Position Vectors Description: ± Includes Math Remediation. In this tutorial, students will find the position vectors for two points that have a common origin, find the position vector between the two points, and determine the force acting along a given direction. Learning Goal: To find a position vector between two arbitrary points. As shown, two cables connect three points. is below by a distance and connected to by a cable long. Cable forms an angle with the positive y axis. is above and the distances and are and , respectively. Part A - Position vector from A to B Using the dimensions in the figure, find the position vector from to in component form. Express your answers, separated by commas, to three significant figures. Hints (3) Hint 1. Position vectors Position vectors give information about the distance between two points and the direction of their relative positions. From previous tutorials, we know that Cartesian coordinates give both magnitude and direction of vectors; therefore, position vectors can be expressed as Cartesian vectors with three-dimensional components. To best illustrate the procedure for finding position vectors, picture two points lying along a number line; the position vector between these two points is their difference. The position vector’s direction is such that it points from the starting point toward the ending point. The position vector’s sign is then given by the direction in which it points; thus if the value increases from the start to the end the sign is positive. If the value decreases from the start to the end the sign is negative. So if two points lying in the x–y plane have coordinates and , the position vector from to is . Naturally, the position vector from to is the negative of the position vector from to . Care should be taken to determine the correct sign for each component. C A = 1.90 ft C z A 9.3 6 f t A C = 27. 0 B 8.5 0 f t C B x B y 7.7 0 f t 4.7 0 f t A B (,) x 1 y 1 (,) x 2 y 2 1 2 {-,-} x 2 x 1 y 2 y 1 2 1 1 2 MasteringEngineering: Lecture 3 https://session.masteringengineering.com/myct/itemView?view=solution&... 1 of 6 1/27/2015 1:21 PM
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Position VectorsDescription: ± Includes Math Remediation. In this tutorial, students will find the position vectors for two points thathave a common origin, find the position vector between the two points, and determine the force acting along a givendirection.
Learning Goal:
To find a position vector between two arbitrary points.
As shown, two cables connect three points. is below by a distance and connected to by a cable long. Cable forms an angle with the positive y axis. is above and the distances
and are and , respectively.
Part A - Position vector from A to B
Using the dimensions in the figure, find the position vector from to in component form.
Express your answers, separated by commas, to three significant figures.
Hints (3)
Hint 1. Position vectors
Position vectors give information about the distance between two points and the direction of their relativepositions. From previous tutorials, we know that Cartesian coordinates give both magnitude and direction ofvectors; therefore, position vectors can be expressed as Cartesian vectors with three-dimensionalcomponents. To best illustrate the procedure for finding position vectors, picture two points lying along anumber line; the position vector between these two points is their difference. The position vector’s directionis such that it points from the starting point toward the ending point. The position vector’s sign is then givenby the direction in which it points; thus if the value increases from the start to the end the sign is positive. Ifthe value decreases from the start to the end the sign is negative.
So if two points lying in the x–y plane have coordinates and , the position vector from to is . Naturally, the position vector from to is the negative of the position vector
from to . Care should be taken to determine the correct sign for each component.
For the position vector found in Part A, find the unit direction vector acting in the same direction. Express your answerin component form.
Express your answers, separated by commas, to three significant figures.
Hints (2)
Hint 1. Unit direction vector
A unit direction vector is a unit vector with the same direction and sign as the position vector. Like all unitvectors, its magnitude is one. The unit direction vector is found by dividing the position vector componentsby the magnitude of the position vector:
.
Hint 2. Calculate the magnitude of the position vector
The magnitude of a vector with known components is found through the application of the Pythagoreantheorem. Find the magnitude of the position vector from to .
Express your answer to three significant figures.
Hints (1)
Hint 1. Equation for the magnitude of a vector
The formula for the magnitude of a vector is
.
ANSWER:
ANSWER:
= ( + ) i+ ( − ) j+ ( − l) krAB A x B x A y B y A z
= (− − ) i+ ( − ) j+ (l− − ) krAB B x A x B y A y Cz A z
The length of cable is , and the cable forms the angle with the y axis. Given this informationand the dimensions provided in the figure, find the position vector from to . Express the position vector incomponent form.
Express your answers, separated by commas, to three significant figures.
Hints (2)
Hint 1. Finding the x component of the position vector AC
The x component of the position vector is not explicitly given. It must be calculated from the length of cable and other information given.
From this diagram, you can see that the side of the triangle from point to the y axis forms a right anglewith the y axis. Using right-triangle trigonometry and the Pythagorean theorem the x and y components canbe determined.
Hint 2. Calculate the x component of the position vector
Using the known values for the length of the cable, , the angle cable makes with the positive yaxis, , and the z component, determine the x component.
