Force Resultants: Comparing Rules

[Bassel AbdulSabour]

I just want to know the difference between those rules:

1. R^2 = F1^2 * F2^2 + 2*F1*F2*COS(the angle between F1 and F2)

2. The second is about the parallelogram rule, it says that the two vectors are added and their summation is the magnitude of the resultant.

Which one is correct?

 

[tnich]

I just want to know the difference between those rules:

1. R^2 = F1^2 * F2^2 + 2*F1*F2*COS(the angle between F1 and F2)

2. The second is about the parallelogram rule, it says that the two vectors are added and their summation is the magnitude of the resultant.

Which one is correct?

If I correctly understand you, both are correct. You seem to have the sign wrong in the equation. $R^2=F_1^2+F_2^2-2F_1F_2\cos(\theta)$

 

[tnich]

If I correctly understand you, both are correct. You seen to have the sign wrong in the equation. $R^2=F_1^2+F_2^2-2F_1F_2\cos(\theta)$

Here is a diagram showing my understanding of the problem.

 

[Orodruin]

Here is a diagram showing my understanding of the problem.
View attachment 230677

That would typically not be ”the angle between the forces”. The angle between two forces in the same direction would typically be zero, whereas your convention would be pi.

 

[tnich]

That would typically not be ”the angle between the forces”. The angle between two forces in the same direction would typically be zero, whereas your convention would be pi.

I agree. I am trying to interpret what the OP has written.

 

[robphy]

Rule 1 is the dot-product [law of cosines] (which is a metrical statement).
Note that the angle-between-the-vectors (tails together, as in the parallelogram method of addition) is not the interior angle in the triangle (in the tail-to-tip method of addition).

The parallelogram rule for adding vectors is true, independent of the metric.
That tells you how to add two vectors... with the tails together, construct a parallelogram, and draw from the common tail to the opposite corner.
That's the resultant vector.
Getting the magnitude of the vector-sum is a different step [see rule 1].

 

[sophiecentaur]

Here is a diagram showing my understanding of the problem.
View attachment 230677

That's the old "Cos Rule" which we all did at school. Using the Supplementary Angle (as with vectors, you just get a change of sign.
Cos(x) = -Cos(π-x)