Determine the resultant of the coplanar concurrent force system shown. Compute the magnitude,...

Question:

Determine the resultant of the coplanar concurrent force system shown. Compute the magnitude, sense, and angle of inclination with the {eq}x {/eq} axis. Use the method of components.

Force system:

When several force vectors are present in a system, then it is called a force system.

If the force vectors meet at a point then they are called concurrent.

The vectors which lie in a single plane are called coplanar vectors.

The force vector can be written in component form along the two mutually perpendicular axes.

Answer and Explanation:

The following vectors are given with their respective vectors,

200 lb at 30 degrees in the anticlockwise direction. with x-axis,

100 lb at 45 degrees in clockwise with the positive x-axis,

50 lb along the negative y-axis,

400 lb along the negative x-axis,

300 lb at 60 degrees in the clockwise direction with the negative x-axis.

Now the given vectors can be written in the vector form as,

{eq}200\ cos\ 30^o \hat i+ 200\ sin\ 30^o\hat j \\ 100\ cos\ 45^o\hat i -100\ sin\ 45^o\hat j\\ - 50\hat j\\ - 400\hat i\\ -300\ cos\ 60^o\hat i + 300\ sin\ 60^o\hat j \\ {/eq}

The resultant of all vectors is

{eq}\vec R = 200\ cos\ 30^o \hat i+ 200\ sin\ 30^o\hat j + 100\ cos\ 45^o\hat i -100\ sin\ 45^o\hat j - 50\hat j - 400\hat i -300\ cos\ 60^o\hat i + 300\ sin\ 60^o\hat j \\ \vec R = 173.2\hat i + 100\hat j + 70.7\hat i - 70.7\hat j - 50\hat j - 400\hat i -150\hat i + 259.8\hat j \\ \vec R = -306.1\hat i +239.1\hat j\\ {/eq}

The magnitude of the resultant is, {eq}R = \sqrt{306.1^2+ 239.1^2} = 388.4\ lb {/eq}

Diagram

The angle is, {eq}tan^{-1}\dfrac{239.1}{-306.1} = -37.99^o {/eq} with the negative x axis.

or {eq}142^o {/eq} with the positive x axis in the counterclockwise drection


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Practice Adding & Subtracting Vectors

from High School Physics: Homework Help Resource

Chapter 3 / Lesson 23
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