Two-Force Resultants: Key Formulas and Quick Checks

Two Forces: Resultant and Included Angle

For two forces $F1$ and $F2$ with the included angle $\theta$, the resultant magnitude $R$ satisfies R2=F<em>12+F</em>22+2F<em>1F</em>2cos⁡θ.R^2 = F<em>1^2 + F</em>2^2 + 2F<em>1F</em>2\cos\theta. The resultant ranges from $R{\min} = |F1 - F2|$ (at $\theta = 180^\circ$) to $R{\max} = F1 + F2$ (at $\theta = 0^\circ$). To find the angle from a given resultant, use cos⁡θ=R2−F<em>12−F</em>222F<em>1F</em>2.\cos\theta = \frac{R^2 - F<em>1^2 - F</em>2^2}{2F<em>1F</em>2}. An angle between a force and the resultant is given by the triangle relation cos⁡ϕ=F<em>12+R2−F</em>222F<em>1R.\cos\phi = \frac{F<em>1^2 + R^2 - F</em>2^2}{2F<em>1R}. If the forces are perpendicular, $\theta = 90^\circ$, then R=F</em>12+F<em>22,ϕ=tan⁡−1(F</em>2F1).R = \sqrt{F</em>1^2 + F<em>2^2}, \quad \phi = \tan^{-1}\left(\frac{F</em>2}{F_1}\right).

  • Example (22): If $F1 = F2 = 6$ N and $R = 6$ N, then cos⁡θ=36−36−362⋅6⋅6=−12⇒θ=120∘.\cos\theta = \frac{36 - 36 - 36}{2\cdot 6 \cdot 6} = -\frac{1}{2} \Rightarrow \theta = 120^\circ.
  • Example (23): If $F1 = 6$ N, $F2 = 8$ N, and $R = 2$ N, then cos⁡θ=4−36−642⋅6⋅8=−1⇒θ=180∘.\cos\theta = \frac{4 - 36 - 64}{2\cdot 6 \cdot 8} = -1 \Rightarrow \theta = 180^\circ.

Maximum and Minimum Resultant

For two forces, the maximum resultant occurs when they are aligned: R<em>max⁡=F</em>1+F<em>2.R<em>{\max} = F</em>1 + F<em>2. The minimum occurs when they are opposite: R</em>min⁡=∣F<em>1−F</em>2∣.R</em>{\min} = |F<em>1 - F</em>2|. These extrema are independent of orientation.

Angle Between Resultant and a Force

The angle between the resultant $R$ and the force $F1$ iscos⁡ϕ=F</em>12+R2−F<em>222F</em>1R.\cos\phi = \frac{F</em>1^2 + R^2 - F<em>2^2}{2F</em>1R}. If the forces form an angle $\theta$, another form istan⁡ϕ=F<em>2sin⁡θF</em>1+F<em>2cos⁡θ.\tan\phi = \frac{F<em>2\sin\theta}{F</em>1 + F<em>2\cos\theta}. When $F1 \perp F2$ ($\theta = 90^\circ$), ϕ=tan⁡−1(F</em>2F1).\phi = \tan^{-1}\left(\frac{F</em>2}{F_1}\right).

Three Forces: Max and Min Resultants

For three coplanar forces $F1, F2, F3$, the maximum resultant is when all three are in the same direction: R</em>max⁡=F<em>1+F</em>2+F<em>3.R</em>{\max} = F<em>1 + F</em>2 + F<em>3. The minimum possible resultant occurs when they oppose as much as possible; a general expression is R</em>min⁡=max⁡(0,  ∣F<em>max⁡−(F</em>sum of others)∣).R</em>{\min} = \max\left(0, \; |F<em>{\max} - (F</em>{\text{sum of others}})|\right). If the largest force exceeds the sum of the others, a nonzero remainder remains; otherwise the resultant can be zero.

Quick Checks and Common Rules

  • If the resultant equals the sum or difference of magnitudes, the angle is $0^\circ$ or $180^\circ$ respectively (verify with the cosine law).
  • If the resultant is zero, the two forces are equal in magnitude and opposite in direction.