Moment Balance

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Question statement

{applet}

Three forces, $A = \var{A F_units}$, $C = \var{C F_units}$ and $D = \var{F F_units}$, act on the rigid, bent bar which can pivot freely on a pin at $B$.

The segment lengths are: $AB = \var{ab D_units}$, $BC = \var{bc D_units}$, and $CD= \var{cd D_units}$.

Give any introductory information the student needs.

			Name	Type	Generated Value

r_a	vector	vector(-2.5,0,0)
r_c	vector	vector(0,1,0)
r_d	vector	vector(1,1,0)
f_a	vector	vector(-21.6506350946,-12.5,0)
f_c	vector	vector(-50,86.6025403784,0)
M_A	vector	vector(0,0,31.25)
M_C	vector	vector(0,0,50)
M_D	vector	vector(0,0,-81.25)
M	number	-81.25

			Name	Type	Generated Value

F_min	vector	vector(40.625,-40.625,0)
f_perp	number	57.4524259714
dperp	number	0.65
alpha1	number	27.3625513094
alpha2	number	152.6374486906
phi	number	45
theta1	number	-107.6374486906
theta2	number	17.6374486906

			Name	Type	Generated Value

applet	ggbapplet	HTML node
params	dict	Dictionary with 8 entries
ab	number	2.5
bc	number	1
cd	number	1
theta_A	integer	210
theta_C	integer	120
f_units	quantity	quantity(1, "lbs")
D_units	quantity	quantity(1, "ft")
A	integer	25
C	integer	100
F	integer	125

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(-ab,0,0)

JME function reference

Description

positon vector

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `vector`

vector(-2.5,0,0)

← Depends on:

→ Used by:

Parts

Gap-fill
1. Gap $\theta$ Angle quantity 2020
  1. Alt $\theta'$
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Find an angle $\theta$ that will make the sum of the moments about point $B=0$.

$\theta = $ $\theta$ (Measure $\theta$ CW from the positive $x$ axis.)

Advice

From the geometry of triangle $BCD$:

$\phi = \tan^{-1}\left( \dfrac{BC}{CD}\right) =\var{precround(phi,1)}°$

$d = \sqrt{BC^2 +CD^2} = \var{siground(abs(r_d),4) D_units}$

From a moment balance around point $B$, the clockwise moment equals the counterclockwise moments:

$\begin{align}
M_D &= M_A + M_C \\
&= (\var{AB})\ A_y + (\var{BC})\ C_x\\
&= \simplify[!collectNumbers]{({siground(M_A[2],4)} +{siground(M_C[2],4)})}\\
&= \var{siground(-M D_units F_units,4)} \text{ clockwise}
\end{align}$

With the required moment $M_D$ and the magnitude of force $D$ known, the perpendicular distance and the angle between segment $BD$ and force $D$ are:

$d_\perp = M_D / D = \dfrac{ \var{siground(-M D_units F_units,4)}}{\var{F F_units}} = \var{siground(dperp,4) D_units}$

$\alpha =\sin^{-1}\left( \dfrac{d_\perp}{d}\right)=\var{precround(alpha1,1)}°$

Since $\sin \alpha = \sin (180°-\alpha)$ another possible angle is:

$\alpha' = (180° - \var{precround(alpha1,1)}) = \var{precround(alpha2,1)}°$

Finally the angle from the positive $x$ axis to force $D$ can be determined. There are two possible solutions:

$\theta =180° - \alpha - \phi = \var{precround(-theta1,1)}° $ or
$\theta = 180 - \alpha' - \phi = \var{precround(-theta2,1)}$

Note: You can use the red slider on the diagram to switch between the two possible solutions.

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Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Angle quantity 2020
Angle quantity 2020		Hasn't run yet

This question is used in the following exam:

Chapter 4 Exercises by William Haynes in Engineering Statics.

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Checkpoint description

William Haynes 1 year, 1 month ago

William Haynes 1 year, 1 month ago

William Haynes 1 year, 1 month ago

William Haynes 1 year, 1 month ago

Error in variable testing condition

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Generated value: `vector`

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Moment Balance

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 1 year, 1 month ago

William Haynes 1 year, 1 month ago

William Haynes 1 year, 1 month ago

William Haynes 1 year, 1 month ago

Error in variable testing condition

Error

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Generated value: vector

Parts

Gap-fill

Generated value: `vector`