Determine the magnitude and direction of the resultant of...

Determine the magnitude and direction of the resultant of the three coplanar forces given below, when they act at a point. Force $A, 10 \mathrm{~N}$ acting at $45^{\circ}$ from the positive horizontal axis. Force $B, 87 \mathrm{~N}$ acting at $120^{\circ}$ from the positive horizontal axis. Force $C, 15 \mathrm{~N}$ acting at $210^{\circ}$ from the positive horizontal axis. The space diagram is shown in Fig. $22.10$. The forces may be written as complex numbers. The resultant force $$ \begin{aligned} =& f_{A}+f_{B}+f_{C} \\ =& 10 \angle 45^{\circ}+8 \angle 120^{\circ}+15 \angle 210^{\circ} \\ =& 10\left(\cos 45^{\circ}+j \sin 45^{\circ}\right)+8\left(\cos 120^{\circ}\right.\\ &\left.+j \sin 120^{\circ}\right)+15\left(\cos 210^{\circ}+j \sin 210^{\circ}\right) \\ =&(7.071+j 7.071)+(-4.00+j 6.928) \\ & \quad+(-12.99-j 7.50) \\ =&-9.919+j 6.499 \end{aligned} $$ Magnitude of resultant force $$ =\sqrt{\left[(-9.919)^{2}+(6.499)^{2}\right]}=11.86 \mathrm{~N} $$ Direction of resultant force $$ =\tan ^{-1}\left(\frac{6.499}{-9.919}\right)=146.77^{\circ} $$ (since $-9.919+j 6.499$ lies in the second quadrant). Thus force $A, f_{A}=10 \angle 45^{\circ}$, force $B, f_{B}=8 \angle 120^{\circ}$ and force $C, f_{C}=15 \angle 210^{\circ}$

Determine the magnitude and direction of the resultant of the three coplanar forces given below, when they act at a point.
Force $A, 10 \mathrm{~N}$ acting at $45^{\circ}$ from the positive horizontal axis.
Force $B, 87 \mathrm{~N}$ acting at $120^{\circ}$ from the positive horizontal axis.
Force $C, 15 \mathrm{~N}$ acting at $210^{\circ}$ from the positive horizontal axis.
The space diagram is shown in Fig. $22.10$. The forces may be written as complex numbers.
The resultant force
$$
\begin{aligned}
=& f_{A}+f_{B}+f_{C} \\
=& 10 \angle 45^{\circ}+8 \angle 120^{\circ}+15 \angle 210^{\circ} \\
=& 10\left(\cos 45^{\circ}+j \sin 45^{\circ}\right)+8\left(\cos 120^{\circ}\right.\\
&\left.+j \sin 120^{\circ}\right)+15\left(\cos 210^{\circ}+j \sin 210^{\circ}\right) \\
=&(7.071+j 7.071)+(-4.00+j 6.928) \\
& \quad+(-12.99-j 7.50) \\
=&-9.919+j 6.499
\end{aligned}
$$
Magnitude of resultant force
$$
=\sqrt{\left[(-9.919)^{2}+(6.499)^{2}\right]}=11.86 \mathrm{~N}
$$
Direction of resultant force
$$
=\tan ^{-1}\left(\frac{6.499}{-9.919}\right)=146.77^{\circ}
$$
(since $-9.919+j 6.499$ lies in the second quadrant).
Thus force $A, f_{A}=10 \angle 45^{\circ}$, force $B, f_{B}=8 \angle 120^{\circ}$ and force $C, f_{C}=15 \angle 210^{\circ}$

Key Concepts

Resultant Direction Calculation

The direction of the resultant vector is found using the inverse tangent function, which computes the angle from the ratio of the vertical to the horizontal component. Proper determination of the angle requires analyzing the signs of the components to ensure the correct quadrant is identified.

Conversion between Polar and Rectangular Forms

Vectors can be expressed in polar form, involving a magnitude and an angle, or in rectangular form as horizontal and vertical components. Converting between these forms is crucial for simplifying calculations, especially when adding multiple forces or finding a resultant vector.

Resultant Magnitude Calculation

The magnitude of the resultant vector is determined by computing the square root of the sum of the squares of its horizontal and vertical components. This uses the Pythagorean theorem and provides a measure of the overall effect of combined forces.

Vector Addition

Vector addition is the process of combining two or more vectors into a single resultant vector. This operation takes into account both the magnitude and direction of each vector, which is essential in determining the overall effect when forces act simultaneously at a point.

Representation of Forces as Vectors

Forces are often represented as vectors because they have both magnitude and direction. This representation allows for the analysis and combination of forces using vector operations, facilitating the study of equilibrium and dynamics in physical systems.

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