Introduction
Physicists love to look at something complicated and find in it, something simple and familiar. Here is an example. If you flip a baseball bat into the air, its motion as it turns, is clearly more complicated than that of a non-spinning tossed ball, which moves like a particle.
Center of Mass
The centre of mass is an imaginary point where one can assume the entire mass of the given system or object to be positioned.
Generalization to N - Particles
Consider a system of N particles of masses m1, m2, m3,------------ mN.
Momentum Conservation and Motion of Center of Mass
If the external forces acting on the system add up to zero, the centre of mass moves with constant velocity.
Motion of Center of Mass
Consider two particles A and B of masses m1 and m2, respectively. Take the line joining A and B as the X-axis. Let the coordinates of the particles at time 't' be x1 and x2.
Center of Mass of a Rigid Body
The centre of mass of a rigid body is a point whose position is fixed with respect to the body as a whole. The point may or may not lie in the body. The position of the centre of mass of a rigid body depends on.
Definition of Angular Velocity
Angular velocity of a rotating rigid body is the rate of change of angle swept.
Definition of Torque
Torque is the product of the magnitude of the force and the lever arm of the force.
Torque in Two and Three Dimensions
Torque in Two and Three dimension, then the torque acting on the particle with respect to the origin.
Angular Momentum
Angular momentum bears the same relation to linear momentum that torque does to force.
Physical Meaning of Angular Momentum
The angular momentum, which is nothing but moment of linear momentum, can be expressed in terms of the lever arm for momentum.
Conservation of Angular Momentum
We already know two powerful conservation laws, namely, the conservation of energy and the conservation of linear momentum. Let us study another conservation law, namely the law of conservation of angular momentum.
Illustrations of the Law of Conservation of Angular Momentum
Let us consider a student seated on a stool that can rotate freely about a vertical axis. The student, who is set into rotation at a modest initial angular speed wi, holds two dumbbells in an outstretched hand. His angular momentum lies along the vertical rotation axis, pointing upwards.
Geometrical Meaning and Conservation of Angular Momentum
In the case of a planet, moving around the sun in an elliptical orbit with the sun at one of the foci of the ellipse, the gravitational force always acts along the line joining the planet with the sun.
Equilibrium of Rigid Bodies
A rigid body is said to be in equilibrium if, both the linear and angular momentum of a rigid body have a constant value. For the equilibrium of a rigid body, the body need not be at rest. However, if it is at rest, it is called the static equilibrium.
Rigid Body Rotation and Equation of Rotation of Motion
Consider a pulley fixed at a typical Indian well on which a rope is wound with one end attached to the bucket. When the bucket is released, the pulley starts rotating. As the bucket goes down, the pulley rotates more rapidly till the bucket goes into the water.
Comparison of Linear and Rotational Motion
When a rigid body such as a merry-go-round rotates around an axis, each particle in the body moves in its own circle around that axis. Since the body is rigid, all particles make one revolution in the same amount of time. i.e., they all have the same angular speed w.
Moment of Inertia and its Physical Meaning
Consider a rigid body rotating about a fixed axis AB. Consider a particle 'p' of mass 'm' rotating in a circle of radius 'r'.
Radius of Gyration
It is the distance from the axis of rotation at which, if the whole mass of the body were to be concentrated, the moment of inertia would be the same as that with the actual distribution of mass. It is denoted by K.
Moment of Inertia of Continuous Mass Distribution
If a body is assumed to be continuous, one can use the technique of integration to obtain its moment of inertia about a given line.
Parallel Axis Theorem
The moment of inertia of a body about an axis is equal to its moment of inertia about a parallel axis through its centre of gravity plus the product of the mass of the body and the square of the perpendicular distance between the two parallel axes.
Perpendicular Axis Theorem
This is applicable only to a plane lamina. The moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about any two mutually perpendicular axes, passing in its own plane, intersecting each other at the point through which the perpendicular axis passes.
Moment of Inertia of Circular Ring, Disc, Cylinder, Sphere and Thin Straight Rod
The weight Mg of the cylinder, acting vertically downwards through the centre of mass of the cylinder.
Examples of Binary Systems in Nature
A binary system is one in which two heavenly bodies revolve about a common centre of mass. (The word binary means two).
Moment of Inertia of a Diatomic Molecule
In stable equilibrium position, the two atoms in a diatomic molecule are separated by a certain distance r0. This distance is called intermolecular distance or bond length.
Summary
Principle of conservation of linear momentum states that the linear momentum of a system remains constant if the external forces acting on the system add up to zero.
