Get ready for better learning with NCERT & CBSE Systems of Particles and Rotational Motion Notes Class 11 Physics. No need to roam anywhere we will provide you with all the study content you need. Let’s study smartly. PDF view is also there for your suitability.
Systems of Particles and Rotational Motion Worksheet
INTRODUCTORY CONCEPTS
Particle. A particle is defined as an object whose mass is finite but whose size and internal structure can be neglected.
System A system is a collection of a very large number of particles which mutually interact with one another. A body of finite size can be regarded as a system because it is composed of a large number of particles interacting with one another.
Internal forces. The mutual forces exerted by the particles of a system on one another are called internal forces. These forces are responsible for holding together the particles as a single object.
External forces. The outside force exerted on an object by any external agency is called an external force. Such a force changes the velocity of an object.
CENTRE OF MASS
Centre of mass. Newton’s laws of motion are applicable to point objects. The introduction of the concept of the centre of mass enables us to apply them equally well to the motion of finite or extended objects.
The centre of mass of a body is a point where the whole mass of the body is supposed to be concentrated for describing its translatory motion.
The centre of mass of a system of particles is that single point which moves in the same way in which a single particle having the total mass of the system and acted upon by the same external force would move.
If a single force acts on a body and the line of action of the force passes through the centre of mass, the body will have only linear acceleration and no angular acceleration. For example, consider a hammer resting on a plane surface. If a force P is applied on the hammer in such a way that its line of action passes
through the centre of mass of the hammer, then the hammer moves along a straight line path, as shown in Fig. 7.1(a). But when a force R is applied along a line not passing through its centre of mass, then the hammer rotates about its centre of mass, as shown in Fig. 7.1(b).
Centre of mass vs. centre of gravity. The centre of mass of body is point where whole mass of the body may be assumed to be concentrated for describing its translatory motion. On the other hand, the centre of gravity is a point at which the resultant of the gravitational forces on all the particles of the body acts
i.e., a point where whole weight may be assumed to act. In a uniform gravitational field such as that of the earth on a small body, the centre of gravity coincides with the centre of mass. But in the case of Mount Everest, the centre of gravity lies a little below its centre of mass because the gravitational force decreases with altitude.
CENTRE OF MASS OF A TWO-PARTICLE SYSTEM
Centre of mass of a two particle-system. Consider a system of two particles P1 and P2 of masses nq and nq.
Let r1→ and r2→ be their position vectors with respect to the origin O, as shown in Fig. 7.2.
The position vector R→cm of the centre of mass C of the two-particle system is given by
R→cm = m1r→1 + m2r→2 / m1 + m2
Discussion. (i) The above equation shows that the position vector of a system of particles is the weighted average of the position vectors of the particles making the system, each particle making a contribution proportional to its mass.
(ii) We can write the above equation as
(m1 + m2) R→cm = m1r→1 + m2r→2
Thus the product of the total mass of the system and the position vector of its centre of mass is equal to the sum of the products of individual masses and their respective position vectors.
(iii) If m1 = m2=m (say), then R→cm = r→1 + r→2 / 2
Thus the centre of mass of two equal masses lies exactly at the centre of the line joining the two masses.
(iv) If (x1, y1) and (x2, y2) are the coordinates of the locations of the two particles, the coordinates of their centre of mass are given by
Systems of Particles and Rotational Motion Notes Class 11 Physics
EXAMPLES OF BINARY SYSTEMS IN NATURE
(i) Binary stars. Two stars bound to each other by the gravitational force and orbiting around their common centre of mass are called binary stars. Fig. 7.19(a) shows binary stars S1 and S2 of equal mass moving in circular orbits around their common centre of mass, which is at rest.
Fig. 7.19 Orbits of binary stars of equal mass, when their (a) CM is at rest, (b) CM is in uniform motion
When no external force acts on the system, the centre of mass of the double star moves like a free particle. The orbits of the two stars are slightly complicated, as shown in Fig. 7.19(b). But these are just the combination of two motions : (i) the uniform motion of the centre of mass CM in a straight line and (ii) the circular orbits of the two stars around their CM. However, the two stars always remain on the opposite sides of the CM.
(ii) Diatomic molecule. A symmetric diatomic molecule like 02 is also an example of binary system. The internal binding force between the two oxygen atoms is due to the chemical bond which can be regarded as a spring. When there is no external force (i.e., no collisions between the molecules themselves or with the walls of the vessel), the centre of mass of the molecule moves with uniform velocity in a straight line, as shown in Fig. 7.19(c). The molecule can also have vibrational and rotational motions again due to the internal forces. Even then the centre of mass moves like a free particle.
Fig. 7.19 (c) Uniform motion of the CM of moving, vibrating symmetric diatomic molecule 02.
(Hi) Earth-moon system. The moon moves around the earth in a circular orbit and the earth moves around the sun in an elliptical orbit. It will be more correct to say that the centre of mass of the earth-moon system moves around the sun in an elliptical orbit, not the earth and moon themselves. As the mass of the earth is nearly 80 times the mass of the moon, so the centre of mass divides the earth-moon (E-M) line in the ratio 1: 80. In fact, this point lies inside the earth, as shown in Fig. 7.19(d).
Here the mutual forces of gravitation between the earth and moon are internal forces while the Sun’s attraction of both earth and moon are the external forces acting on the centre of mass of the earth-moon system.
Systems of Particles and Rotational Motion Notes Class 11 Physics