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Parallelogram Law Of Vectors

Vijaya Kumar Jayanthi
06/11/2017 0 0

If  two vectors acting simultaneously on a particle are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then their resultant is completely represented in magnitude and direction by the diagonal of that parallelogram drawn from that point.

Parallelogram Law of Vectors explained:

Let two vectors P and Q act simultaneously on a particle O at an angle «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»§#952;«/mi»«/math». They are represented in magnitude and direction by the adjacent sides OA and OB of a parallelogram OACB drawn from a point O.Then the diagonal OC passing through O, will represent the resultant R in magnitude and direction.

On a Gravesand's apparatus, if the body of unknown weight (say S) is suspended from the middle hanger and balancing weights P and Q are suspended from othe two hangers then,

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mi mathvariant=¨normal¨»R«/mi»«mo»§#8594;«/mo»«/mover»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mover accent=¨true¨»«mi mathvariant=¨normal¨»P«/mi»«mo»§#8594;«/mo»«/mover»«mo»§nbsp;«/mo»«mo»+«/mo»«mover accent=¨true¨»«mi mathvariant=¨normal¨»Q«/mi»«mrow»«mo»§#8594;«/mo»«mo»§nbsp;«/mo»«/mrow»«/mover»«/math»

The unknown weight can be calculated from the above equation .

On a Gravesand's apparatus, if the body of unknown weight (say S) is suspended from the middle hanger and balancing weights.

 And Q are suspended from the other two hangers then,

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mi mathvariant=¨normal¨»P«/mi»«mo»§#8594;«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mi mathvariant=¨normal¨»Q«/mi»«mo»§#8594;«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mi mathvariant=¨normal¨»S«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mn»0«/mn»«/math»

Now construct a parallelogram OACB by assuming a scale (say 1cm = 50 gwt) corresponding to the weights P and Q. The diagonal of the parallelogram OC will give the resultant vector. The weight of the unknown body,

S = OC * scale

If W is the actual weight of the body, then the percentage error in the experiment can be calculated using the equation,

Percentage error = (Actual weight - caliculated weight) * 100/Actual weight.

                        = (W - S) * 100/W

 

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