Constraints
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The document discusses constraints in the context of solving differential equations for the motion of n-particles, defining constraints as restrictions on positions and velocities. It categorizes constraints into several types, including holonomic, non-holonomic, scleronomic, and rheonomic, with examples provided for each type. The document elaborates on the differences between these classifications, particularly highlighting how certain conditions can depend on time or can be expressed as equations connecting particle coordinates.
Constraints
- 2. CONSTRAINTS TOPICS TO BE DISCUSSED • DEFINITION OF CONSTRAINTS • EXAMPLES OF CONSTRAINTS • TYPES OF CONSTRAINTS WITH EXAMPLES
- 3. CONSTRAINTS In order to solve a set of differential equations for the motion of a system of n-particles, we have to impose certain restrictions on the positions and velocities of the particles of the system. Such geometrical or kinematical restrictions on the motion of a particle or system of particles are called constraints. EXAMPLES OF CONSTRAINED MOTION 1. The motion of a rigid body restricted by the condition that the distance between any of its two particles remains unchanged. 2. The beads of an abacus constrained to one dimensional motion by the supporting wires. 3. Motion of the gas molecules within a container restricted by the walls of the vessel. CLASSIFICATION OF CONSTRAINTS (1) (a) Scleronomic: constraint relations do not explicitly depend on time. (b) Rheonomic: constraint relations depend explicitly on time. (2) (a) Holonomic: conditions of constraint can be expressed as equations connecting the coordinates of theparticles.
- 4. (b) Non holonomic: constraint relations are not holonomic. (3) (a)Conservative: total mechanical energy of the system is conserved while per-forming, the constrained motion. Constraint forces do not do any work. (b) Dissipative: constraint forces do work and total mechanical energy is not conserved. (4) (a)Bilateral: at any point on the constraint surface both the forward and back-ward motions are possible. Constraint relations are not in the form of inequalities but are in the form of equations. (b) Unilateral: at some points no forward motion is possible. Constraint relations are expressed in the form of inequalities. HOLONOMIC CONSTRAINTS For a constraint to be holonomic its conditions must be expressible as equations connecting the coordinates of the particles (and , if possible, time also) i.e. in the form of equation: F(r1 ,r2,………,rn, t)=0 Where r 1,r2,………..,rn represent the position vectors of the particles of a system and t the time. Hence, a holonomic constraint depends only on the coordinates and time. It does not depend on the velocities.
- 5. EXAMPLES OF HOLONOMIC CONSTRAINTS • CONSTRAINTS INVOLVED IN THE MOTION OF RIGID BODIES: l ri – rjI2= Cij 2 where Cij are constants and ri and rj are the position vectors of ith and jth particles. • Constraints involved in the motion of the point mass of simple pendulum | r– a | = l2
- 6. NON-HOLONOMIC CONSTRAINTS • If the conditions of the constraints cannot be expressed as equations connecting the coordinates of particles of the system, they are called non-holonomic constraints. • The conditions of these constraints are expressed in the form of inequalities. Examples of non-holonomic constraints •constraints involved in the motion of a particle placed on the surface of solid sphere i.e. r2 –a2 >0 where a is the radius of thesphere. •constraint on an object rolling on a rough surface without slipping. •constraints involved in the motion of gas molecules in a container i.e. r3 <=a3
- 7. AN EXAMPLE TO ILLUSTRATE THE DIFFERENCE BETWEEN HOLONOMIC AND NON- HOLONOMIC CONSTRAINTS The motion of a particle constrained to lie on the surface of a sphere is subject to a holonomic constraint. But if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic.
- 8. SCLERONOMIC CONSTRAINTS • The constraints which are independent of time are called scleronomic constraints e.g. a bead sliding on a rigid curved wire fixed in space RHEONOMIC CONSTRAINTS The constraints which contain time explicitly are called rheonomic constraints. e.g. 1) a bead sliding on a rigid curve wire moving in some prescribed fashion. 2)if we construct a simple pendulum whose length changes with time i.e. l=l(t) then the constraints expressed by the equations are time dependent, hence, rheonomic.
- 9. THANX