# assignment 2 point equilibrium coplanar force systems

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Assignment 2: Point Equilibrium
Due: 11:59pm on Thursday, September 24, 2009

Coplanar Force Systems
Description: Learning Goal: To understand how to establish a particle's free-body diagram in a coplanar force system and to apply the equations of equilibrium to solve for
unknowns. In this tutorial, students identify terms to describe a force system, draw a free-body diagram to relate the forces in a force system, and use the equilibrium conditions to
find unknown quantities. (vocab applet) (vector applet)

Learning Goal: To understand how to establish a particle's free-body diagram in a coplanar force system and to apply the equations of equilibrium to solve for unknowns.
In a coplanar force system, a particle is subjected to forces that lie in a single plane. If that plane is the x–y plane, then the conditions of equilibrium are met when

For this vector equation to be satisfied, the force vector's x and y components must be equal to zero:

Part A
Three identical objects in three different systems each have three forces acting on them. What word best completes the following sentences that describe each force system?

Hint A.1           How to approach the problem
Learning the terminology used to describe force systems helps explain the differences encountered in these systems. The term "concurrent" is used when the lines of action of the
forces of interest intersect at one point. Conversely, the term "nonconcurrent" is used when the lines of action of the forces of interest do not intersect at one point. The term
"collinear" is used when the forces of interest have the same line of action. The term "parallel" is used when all of the lines of action of the forces of interest never intersect. The
term "perpendicular" is used when the lines of action of two or more forces of interest intersect at 90 angles.

Match the words in the left column to the appropriate blanks in the sentences on the right. Make certain each sentence is complete before submitting your answer.

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By identifying the properties of a force system, basic assumptions can be made that simplify the analysis of the system. Coplanar force systems require force balances in only
two dimensions, eliminating the need for a third balance equation. Collinear force systems are one dimensional and the forces can be combined as scalars. Concurrent force
systems intersect at a single point and do not require a moment analysis. In a parallel force system, the forces can be combined as scalars but any separation will necessitate an
additional moment analysis of the system.

Part B
A tugboat tows a ship at a constant velocity. The tow harness consists of a single tow cable attached to the tugboat at point A that splits at point B and attaches to the ship at points C
and D. The two rope segments BC and BD angle away from the center of the ship at angles of = 33.0 and = 20.0 , respectively. The tugboat pulls with a force of 2050 .
What are the tensions        and        in the rope segments BC and BD?
Hint B.1          How to approach the problem
Whereas the tow harness system is three dimensional, all the ropes, and, therefore, the tensions in those ropes are coplanar. They exist entirely in a plane and can be treated as
two-dimensional forces. In this case, you can treat the rope segment AB as lying along the y axis with pointing in the negative y direction. Any forces that are perpendicular to
the rope segment AB will be in the x direction. Determine all of the forces in each direction in the plane and apply the conditions for equilibrium. This will require a system of
equations that can be used to find the two unknown quantities: the tensions in the rope segments BC and BD.

Hint B.2       Draw the free-body diagram
Complete the free-body diagram of B by drawing the forces that act on it at the origin of the axes, as if viewing the system from above the tow harness.
Draw your vectors starting at point B. The orientation of the vectors will be graded. The exact length of the vectors will not be graded. Assume that              points in the
negative y direction.

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Hint B.3       Find an expression for

Apply the conditions for equilibrium to          and rearrange the equation to solve for       in terms of the tension      and the angles   and    .

Hint B.3.1         Find an expression for

What is the sum of all the forces in the x direction in terms of the tension     in rope segment BC, the tension          in rope segment BD, and the angles    and   ?

=

Hint B.3.2         The conditions for equilibrium
The conditions for equilibrium require that the sum of the forces in each direction be zero:

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Hint B.4          Find an expression for

Apply the conditions for equilibrium to          and use your equation for       to rearrange the equation to solve for       in terms of the tension      in rope segment BA and
the angles     and   .

Hint B.4.1          Find an expression for

What is the sum of all the forces in the y direction in terms of the tension         in rope segment BC, the tension          in rope segment BD, the tension     in rope segment BA,
and the angles      and    ?

Express your answer in terms of                ,     ,       , , and   .

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Hint B.4.2          The conditions for equilibrium
The conditions for equilibrium require that the sum of the forces in each direction be zero:

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Hint B.5         Determine

What is the tension             in the rope segment BA?

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Express your answers numerically in pounds to three significant figures separated by a comma.

,           =

The analysis of this problem has been simplified by realizing that the forces are coplanar. Whereas the tow harness is three dimensional, all the ropes and tensions are in a
single plane. This reduces the problem to finding the sum of the forces in only two dimensions. The problem is further simplified by identifying the forces as concurrent. All
the forces intersect at a single point, requiring the forces to be summed only at that point. After the system is simplified, the free-body diagram was drawn to determine all the
forces that interact at point B. The x and y components of the forces were then summed:

Because         is given in the problem, there are two equations and two unknowns,            and       . By applying the conditions of equilibrium, the system of equations can be
solved to find these unknowns.

Part C
Rope BCA passes through a pulley at point C and supports a crate at point A. Rope segment CD supports the pulley and is attached to an eye anchor embedded in a wall. Rope
segment BC creates an angle of = 55.0 with the floor and rope segment CD creates an angle with the horizontal. If both ropes BCA and CD can support a maximum tensile
force        = 100       , what is the maximum weight            of the crate that the system can support? What is the angle    required for equilibrium?
Hint C.1          How to approach the problem
The system of forces is coplanar; thus, there are two equations of force equilibrium that can be solved to find the two unknowns: the maximum weight           and the angle .

