Equilibrium of a particle: four forces

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Question statement

{applet} {alpha} {beta} {gamma}

Four forces with magnitudes $A$ = {qty(maga,units)}, $B$ = {qty(magb,units)}, and $C$ = {qty(magC,units)}, act on a particle in the directions shown.

Determine the magnitude and direction of force $\mathbf{D}$ required for equilibrium.

Give any introductory information the student needs.

			Name	Type	Generated Value

alpha	integer	70
beta	integer	340
gamma	integer	230
units	string	N
maga	integer	10
magb	integer	95
magc	integer	55
debug	boolean	false
applet	ggbapplet	HTML node
params	dict	Dictionary with 6 entries

			Name	Type	Generated Value

A	vector	Vector with 2 components
B	vector	Vector with 2 components
C	vector	Vector with 2 components
D	vector	Vector with 2 components
magd	quantity	quantity(86.845999443740993, "N")
theta	number	131.3168432414

			Name	Type	Generated Value

			dx	number	-57.3376818752
			dy	number	65.2274317796

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(0..90#5)

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Generated value: `integer`

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A
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$D$ = Magnitude ({siground(magd,4)}) acting at Direction measured Sign from the Reference Direction. ({siground(theta,4)}°)

Advice

Begin by drawing a free body diagram of the particle. Since the direction of force $\mathbf{D}$ is unknown you will have to make an assumption about which way it points.

Set up and solve the equilibrium equations based on your assumed direction for $\mathbf{D}$

$\begin{align}
\Sigma F_x &= 0\\
A_x + B_x + C_x + D_x &= 0\\
\simplify[!collectNumbers]{{enground(A[0])} + {enground(B[0])} + {enground(C[0])} - D_x} &= 0 \\
D_x &= \var{qty(enground(A[0] +B[0] +C[0]),units)}&(1)\\\\
\Sigma F_y &= 0\\
A_y + B_y + C_y + D_y &= 0\\
\simplify[!collectNumbers]{{enground(A[1])} + {enground(B[1])} + {enground(C[1])} + D_y } &= 0\\
D_y &= \var{qty(- enground(A[1] +B[1] +C[1]),units)} & (2)
\end{align}$

If your calculations result in a negative value for $D_x$ or $D_y$, then your assumed direction is opposite of the component's actual direction.

Draw a sketch showing $D_x$, $D_y$, and $D$ pointing in their actual directions and define angle $\theta$.

Resolve the components using right triangle trigonometry to find $D$ and $\theta$.

$D = \var{enground(abs(D))}$ {units},

$\theta = \var{siground(theta,4)}$° measured counterclockwise from the $x$-axis.

Your value of $\theta$ will depend on the reference angle you choose. Choose one that makes $\theta$ less than 90° to avoid sign errors.

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This question is used in the following exams:

Exam 1 Practice Fall 2020 by William Haynes in Online Exams.
Chapter 03 Exercises by Michael Proudman in Engineering Statics.
05. Equilibrium of a Particle by Michael Proudman in Engineering Statics.
Statics - Equilibrium by Robert Birch in Robert's workspace.
Statics - Chapter 3 Exercises by Jay McCormack in studio 4.
exam prova 1 by Laura Lamberti in Laura's workspace.
Chapter 3 Exercises by William Haynes in Engineering Statics.
Homework #2 by Mallory Barkdull in Mallory's workspace.
05. Equilibrium of a Particle by William Haynes in Engineering Statics.

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Equilibrium of a particle: four forces

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Generated value: `integer`