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### Constrains and constrained motion

• A constrained motion is a motion which can not proceed arbitrary in any manner.
• Particle motion can be restricted to occur (1) along some specified path (2) on surface (plane or curved) arbitrarily oriented in space.
• Imposing constraints on a mechanical system is done to simplify the mathematical description of the system.
• Constraints expressed in the form of equation f(x1,y1,z1,......,xn,yn,zn :t)=0 are called holonomic constraints.
• Constraints not expressed in this fashion are called non-holonomic constraints.
• Scleronomic conatraints are independent of time.
• Constraints containing time explicitely are called rehonomic.
• Therefore a constraint is either
"Scleronomic where constraints relations does not depend on time or rheonomic where constraints relations depends explicitly on time "
and either
"holonomic where constraints relations can be made independent of velocity or non-holonomic where these relations are irreducible functions of velocity"

Constraints types of some physicsl systems are given below in the table

### How to simplify circuits with resistors

1. In any given circuit first of all recognize the resistances connected in series then by summing the individual resistances draw a new, simplified circuit diagram.

For series combination of resistances equivalent resistance is given by the equation
Req= R1 + R2+R3
The current in each resistor is the same when connected in parallel combination.

2. Then recognize the resistances connected in parallel and find the equivalent resistances of parallel combinations by summing the reciprocals of the resistances and then taking the reciprocal of the result. Draw the new, simplified circuit diagram.
(1/R)=(1/R1)+(1/R2)+(1/R2)

Remember that for resistors connected in parallel combination ‘The potential difference across each resistor is the same’.

3. Repeat the first two steps as required, until no further combinations can be made using resistances. If there is only a single battery in the circuit, this will usually result in a single equivalent resistor in series with the battery.

4. Use Ohm’s Law, V= IR, to determine the current in the equivalent resistor. Then work backwards through the diagrams, applying the useful facts listed in step 1 or step 2 to find the currents in the other resistors. (In more complex circuits, Kirchhoff’s rules can be applied).