Express your answer to three significant figures and include the appropriate units.
Find the position vector from to . Express your answer in component form.
Express your answers, separated by commas, to three significant figures.
Hints (2)
Hint 1. Determining the position vector between two points
The position between two arbitrary points can be found using several methods.
If the two points are already defined (position vectors are known) with respect to a commonthird point, then the position vector between the two arbitrary points can be found from theknown position vectors using vector addition. Recall that vectors add tip-to-tail. To add vectorstip-to-tip or tail-to-tail requires a sign change.
1.
A second, simpler method is to write the Cartesian coordinates of the points with point atthe origin. The position vector is the difference between points and .
2.
Lastly, a third method is to determine the displacement for each coordinate direction betweenthe two points, then determine the correct sign of the component vector. For example, if thefirst point is and the second point is , then the displacement in the xdirection is positive , etc.
3.
For the first method, recognize that point is the common point for which and are already defined.The position vector between the and then can be written in terms of the previous position vectorsfound in Part A and Part C: .
Hint 2. Calculate the x component of the position vector from B to C
Using the vector components for position vectors and found in Parts A and B, determine the xcomponent of position vector . Recall the equation ; this equation can be evaluatedby summing the Cartesian components for each direction.
Express your answer to three significant figures.
ANSWER:
ANSWER:
Part E - Tension in the cable
A cable is attached between and . The cable is attached in such a way that it has a tension of along itslength. The tensile force will attempt to pull toward . The force exerted on has a z component with a magnitudeof . What is the magnitude of the tension in the cable, ?
Express your answer to three significant figures and include the appropriate units.
The geometry of the cable will determine the components of force for a given magnitude. Notice that theangle between the position vector and its components is the same for the force and its components. Byfinding the angle between the position vector and one of its components, the tensile force can be foundusing the known force component.
Hint 2. Find the cosine of the angle between the cable and the positive z axis
The angle, , is the same between the z components of both the position vector and the tensile force. Thefirst step in solving for the tension is to solve for the cosine of the angle. What is the cosine of the anglebetween the z axis and the cable?
Express your answer to three significant figures.
Hints (1)
Hint 1. Direction of the z component
Recall that is defined between the cable and the positive z axis. However, because the zcomponent of the tension acts in the negative direction, it has a negative value when calculating
The tower is held in place by three cables. If the force ofeach cable acting on the tower is shown, determine themagnitude and coordinate direction angles of theresultant force. Take , .y = 15 mx = 20 m
a, b, g
SOLUTION
Ans.
Ans.
Ans.
Ans.g = cos-1a -1466.711501.66
b = 168°
b = cos-1a -16.821501.66
b = 90.6°
a = cos-1a 321.661501.66
b = 77.6°
= 1501.66 N = 1.50 kN
FR = 2(321.66)2 + (-16.82)2 + (-1466.71)2
= {321.66i - 16.82j - 1466.71k} N
FR = FDA + FDB + FDC
FDC = 600a1634
i -1834
j -2434
kb N
FDB = 800a -625.06
i +4
25.06 j -
2425.06
kb N
FDA = 400a 2034.66
i +15
34.66 j -
2434.66
kb N
x
z
yxy6 m
4 m
18 m
C
A
D
400 N 800 N600 N
24 m
O16 m
B
2–111.
The window is held open by chain AB. Determine thelength of the chain, and express the 50-lb force acting at Aalong the chain as a Cartesian vector and determine itscoordinate direction angles.
SOLUTION
Unit Vector: The coordinates of point A are
Then
Ans.
Force Vector:
Ans.
Coordinate Direction Angles: From the unit vector obtained above, we have
Ans.
Ans.
Ans.cos g = 0.8748 g = 29.0°
cos b = -0.2987 b = 107°
cos a = -0.3814 a = 112°
uAB
= 5-19.1i - 14.9j + 43.7k6 lb
F = FuAB = 505-0.3814i - 0.2987j + 0.8748k6 lb
= -0.3814i - 0.2987j + 0.8748k
uAB =rAB
rAB=
-3.830i - 3.00j + 8.786k10.043
rAB = 21-3.83022 + 1-3.0022 + 8.7862 = 10.043 ft = 10.0 ft
F = (-300 sin 30° sin 30°i + 300 cos 30°j + 300 sin 30° cos 30°k)
Coplanar Force SystemsDescription: In this tutorial, students will use the spring equation to solve for unknown forces in two dimensions andfind the new length of the springs and will also find an unknown spring constant from an applied force and the system’sgeometry.
Learning Goal:
To use the equations of equilibrium to find unknown forces in two dimensions; understand the relationship between aspring’s unloaded length, its displacement, and its loaded length; and use the spring equation to solve problems involvingmultiple springs.
