Frame: A-frame Difficulty 2

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Created this as a copy of Frame: A-frame Difficulty 1.

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

There are 7 other versions that do you not have access to.

Question statement

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

The A-frame shown supports a {description} 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(0, "N")
M_E	quantity	quantity(2000, "N*m")
alpha	number	51.3401917459
beta	number	50.1944289077
magA	quantity	quantity(261.3520096911367907337320550712144354806, "N")
magC	quantity	quantity(0, "N")
x1	quantity	quantity(4, "m")
y1	quantity	quantity(5, "m")
x2	quantity	quantity(5, "m")
y2	quantity	quantity(6, "m")
description	string	2000 N⋅m counter-clockwise mom

			Name	Type	Generated Value

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

			Name	Type	Generated Value

B	vector	vector(4,5)
C	vector	vector(9,-1)
fd	integer	0
me	integer	2000
w	integer	0
units	list	[ "N", "m" ]
ggb_load	list	Nested 3×2 list
Load	list	[ 0, 2000 ]
version	integer	1
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(0, "N")

← Depends on:

units
w

This variable doesn't seem to be used anywhere.

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

Strategy:

Draw a free body diagram of the entire frame and each of its legs. Note that member {if(Fd=0,"AB","BC")} is a two-force body and the direction of the line of action of force {if(Fd=0,"A","C")} is known; that means that FBD {if(Fd=0,'III','II')} has three unknowns. FBD I has four unknowns, and FBD {if(Fd=0,'II','III')} has no knowns.

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

Known:

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

Determine necessary angles:

$\alpha = \tan^{-1}\left(\frac{\var{y1}}{\var{x1}}\right) = \var{siground(alpha,4)}$°

$\beta = \tan^{-1}\left(\frac{\var{y2}}{\var{x2}}\right) = \var{siground(beta,4)}$°

Start with FBD {if(Fd=0,'III','II')} which has three unknowns, and take moments at point {if(Fd=0,'C','A')} to find force {if(Fd=0,"A","C")}.

II: $\Sigma M_A = 0$

$\simplify{{sign(-fd)} D } (\var{y1/2}) + C_x (\var{y1}) + C_y ( \var{x1}) = 0$

$\simplify{{sign(-fd)} D} (\var{y1/2}) + (C \cos \beta) (\var{y1}) + (C \sin \beta) (\var{x1}) = 0$

$ C = \dfrac{\simplify{{sign(fd)} D } (\var{y1/2}) }{ (\var{y1}) \cos \beta + (\var{x1}) \sin \beta} = \dfrac{ \var{ F_D y1 /2 }}{\var{display(y1 * cos(radians(beta)) + x1 * sin(radians(beta)))}} = \var{display(magC)}$

$C_x = C \cos \beta = \var{display(qty(-vecC[0],units[0]))} = \var{display(abs(qty(vecC[0],units[0])))} \var{if(vecC[0]>0,latex('\\rightarrow'),latex('\\leftarrow'))}$

$C_y = C \sin \beta = \var{display(qty(vecC[1],units[0]))} = \var{display(abs(qty(vecC[1],units[0])))} \var{if(vecC[1]<0,latex('\\downarrow'),latex('\\uparrow'))}$

II: $\Sigma M_C = 0$

$\simplify{M_E } + A_x (\var{y2}) + A_y ( \var{x2}) = 0$

$\simplify{M_E} + (A \cos \alpha) (\var{y2}) + (A \sin \alpha) (\var{x2}) = 0$

$ A = \dfrac{\simplify{- M_E }}{ (\var{y2}) \cos \alpha + (\var{x2}) \sin \alpha} = \dfrac{ \var{M_E }}{\var{display(y2 * cos(radians(alpha)) + x2 * sin(radians(alpha)))}} = \var{display(magA)}$

$A_x = A \cos \alpha = \var{display(qty(vecA[0],units[0]))} = \var{display(abs(qty(vecA[0],units[0])))} \var{if(vecA[0]<0,latex('\\leftarrow'),latex('\\rightarrow'))}$

$A_y = A \sin \alpha = \var{display(qty(vecA[1],units[0]))} = \var{display(abs(qty(vecA[1],units[0])))}\var{if(vecA[1]<0,latex('\\downarrow'),latex('\\uparrow'))}$

Apply $\Sigma F_x = 0$ and $\Sigma F_y=0$ to FBD I to find the reactions at {if(fd=0,'C','A')}.

$\begin{align} \Sigma F_x &= 0\\ A_x - C_x &=0\\ C_x &= A_x\\ &= (\var{display(qty(vecA[0],units[0]))}) \\ &= \var{display(abs(qty(vecC[0],units[0])))} \var{if(vecC[0]<0,latex('\\leftarrow'),latex('\\rightarrow'))} \end{align}$

$\begin{align} \Sigma F_y &= 0\\ C_y +A_y & = 0\\ C_y &= - A_y \\&= -(\var{display(qty(vecA[1],units[0]))})\\ &= \var{display(abs(qty(vecC[1],units[0])))} \var{if(vecC[1]>0,latex('\\downarrow'),latex('\\uparrow'))} \end{align}$

$\begin{align} \Sigma F_x &= 0\\ \simplify{ A_x + {sign(fd)} D } - C_x &= 0\\ A_x &= \simplify{C_x - {sign(fd)} D } \\&= \simplify[!collectNumbers]{({-siground(vecC[0],4)}) + ({-vecD[0]})}\\ &= \simplify{({-siground(vecC[0],4)}) + ({-vecD[0]})}\\&= \var{display(abs(qty(vecA[0],units[0])))} \var{if(vecA[0]<0,latex('\\leftarrow'),latex('\\rightarrow'))} \end{align}$

$\begin{align}\Sigma F_y &= 0\\ A_y + C_y &=0\\A_y &= -C_y \\ &= -( \var{display(qty(vecC[1],units[0]))}) \\ &= \var{display(abs(qty(vecA[1],units[0])))} \var{if(vecA[1]<0,latex('\\downarrow'),latex('\\uparrow'))} \end{align}$

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This question is used in the following exams:

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 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

William Haynes 6 years, 4 months ago

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Frame: A-frame Difficulty 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

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

William Haynes 6 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: quantity

Parts

Gap-fill

Generated value: `quantity`