Parallelogram Law of Vectors.

Objective

Our objective is to find the weight of a given body using the Parallelogram Law of Vectors.

Theory

What does the Parallelogram Law of Vectors state?

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»

or

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»R«/mi»«mo»=«/mo»«msqrt»«mrow»«msup»«mi mathvariant=¨normal¨»P«/mi»«mn»2«/mn»«/msup»«mo»§nbsp;«/mo»«mo»+«/mo»«msup»«mi mathvariant=¨normal¨»Q«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»P«/mi»«mi mathvariant=¨normal¨»Q«/mi»«mi mathvariant=¨normal¨»cos«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»§#920;«/mi»«/mrow»«/msqrt»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»1«/mn»«mo»)«/mo»«/math»                                  

The unknown weight can be calculated from the equation (1).

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 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,

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»S«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»OC«/mi»«mo»§nbsp;«/mo»«mo»§#215;«/mo»«mi mathvariant=¨normal¨»Scale«/mi»«mo»§nbsp;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«/math»

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

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»Percentage«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»error«/mi»«mo»=«/mo»«mfrac»«mfenced»«mrow»«mi mathvariant=¨normal¨»A«/mi»«mi mathvariant=¨normal¨»c«/mi»«mi mathvariant=¨normal¨»t«/mi»«mi mathvariant=¨normal¨»u«/mi»«mi mathvariant=¨normal¨»a«/mi»«mi mathvariant=¨normal¨»l«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»w«/mi»«mi mathvariant=¨normal¨»e«/mi»«mi mathvariant=¨normal¨»i«/mi»«mi mathvariant=¨normal¨»g«/mi»«mi mathvariant=¨normal¨»t«/mi»«mi mathvariant=¨normal¨»h«/mi»«mi mathvariant=¨normal¨»t«/mi»«mo»§nbsp;«/mo»«mo»-«/mo»«mi mathvariant=¨normal¨»C«/mi»«mi mathvariant=¨normal¨»a«/mi»«mi mathvariant=¨normal¨»c«/mi»«mi mathvariant=¨normal¨»u«/mi»«mi mathvariant=¨normal¨»l«/mi»«mi mathvariant=¨normal¨»a«/mi»«mi mathvariant=¨normal¨»t«/mi»«mi mathvariant=¨normal¨»e«/mi»«mi mathvariant=¨normal¨»d«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»w«/mi»«mi mathvariant=¨normal¨»e«/mi»«mi mathvariant=¨normal¨»i«/mi»«mi mathvariant=¨normal¨»g«/mi»«mi mathvariant=¨normal¨»h«/mi»«mi mathvariant=¨normal¨»t«/mi»«/mrow»«/mfenced»«mrow»«mi mathvariant=¨normal¨»A«/mi»«mi mathvariant=¨normal¨»c«/mi»«mi mathvariant=¨normal¨»t«/mi»«mi mathvariant=¨normal¨»u«/mi»«mi mathvariant=¨normal¨»a«/mi»«mi mathvariant=¨normal¨»l«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»w«/mi»«mi mathvariant=¨normal¨»e«/mi»«mi mathvariant=¨normal¨»i«/mi»«mi mathvariant=¨normal¨»g«/mi»«mi mathvariant=¨normal¨»h«/mi»«mi mathvariant=¨normal¨»t«/mi»«/mrow»«/mfrac»«mo»§#215;«/mo»«mo»§nbsp;«/mo»«mn»100«/mn»«mo»§nbsp;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»3«/mn»«mo»)«/mo»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»Percentage«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»error«/mi»«mo»=«/mo»«mfrac»«mfenced»«mrow»«mi mathvariant=¨normal¨»W«/mi»«mo»-«/mo»«mi mathvariant=¨normal¨»S«/mi»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»W«/mi»«/mfrac»«mo»§#215;«/mo»«mn»100«/mn»«/math»

Learning Outcomes

  • Students learn what is parallelogram law of vectors.
  • They become familiar with the Gravesands apparatus.
  • Students are able to find the unknown weight of an object using the parallelogram law of vectors.

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