Hint C.2       Draw the free-body diagram of C
Complete the free-body diagram of C by drawing the forces that act on the pulley.
Draw your vectors starting at the center of the pulley. The orientation of the vectors will be graded. The exact length of the vectors will not be graded.

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Hint C.3       Relate the tensions and the weight
Which of the following is true about the tension       in rope segment CA, the tension         in rope segment CB, the tension     in rope segment CD, and the weight of the crate
?

The two rope segments CA and CB are parts of the same rope. Because the tension is the same throughout the rope, the tensions in the two segments are the same (
). The only two forces that act at point A are the weight and the tension      in rope segment CA; therefore, the magnitudes of these two forces must be
equal if the system is in equilibrium.

Hint C.4         Find an expression for

Apply the conditions for equilibrium and express the maximum weight of the crate              in terms of the known values—the maximum rope tension          and the angle   .

Hint C.4.1         Find an expression for

What is the sum of all the forces in the x direction in terms of the tension      in rope segment CB, the tension       in rope segment CD, and the angles    and   ?

=

Hint C.4.2         Find an expression for

Keeping in mind that                , what is the sum of all the forces in the y direction in terms of the tension   in rope segment CB, the tension     in rope segment CD,
and the angles    and   ?
Hint C.4.2.1        The tension in rope BCA
Because the rope segments CB and CA are part of the same rope, the tension throughout the rope is the same:

=

Hint C.4.3          The conditions for equilibrium
The conditions for equilibrium require that the sum of the forces in each direction be zero:

Use the trigonometric identity                           and the equilibrium conditions to express             in terms of       and   ..

=

Hint C.5         Find an expression for

Use the condition for equilibrium in the x direction to find an expression for the angle          in terms of the maximum weight of the crate            , the maximum rope tension       ,
and the angle    .

=

Express your answers numerically in pounds and degrees to three significant figures separated by a comma.

,       =

The analysis of this problem has been simplified by realizing that the forces are coplanar. This reduces the problem to finding the sum of the forces in only two dimensions.
The problem is further simplified by identifying the forces as concurrent. All the forces intersect at a single point, requiring the forces to be summed only at that point. After
the system is simplified, the free-body diagram was drawn to determine all the forces that interact at point C. The x and y components of the forces were then summed:

Because CB and CA are part of the rope BCA,                         . The tension in this rope must also be equal to the weight of the crate,         , so                      . The
maximum tension for the ropes and the angle          are given in the problem. There are then two equations and two unknowns,                   and . By applying the conditions of
equilibrium, the system of equations can be solved to find these unknowns.

Problem 3.1 •
Description: (a) Draw a free-body diagram of the ring at B. (b) Determine the force in cord BA for equilibrium of the m crate. Cord BC remains horizontal due to the roller at C,
and BA has a length of 1.5 m.Set y = 0.75 m. (c) Determine the force in cord BC.
Part A
Draw a free-body diagram of the ring at B.
Draw your vectors starting from the black dot. The orientation of your vectors will be graded. The exact length

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Part B
Determine the force in cord BA for equilibrium of the 220     crate. Cord BC remains horizontal due to the roller at C, and BA has a length of 1.5   .Set              .

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Part C
Determine the force in cord BC.

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Problem 3.5 •
Description: The members of a truss are connected to the gusset plate. (a) Draw a free-body diagram of the gusset plate at O. (b) If the forces are concurrent at point O, determine
the magnitude of F_evec for equilibrium. Take theta=## degree(s). (c) Determine ...
The members of a truss are connected to the gusset plate.

Part A
Draw a free-body diagram of the gusset plate at O.

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Part B
If the forces are concurrent at point O, determine the magnitude of   for equilibrium. Take   = 32 .

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Part C
Determine the magnitude of       .

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Problem 3.12
Description: (a) Draw a free-body diagram of the ring at A. (b) If block B weighs W_B and block C weighs W_C, determine the required weight of block D for equilibrium. (c)
Determine the angle theta.

Part A
Draw a free-body diagram of the ring at A.
Draw your vectors starting from the black dot. The orientation of your vectors will be graded. The exact length

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Part B
If block B weighs 210      and block C weighs 120    , determine the required weight of block D for equilibrium.

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Part C
Determine the angle .

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Problem 3.45 •
Description: (a) Draw a free-body diagram of the ring at A. (b) Determine the tension in cable AB in order to support the m crate in the equilibrium position shown. (c) Determine
the tension in cable AC. (d) Determine the tension in cable AD.

Part A
Draw a free-body diagram of the ring at A.
Draw your vectors starting from the black dot. The orientation of your vectors will be graded. The exact length

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Part B
Determine the tension in cable AB in order to support the 100        crate in the equilibrium position shown.

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Part C
Determine the tension in cable AC.

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Part D
Determine the tension in cable AD.

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Problem 3.57 •
Description: The ends of the three cables are attached to a ring at A and to the edge of the uniform plate. (a) Draw a free-body diagram of the ring at A. (b) Determine the largest
mass the plate can have if each cable can support a maximum tension of T_max.
The ends of the three cables are attached to a ring at A and to the edge of the uniform plate.
Part A
Draw a free-body diagram of the ring at A.

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Part B
Determine the largest mass the plate can have if each cable can support a maximum tension of 11   .

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Score Summary:
Your score on this assignment is 0%.
You received 0 out of a possible total of 6 points.

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Description: assignment 2 point equilibrium coplanar force systems
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