Frame: A-frame Difficulty 4

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Created this as a copy of Frame: A-frame Difficulty 3.

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 {display(abs(F_d))} force $D$ at the midpoint of member AB and a {display(abs(M_E)) } {if(me>0,"counter-","")}clockwise moment $M_E$ at E as shown.

Determine the scalar components of the reactions at pins A and C.

Note: the sign of a scalar component indicates the sense of the force. A positive sense indicates the force points towards the positive end of the corresponding axis.

Give any introductory information the student needs.

			Name	Type	Generated Value

F_w	quantity	quantity(0, "N")
F_d	quantity	quantity(500, "N")
M_E	quantity	quantity(1000, "N*m")
alpha	number	36.8698976458
beta	number	11.309932474
x1	quantity	quantity(4, "m")
y1	quantity	quantity(3, "m")
x2	quantity	quantity(5, "m")
y2	quantity	quantity(1, "m")
description	string	500 N force $D$ at the midpoin
Cy	quantity	quantity(-118.421052631578945, "N")
Cx	quantity	quantity(407.89473684210526, "N")
Bx	quantity	quantity(407.89473684210526, "N")
By	quantity	quantity(-118.421052631578945, "N")
det	number	19
detx	number	7750
dety	number	-2250

			Name	Type	Generated Value

vecW	vector	vector(0,0)
vecA	vector	Vector with 2 components
vecC	vector	Vector with 2 components
vecD	vector	vector(500,0)
sigma_f	vector	vector(0,0)

			Name	Type	Generated Value

B	vector	vector(4,3)
C	vector	vector(9,2)
fd	integer	500
me	integer	1000
w	integer	0
units	list	[ "N", "m" ]
ggb_load	list	Nested 3×2 list
debug	boolean	true

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(0, "N")

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

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

Advice

Strategy:

Draw a free body diagram of the entire frame and each of its legs. Note that members AB or BC are not two-force bodies; therefore the direction of the forces at A and C don't align with the members and should be represented with x and y components. Since all of the free body diagrams have four unknowns, there is no simple way to find one unknown using just one equation. The general approach in this case is to write two equations which contain the same two unknowns, then solve them simultaneously.

{geogebra_applet('p6rz9bqx',[["B",B],["C",C]] + ggb_load )}

Known:

$M_E = \var{display(abs(M_E))}$ and $D = \var{display(abs(F_D))}$

Determine necessary angles:

$\begin{align}\alpha &= \tan^{-1}\left(\frac{\var{y1}}{\var{x1}}\right) = \var{siground(alpha,4)}° & \beta &= \tan^{-1}\left(\frac{\var{y2}}{\var{x2}}\right) = \var{siground(beta,4)}°\end{align}$

Use FBDs II and III to generate two independent equations containing $B_x$ and $B_y$:

$\begin{align} \textrm{II: } \Sigma M_A &= 0 & \textrm{III: } \Sigma M_C &= 0\\B_x (\var{y1}) + B_y (\var{x1}) &= \simplify{{sign(fd)} D ({y1/2})}& - B_x (\var{y2}) + B_y (\var{x2}) &= \simplify{ - {sign(me)} M_E } \\ \var{scalar(y1)} B_x + \var{scalar(x1)}B_y& = \var{fd scalar( y1/2) }\qquad (1) & - \var{scalar(y2)} B_x + \var{Scalar(x2)} B_y &= \var{scalar(-M_E ) }\qquad (2)\end{align}$

Solve equations (1) and (2) simultaneously to find $B_x$ and $B_y$:

$\qquad B_x = \var{display(Bx)} \qquad\qquad\qquad\qquad\qquad B_y = \var{display(By)}$

From FBD III:

$\begin{align} \textrm{III: }\Sigma F_x &= 0 & \textrm{III: }\Sigma F_y &= 0 \\ C_x &= B_x & C_y & = B_y \\ &= \var{display(abs(Cx))} \var{if(vecC[0]<0,latex('\\leftarrow'),latex('\\rightarrow'))}& &= \var{display(abs(Cy))} \var{if(vecC[1]<0,latex('\\downarrow'),latex('\\uparrow'))} \end{align}$

From FBD II:

$\begin{align} \textrm{II: }\Sigma F_x &= 0 & \textrm{II: } \Sigma F_y &= 0\\ A_x &= \simplify{B_x - {sign(fd)} D } & A_y &= -B_y \\&= \simplify[!collectNumbers]{({display(-qty(vecC[0],units[0]))}) + ({-F_D})}& &= - ( \var{display(qty(vecC[1],units[0]))}) \\ &=\var{display(abs(qty(vecA[0],units[0])))} \var{if(vecA[0]<0,latex('\\leftarrow'),latex('\\rightarrow'))} & &= \var{display(abs(qty(vecA[1],units[0])))} \var{if(vecA[1]<0,latex('\\downarrow'),latex('\\uparrow'))}\end{align}$

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This question is used in the following exams:

Exam 2 Practice Fall 2020 by William Haynes in Online Exams.
Chapter 06 Exercises by Michael Proudman in Engineering Statics.
32. Frames and Machines 2 by Michael Proudman in Engineering Statics.
Homework 18: Frames by Sarah Sojka in Statics at Randolph College.
32. Frames and Machines 2 by William Haynes in Engineering Statics.
Chapter 6 Exercises by William Haynes in Engineering Statics.

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William Haynes 2 years, 11 months ago

William Haynes 2 years, 11 months ago

William Haynes 2 years, 11 months ago

William Haynes 2 years, 11 months ago

William Haynes 2 years, 11 months ago

William Haynes 5 years, 7 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 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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Frame: A-frame Difficulty 4

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 2 years, 11 months ago

William Haynes 2 years, 11 months ago

William Haynes 2 years, 11 months ago

William Haynes 2 years, 11 months ago

William Haynes 2 years, 11 months ago

William Haynes 5 years, 7 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 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`