Frame: A-frame Difficulty 1

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Created this as a copy of Frame: A-frame.

Name	Status	Author	Last Modified
Frame: A-frame Difficulty 1	Ready to use	William Haynes	28/02/2023 23:29
Frame: A-frame Difficulty 2	Ready to use	William Haynes	21/03/2021 14:25
Frame: A-frame Difficulty 3	Ready to use	William Haynes	30/03/2022 19:14
Frame: A-frame Difficulty 4	Ready to use	William Haynes	31/03/2022 01:05
Keith's copy of Frame: A-frame Difficulty 1	draft	Keith Tarnowski	03/06/2019 11:52
Katrin's copy of Frame: A-frame Difficulty 1	draft	Katrin Thomson	03/06/2019 11:53

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Question statement

{geogebra_applet('qe7xhnse',[["B",B],["C",C],['units','"'+units[1]+'"']]+ggb_load )}

The A-frame shown supports a {F_w} load as shown. Determine the scalar components of the reactions at pins A and C.

Give any introductory information the student needs.

			Name	Type	Generated Value

F_w	quantity	quantity(125, "lbs")
alpha	number	36.8698976458
beta	number	14.0362434679
magA	quantity	quantity(156.2499999999999194705932496551176827574, "lbs")
magC	quantity	quantity(128.8470508005518117075881159984300761297, "lbs")
x1	quantity	quantity(4, "ft")
y1	quantity	quantity(3, "ft")
x2	quantity	quantity(4, "ft")
y2	quantity	quantity(1, "ft")

			Name	Type	Generated Value

vecW	vector	vector(0,-125)
vecA	vector	vector(125.0,93.75)
vecC	vector	vector(-125.0,31.25)
sigma_f	vector	vector(0,0)

			Name	Type	Generated Value

B	vector	vector(4,3)
C	vector	vector(8,2)
fd	integer	0
me	integer	0
w	integer	125
units	list	[ "lb", "ft" ]
ggb_load	list	Nested 3×2 list
debug	boolean	false

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
qty(w,units[0])

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `quantity`

quantity(125, "lbs")

← Depends on:

units
w

→ Used by:

magA
magC

Statement
Advice

Parts

Gap-fill

Gap-fill

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

$A_x =$ $A_x$ $A_y =$ $A_y$ $C_x =$ $C_x$ $C_y =$ $C_y$

Advice

{geogebra_applet('uyjv6v8r',[["B",B],["C",C]] + ggb_load )}

Strategy:

Note that members AB and BC are two-force bodies and both act on pin B in known directions. Draw a free body diagram of the two members and of the forces acting on pin at B, and solve it for the magnitudes of A and C using the equilibrium equation method or the force-triangle method

Known:

$W$ = {F_w}

Determine necessary angles:

$\alpha = \tan^{-1}\left(\frac{\var{y1}}{\var{x1}}\right) = \var{siground(alpha,4)}$° This is the angle between force $\textbf{A}$ and the horizontal.

$\beta = \tan^{-1}\left(\frac{\var{y2}}{\var{x2}}\right) = \var{siground(beta,4)}$° This is the angle between force $\textbf{C}$ and the horizontal.

Using the law of sines to solve for forces A and C:

$\dfrac{W}{\sin(\alpha+\beta)} = \dfrac{A}{\sin(90°-\beta)} = \dfrac{C}{\sin(90°-\alpha)}$

$A = W \dfrac{\cos(\beta)}{\sin(\alpha+\beta)}$ = {display(magA)} $\nearrow$

$C = W \dfrac{\cos(\alpha)}{\sin(\alpha+\beta)}$ = {display(magC)} $\nwarrow$

Find the scalar components of the forces at pins A and C.

Note: Scalar components have a sign which indicates direction: positive values indicate $\uparrow$ or $\rightarrow$, negative indicate $\downarrow$ or $\leftarrow$.

$\begin{align} A_x = A \cos(\alpha) &= \var{qty(vecA[0],units[0])}& &C_x = C \cos(\beta) = \var{qty(vecC[0],units[0])} \\ A_y = A \sin(\alpha) &= \var{qty(vecA[1],units[0])} && C_y = C \sin(\beta) = \var{qty(vecC[1],units[0])} \end{align}$

Check

You can use the equations of equilibrium to verify that these answers put point $B$ in equilibrium.

$\begin{align}A_x &\stackrel{?}{=}C_x &A_y + C_y &\stackrel{?}{=}W\\ \var{vecA[0]} &= \var{-vecC[0]} \quad \checkmark & \var{vecA[1]} + \var{vecC[1]} &= \var{W} \quad \checkmark\\ \end{align}$

Give a worked solution to the whole question.

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This question is used in the following exams:

Test Statics Exam by Keith Tarnowski in Keith's workspace.
Chapter 06 Exercises by Michael Proudman in Engineering Statics.
31. Frames and Machines 1 by Michael Proudman in Engineering Statics.
Homework 18: Frames by Sarah Sojka in Statics at Randolph College.
31. Frames and Machines 1 by William Haynes in Engineering Statics.
Chapter 6 Exercises by William Haynes in Engineering Statics.

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William Haynes 4 years, 4 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

Error in variable testing condition

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Generated value: `quantity`

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Gap-fill

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Frame: A-frame Difficulty 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 4 years, 4 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: quantity

Parts

Gap-fill

Generated value: `quantity`