Equilibrium of a rigid body: beam and pulley

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Question statement

{geogebra_applet('nnkbvxb8',[['L',scalar(L)],['ab',scalar(ab)],['r',scalar(r)/12],['alpha',alpha+'°'],['theta',theta+'°']])}

{L} long beam $ABC$ supports a frictionless pulley with {r} radius at point $B$ , located {ab} from the left end. Determine the reactions at $A$ and $C$ when it supports {W} weight $W$ as shown.

Give any introductory information the student needs.

			Name	Type	Generated Value

Cy	quantity	quantity(22.5539741507539453125, "lbs")
magC	quantity	quantity(31.89613612940841661908927259536793266289, "lbs")
Ax	quantity	quantity(135.8424475303351634163638705915833809107, "lbs")
Ay	quantity	quantity(49.6187431316586796875, "lbs")
magA	quantity	quantity(144.620849883401604, "lbs")
dirA	number	20.0655729884
Cx	quantity	quantity(-22.55397415075391341636387059158338091073, "lbs")
vecA	vector	Vector with 2 components
vecC	vector	Vector with 2 components
vecW	vector	vector(0,-125)
vecT	vector	Vector with 2 components
sum	vector	Vector with 2 components
dirC	integer	135
dir_C_from_X	quantity	quantity(134.999999999999972, "deg")

			Name	Type	Generated Value

L	quantity	quantity(16, "ft")
r	quantity	quantity(10, "in")
ab	quantity	quantity(5, "ft")
alpha	integer	45
theta	integer	65
W	quantity	quantity(125, "lbs")
T	quantity	quantity(125, "lbs")
debug	boolean	false

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
ab * W *(1- cos(radians(theta)))/ L

JME function reference

Description

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

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Generated value: `quantity`

quantity(22.5539741507539453125, "lbs")

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Reaction at C Gap-fill
1. Gap angleC Angle quantity 2020
  Add
2. Gap sign Choose one from a list
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3. Gap ref Choose one from a list
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4. Gap magC Engineering Accuracy with units
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Reaction at A Gap-fill
1. Gap angleA Angle quantity 2020
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2. Gap sign Choose one from a list
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3. Gap ref Choose one from a list
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4. Gap magA 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 and direction of the reaction at the roller at $C$.

Force $\mathbf{C}$ has a magnitude of magC

acting at an angle of angleC measured sign from the ref.

C = {magC}, angle={dirC}°

Advice

Known: $L = \var{l} \qquad d = \var{ab} \qquad \theta = \var{theta}° \qquad \alpha = \var{alpha}° \qquad W = T = \var{W}$

1. Draw a free body diagram of the beam. It simplifies the solution to replace the forces of the rope on the pulley with equivalent forces at point $B$. The force at $C$ acts perpendicular to the surface the roller rests on.
  {geogebra_applet('kggckbyp',[['L',scalar(L)],['ab',scalar(ab)],['r',scalar(r)/12],['alpha',alpha+'°'],['theta',theta+'°']])}
2. Take moments about point $A$ to find the reaction at $C$.
  $\begin{align}\Sigma M_A &= 0 \\+ C_y L+ T_y d - W d&= 0 \\ C_y &=\dfrac{ d (W - T \cos \theta)}{L} = \var{display(Cy)}\\C &= \dfrac{C_y}{\cos \alpha} = \var{display(magC)}\end{align}$
3. Apply $F_x=0$ to find $A_x$.
  $\begin{align}F_x&=0\\A_x - T_x - C_x &= 0 \\A_x &= C_x + T_x\\A_x&= C \sin \alpha + T \sin \theta = \var{display(Ax)}\end{align}$
4. Apply $\Sigma F_y = 0$ to find $A_y$.
  $\begin{align}F_y&=0\\A_y + C_y + T_y - W &= 0 \\A_y &= W - C_y - T_y\\A_y&= W - C \cos \alpha - T \cos \theta= \var{display(Ay)}\end{align}$
5. Resolve $A_x$ and $A_y$ to get magnitude and direction of force $\mathbf{A}$.
  $A = \sqrt{(A_x)^2+(A_y)^2} = \var{display(magA)}$
  
  $\phi = \tan^{-1} \left(\dfrac{A_y}{A_x}\right) = \var{precround(dirA,2)}$° from the positive x-axis.
6. Check with $\Sigma \mathbf{F} = 0$.
  $ \mathbf{A} + \mathbf{C} + \mathbf{W} + \mathbf{T} = 0$
  
  $\var{precround(vecA,2)} + \var{precround(vecC,2)} + \var{vecW} + \var{precround(vecT,2)} = \var{precround(sum,2)}$

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This question is used in the following exams:

Chapter 5 Exercises by William Haynes in Engineering Statics.
Chapter 05 Exercises by Michael Proudman in Engineering Statics.
13. Equilibrium of Rigid Bodies by Michael Proudman in Engineering Statics.
Statics by Robert Birch in Robert's workspace.
studio 4 - Chapter 5 Exercises by Jay McCormack in studio 4.
studio 4 - Chapter 5 Exercises by Jay McCormack in Jay's workspace.
Chapter 5 Exercises for studio 4 by Jay McCormack in 23-24 Studio 4 Statics Practice.
13. Equilibrium of Rigid Bodies by William Haynes in Engineering Statics.

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Equilibrium of a rigid body: beam and pulley

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Equilibrium of a rigid body: beam and pulley

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

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William Haynes 2 years ago

William Haynes 4 years, 3 months ago

William Haynes 4 years, 7 months ago

William Haynes 5 years ago

William Haynes 5 years ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 11 months ago

William Haynes 6 years, 6 months ago

William Haynes 6 years, 6 months ago

William Haynes 6 years, 6 months ago

Error in variable testing condition

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Generated value: quantity

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Gap-fill

Generated value: `quantity`