Since forces are vector quantities with both magnitude and direction, they follow the rules of vector addition. This tutorial presents two fundamental methods for determining the resultant force when multiple forces act on an object.
The geometric (or head-to-tail) method provides a visual approach to force addition. When adding forces geometrically:
The magnitude of the resultant can be measured directly from the diagram using an appropriate scale, and its direction can be determined with a protractor.
The analytical method provides greater accuracy and is particularly useful for three or more forces. This approach involves:
Mathematically, for two forces \( \vec{F_1} = (F_{1x}, F_{1y}) \) and \( \vec{F_2} = (F_{2x}, F_{2y}) \), the resultant is:
\( \vec{R} = \vec{F_1} + \vec{F_2} = (F_{1x} + F_{2x}, \; F_{1y} + F_{2y}) \)
The magnitude and direction are then:
\( |\vec{R}| = \sqrt{(F_{1x} + F_{2x})^2 + (F_{1y} + F_{2y})^2} \)
\( \theta_R = \tan^{-1}\left(\frac{F_{1y} + F_{2y}}{F_{1x} + F_{2x}}\right) \), adjusted to the correct quadrant
Forces \( \vec{F_1} \) and \( \vec{F_2} \) have magnitudes 20 N and 30 N with directions 25° and -44° relative to the positive x-axis, respectively. Determine the magnitude and direction of the resultant force \( \vec{R} = \vec{F_1} + \vec{F_2} \).
Solution
Method 1: Geometric/Trigonometric Approach
Using the law of cosines for the triangle formed by the vectors:
\( |\vec{R}|^2 = |\vec{F_1}|^2 + |\vec{F_2}|^2 - 2|\vec{F_1}||\vec{F_2}|\cos(180^\circ - (25^\circ + 44^\circ)) \)
\( |\vec{R}| = \sqrt{20^2 + 30^2 - 2(20)(30)\cos(69^\circ)} \approx 29.5 \, \text{N} \)
Applying the law of sines to find angle \( \alpha \):
\( \frac{\sin(\alpha)}{30} = \frac{\sin(69^\circ)}{29.5} \)
Solving gives \( \alpha \approx 71.7^\circ \). The direction of \( \vec{R} \) relative to the positive x-axis is:
\( \theta_R = 25^\circ + \alpha \approx 96.7^\circ \)
Method 2: Analytical Component Approach
Resolve each force into components:
\( \vec{F_1} = (20\cos 25^\circ, 20\sin 25^\circ) \approx (18.13, 8.45) \)
\( \vec{F_2} = (30\cos(-44^\circ), 30\sin(-44^\circ)) \approx (21.57, -20.85) \)
Sum the components:
\( \vec{R} = \vec{F_1} + \vec{F_2} \approx (-3.44, 29.30) \)
Calculate magnitude and direction:
\( |\vec{R}| = \sqrt{(-3.44)^2 + (29.30)^2} \approx 29.5 \, \text{N} \)
\( \theta_R = 180^\circ - \tan^{-1}\left(\frac{29.30}{3.44}\right) \approx 96.7^\circ \)
The analytical method provides greater precision and efficiency, especially when adding three or more forces.
For video explanations of similar problems, visit: Resultant of Two Forces
Determine the magnitude and direction of force \( \vec{F_2} \) such that the system of three forces is in equilibrium: \( \vec{F_1} + \vec{F_2} + \vec{F_3} = 0 \).
Solution
For equilibrium, the resultant force must be zero. Express each force in component form:
\( \vec{F_1} = (23\cos 30^\circ, 23\sin 30^\circ) \approx (19.92, 11.50) \)
\( \vec{F_2} = (|F_2|\cos\theta, |F_2|\sin\theta) \) (unknown magnitude and direction)
\( \vec{F_3} = (-25\sin 26^\circ, -25\cos 26^\circ) \approx (-10.96, -22.47) \)
Applying the equilibrium condition:
\( 19.92 + |F_2|\cos\theta - 10.96 = 0 \)
\( 11.50 + |F_2|\sin\theta - 22.47 = 0 \)
Solving these equations:
\( |F_2|\cos\theta = 10.96 - 19.92 = -8.96 \)
\( |F_2|\sin\theta = 22.47 - 11.50 = 10.97 \)
Calculate the magnitude:
\( |F_2| = \sqrt{(-8.96)^2 + (10.97)^2} \approx 14.2 \, \text{N} \)
Determine the direction (noting that cosine is negative and sine is positive, placing \( \theta \) in the second quadrant):
\( \theta = 180^\circ - \tan^{-1}\left(\frac{10.97}{8.96}\right) \approx 129.2^\circ \)
What relationship must exist between the magnitudes of forces \( \vec{W} \) (downward) and \( \vec{N} \) (upward) for their sum to be zero?
Solution
Express the forces in component form:
\( \vec{W} = (0, -|W|) \)
\( \vec{N} = (0, |N|) \)
Their sum is:
\( \vec{W} + \vec{N} = (0, -|W| + |N|) \)
For this to equal zero:
\( -|W| + |N| = 0 \quad \Rightarrow \quad |W| = |N| \)
This demonstrates that for two oppositely directed forces along the same line to be in equilibrium, their magnitudes must be equal.