, in which the kinetic energy is converted back into potential energy.
