To find the angle between two equal forces (P) when their resultant is equal to (i) P and (ii) P/2, we can use the concept of vector addition and trigonometry.
(i) When the resultant is equal to P:
In this case, the forces are balanced, and the angle between them would be 180 degrees or π radians. The forces act in opposite directions, canceling each other out.
(ii) When the resultant is equal to P/2:
To determine the angle between the two equal forces, we can use the law of cosines. The formula for the law of cosines is:
c² = a² + b² - 2ab * cos(C)
Where:
c is the length of the resultant vector (P/2)
a and b are the lengths of the two equal forces (P)
C is the angle between the two equal forces
Rearranging the equation, we have:
cos(C) = (a² + b² - c²) / (2ab)
Substituting the values, where a = b = P and c = P/2:
cos(C) = (P² + P² - (P/2)²) / (2P * P)
cos(C) = (2P² - P²/4) / (2P²)
cos(C) = (8P² - P²) / (8P²)
cos(C) = 7P² / (8P²)
cos(C) = 7/8
Taking the inverse cosine of both sides:
C = cos^(-1)(7/8)
Therefore, the angle between two equal forces (P) when their resultant is equal to P/2 is approximately equal to cos^(-1)(7/8) radians or degrees.