# Constraints in Motion (Classical Mechanics) | Types and Example | FAQs

Constraints, Types, Holonomic, Non-Holonomic constraint, Superfluous or Redundant, Rheonomous and Scleronomous, Conservative and Dissipative, Unilater

## Constraints

If the motion of a particle or system is restricted by one or more conditions, then the number of independent ways to move the particle freely is reduced. "The limitations or restrictions on the motion of the system are called constraints and this type of motion of system having constraint is called constraint motion."

### Example

1. Simple pendulum has a constraint that the radius of bead or length of wire is fixed 'l', hence we need only 'θ' as free coordinate.
2. A particle moving on Earth is restricted by the certain limitations that the velocity in z-direction is zero. So, we need now only two coordinates (x and y) as degree of freedom.

### Types of constraints

Constraints are of following types:

#### (i) Holonomic Constraints

If a particle is free to move on the circumference of the circle then only one coordinate needed (i.e. θ) to describe the motion of particle because the radius of the circle remains same

|r| = a

(constraint in circular motion)

r - a = 0 ......(1)

If a particle is free to move on the spherical circumference then among r, θ and Φ only θ and Φ are sufficient because

|r| = a

suppose the constraint is in the form of equation below for N-particles system

f (r1, r2, r3,…,t) = 0 ....(2)

then they are called holonomic constraint.

#### (ii) Non-Holonomic constraint

If the constraints is not expressible in the form of equation (2) then it is called non-holonomic constraints, e.g. the motion of a particle placed on the surface of radius

|r| ≤ a,

r - a ≤ 0

The particle will first slide and then fall freely on earth at that time the distance of particles from the surface will increase, e.g., a gas molecule in a spherical container, motion is described as :

|r| ≤ a

r - a ≤ 0

#### Superfluous or Redundant coordinates

If we throw a particle vertically upward with v velocity then it will move in straight line. The co-ordinate may be defined at a time t.

x = 0,   y = vt - 1/2gt2,  z = 0

the x and z co-ordinates remains 0 for every time. These type of coordinates are called redundant or superfluous co-ordinate. Simply the coordinates of no use are called redundant.

#### Rheonomous and Scleronomous

If the equation of constraint or constraint relation depend explicitly on time (clearly on time) then constraint is called Rheonomous otherwise if constraint relations do not depend on time explicitly, constraints are scleronomous.

#### Conservative and Dissipative

• If total mechanical energy of a system is conserved while performing the Constraint, then Constraint is called conservative. This type of forces of constraint do not do any work. {Simple pendulum}
• If the constraint forces do work and total mechanical energy is not conserved, constraint is called dissipative. e.g. pendulum of variable length, Time dependent or Rheonomous constraints are also dissipative.

#### Unilateral and Bilateral

• If the constraint relation are the form of inequalities then they are called unilateral, e.g. a gas molecules in a spherical container |r| ≤ a
• If the constraint relations are in form of equations then they are called bilateral.

### Force of Constraint

Constraints are always related to a force that restrict the motion of the particle. These forces associated with the constraints are called as forces of constants.

e.g., In case of simple pendulum, constraint force is the tension of string. In case of rigid body the action and reaction between any two particles (i.e. Inertial force) is the constraint force.

Usually the force of constraints act perpendicular to the surface and momentum of the particle parallel to the surface of constraints. In this case work done by force is always 0 and constraint is workless and remark as ideal.

## FAQs

Que : In the following cases, discuss whether the constraint is holonomic or non-holonomic. Specify the constraint force also :

1. Motion of a body on an inclined plane under gravity.
2. A bead on a circular wire.
3. A particle moving on an ellipsoid under the influence of gravity.
4. A pendulum with variable length.

Ans :

1. When a body moves on an incline plane, it is constrained to move on the inclined plane surface. Hence the constraint is holonomic. The force of constraint is the reaction of a plane, acting normal to the inclined surface.
2. The constraint is that the bead remains at a constant distance a, the radius of the circular wire and can be expressed as r = a. Hence the constraint is holonomic. The force of constraint is the reaction of the wire, acting on the bead.
3. The constraint is nonholonomic, because the particle after reaching a certain point will leave the ellipsoid. Force of constraint is the reaction force of the ellipsoid surface on the particle.
4. The constraint is holonomic and rheonomic, because the constraint equation is |r| = l (t). The constraint force is the tension in the string.