Question

Two forces of magnitudes 8 and F. The measure of the angle between them is θ. If their resultant bisects the included angle between them, then F = ?

Answer 1

If the resultant of two forces of magnitudes 8 and F bisects the included angle between them, then
`\( F = 8 \).`

Step-by-step explanation:

When two forces act at an angle, their resultant can be determined using vector addition. In this case, if the resultant bisects the included angle between the forces, it creates two equal angles on either side. Applying the Law of Cosines, we know that the magnitude of the resultant
`\( R \)` is given by
`\( R^2 = F_1^2 + F_2^2 + 2 \cdot F_1 \cdot F_2 \cdot \cos(\theta) \),` where
`\( F_1 \) and \( F_2 \)` are the magnitudes of the forces, and \( \theta \) is the angle between them.

If the resultant bisects the angle, the two resulting angles are
`\( \theta/2 \), and the cosine of \( \theta/2 \) is \( \cos(\theta/2) = \sqrt{(1 + \cos(\theta))/(2)} \).`Substituting this into the Law of Cosines, we get
`\( R^2 = 8^2 + F^2 + 2 \cdot 8 \cdot F \cdot \sqrt{(1 + \cos(\theta))/(2)} \). Since \( R^2 = (2F)^2 = 4F^2 \)` (as it bisects the angle), we can equate the expressions and solve for
`\( F \)`, leading to
`\( F = 8 \).`

Therefore, if the resultant bisects the included angle between two forces of magnitudes 8 and
`\( F \), then \( F \)` must be equal to 8. This result is obtained through the application of vector addition principles and trigonometric identities within the context of force components.