Opposing moments using Verignon's theorem

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Name	Status	Author	Last Modified
Opposing moments using Verignon's theorem	Ready to use	William Haynes	16/02/2024 15:08
Xiaodan's copy of	draft	Xiaodan Leng	10/07/2019 22:09

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Question statement

An L shaped bracket is supported by a frictionless pin at $A$ and a cable between points $B$ and $D$.

{applet}

Determine the tension in the cable required for equilibrium when a {F} force $\mathbf{F}$ is applied to the bracket as shown.

Give any introductory information the student needs.

			Name	Type	Generated Value

A	vector	vector(0,0)
B	vector	vector(8,-4)
D	vector	vector(3,-2)
alpha	integer	40
F	quantity	quantity(65, "lbs")
units	list	[ "ft", "lb" ]
debug	boolean	false

			Name	Type	Generated Value

F_hat	vector	Vector with 3 components
F_y	quantity	quantity(41.781194629625035, "lbs")
d_AB	quantity	quantity(8, "ft")
M_F	quantity	quantity(334.25, "lbs*ft")
beta	number	158.199
rise	number	2
run	number	-5
T_hat	vector	Vector with 3 components
r_ad	vector	vector(3,-2,0)
d_perp_t	quantity	quantity(0.742781352708208, "ft")
d_x	quantity	quantity(8, "ft")
d_y	quantity	quantity(4, "ft")

			Name	Type	Generated Value

T	quantity	quantity(449.9978341961766614727749908989219875373, "lbs")
applet	ggbapplet	HTML node
params	dict	Dictionary with 5 entries

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(0,0)

JME function reference

Description

Position of point A

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

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Generated value: `vector`

vector(0,0)

→ Used by:

d_x
d_y
params
r_ad

This variable doesn't seem to be used anywhere.

Parts

Find Moment Gap-fill
1. Gap M_F Engineering Accuracy with units
  Add
Find Tension Gap-fill
1. Gap T Engineering Accuracy with units
  Add

Gap-fill

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

Determine the magnitude of the moment that the force $\mathbf{F}$ produces about point $A$.

$M_F =$ M_F {M_F}

Advice

Since the object is in equilibrium $\Sigma M_A = 0$ implies that the opposing moments must have equal magnitudes. This is the principle we use to solve the problem. At the start we know the magnitude of $\mathbf{F}$ and the geometry of the system.

1. Begin by drawing a sketch, defining your symbols and determining geometric values.

Let $\alpha$ be the angle force $\mathbf{F}$ makes with the horizontal and $\beta$ be the angle the cable makes with the horizontal.

$\alpha = \var{abs(alpha)}°$, $\beta = \var{siground(if(abs(beta)>90, 180-abs(beta), abs(beta)),4)}°$

Let $d_x$ and $d_y$ represent the horizontal and vertical distances from $A$ to $B$.

$d_x = \var{d_x}$, $d_y = \var{d_y}$

2. Develop expressions for the moments about point $A$ caused by the force $\mathbf{F}$ and the cable. The moment caused by the force can be found completely, but the moment caused by the cable tension can only be expressed as a function of the unknown tension $T$.

$M_F = F_y \cdot d_x = F \sin \alpha \cdot d_x $

$M_T = T_x \cdot d_y \pm T_y \cdot d_x = (T \cos \beta) \, d_y \pm (T \sin \beta) \, d_x = T (\var{scalar(d_y)} \cos \beta \pm \var{scalar(d_x)} \sin \beta) $

When calculating the moment due to the cable $M_T$, if the two component moments act in the same direction, add the terms. If they twist in opposite directions, take their difference. Either way, make the final result positive, because we're comparing magnitudes.

3. After substituting the known values and simplifying, equate the moments to get:

$M_T = M_F$

$ T (\var{format(d_perp_t)}) = \var{format(M_F)} $

$T = \dfrac{\var{format(M_F)}}{\var{format(d_perp_t)}}= \var{format(T)} $

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This question is used in the following exams:

Exam 1 Practice Fall 2020 by William Haynes in Online Exams.
Chapter 04 Exercises by Michael Proudman in Engineering Statics.
08. Verignon's Theorem by Michael Proudman in Engineering Statics.
Studio 4 - Chapter 4 Exercises by Jay McCormack in studio 4.
Chapter 4 Exercises for studio 4 by Jay McCormack in 23-24 Studio 4 Statics Practice.
Chapter 4 Exercises by William Haynes in Engineering Statics.
08. Verignon's Theorem by William Haynes in Engineering Statics.

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Opposing moments using Verignon's theorem

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 1 year, 1 month ago

William Haynes 4 years, 11 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

Error in variable testing condition

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Generated value: vector

Parts

Gap-fill

Generated value: `vector`