As shown, a frictionless pulley hangs from a system of springs and a cable. The pulley is equidistant between the twosupports attaching the springs to the ceiling. The distance between the supports is . The cable cannot stretchand its length between the two springs is 1.8 m.
Part A - Finding an unknown weight
As shown, a mass is hung from the pulley. This mass causes a tensile force of in the cable and the pulley tohang from the ceiling. Assume that the pulley has no mass. What is the weight of the mass?
Express your answer to three significantfigures and include the appropriate units.
Hints (4)
Hint 1. Coplanar force system
A coplanar force system is a collection of forces that act entirely within a particular plane of interest.Therefore, all the forces acting on a point of interest are two dimensional and act only within atwo-dimensional plane (x–y, y–z, x–z). Coplanar force systems allow for a simplified analysis of equilibrium
because only two directions need to be considered. So simplifies to
for the x–y plane and can be evaluated using and .
It is important to continue using a consistent sign convention for all forces, since the sign accounts for thedirection of action of the force. For unknown forces, a direction or sign can be assumed; if the answeryields a negative value the force acts in the opposite direction from what was assumed.
Hint 2. Identify the equation for the angle of action of the tension
To use the equations of equilibrium for the component directions, the forces acting on a particle must bewritten in component form. Before the components of the tensile force in the cable can be written, the anglebetween the force and a coordinate axis needs to be determined from the geometry of the system.In this problem, two separate angles can be used to solve the problem. The first, , is the angle betweenthe positive y axis and the cable. The second, , is the angle between the x axis and the cable.
Identify the correct trigonometric relationships defining the angles in terms of the dimensions and .
ANSWER:
Hint 3. Draw the free-body diagram of the pulley
Draw the free body diagram of the pulley. Recall that the pulley has no mass.
Start all forces in the center of the pulley. Draw all forces as tensile forces, or pulling forces. Allangles are measured from the positive x axis and are positive in the counterclockwise direction.
Find the magnitude of the y component of the tensile force in one side of the cable.
Express your answer to three significant figures and include the appropriate units.
ANSWER:
ANSWER:
Part B - Finding the mass of the pulley
As shown, an object with mass is hung from a pulley and spring system. When the object is hung, thetension in the cable is and the pulley is below the ceiling.
Because the tensile force is greater than the object’s weight, the pulley cannot be massless as assumed. Find themass of the pulley.
figures and include the appropriate units.For this problem, use .
Hints (3)
Hint 1. Coplanar forces
Coplanar force systems have forces that act entirely within a plane. In other words, all forces are twodimensional. The equation of equilibrium for a coplanar system is:
which can be evaluated with the magnitudes as and .
Draw the free-body diagram of the pulley. Recall that the pulley has a mass.
Start all forces in the center of the pulley. Draw all forces as tensile forces, or pulling forces. Allangles are measured from the positive x axis and are positive in the counterclockwise direction.
ANSWER:
Hint 3. Determine the equation of equilibrium
Determine the equation of equilibrium needed to solve for the object of the pulley. For this equation let bethe tension in the cable, be the mass of the hanging weight, and be the mass of the pulley. Let be
the angle between the cable and the positive y axis and be the angle between the cable and the ceiling orhorizontal.
As shown, Spring 3, which has an unknown spring constant , replaces Spring 2. The mass of the weight and pulleyare unchanged: and . However, because of the different spring constant the distance the
pulley hangs below the ceiling is now, . The unloaded length of Spring 3 is ; afterhanging the mass and pulley, the spring’s length is . Determine the spring constant of Spring 3, .
Express your answer to three significantfigures and include the appropriate units.
Hints (3)
Hint 1. Spring force equation
For linear elastic springs, the equation defining the force developed as they stretch is
where is the spring constant, which has units of force over length, and is the deformation or change inlength of the spring.
Hint 2. Find the magnitude of the tension
Given , the mass of the pulley, , and the mass of the hanging object,
, find the magnitude of the tension in the cable.
Express your answer to three significant figures and include the appropriate units. For thisproblem, use .
ANSWER:
Hint 3. Find the displacement in the spring, s
The spring force equation for a linear elastic spring is . In this equation is the spring constantand is the displacement of the spring. To solve for the unknown spring constant of Spring 3, you mustknow the force, , and the displacement of the spring. Given the unloaded and loaded lengths of Spring 3find the displacement in meters.
To use the spring force equation the force in or applied to the spring, , must be known. To findit, draw the free-body diagram of the spring and cable (on the side of the pulley containing Spring3). From the free-body diagram, notice that the tension in the cable is transferred entirely to thespring.
The lift sling is used to hoist a container having a mass of500 kg. Determine the force in each of the cables AB andAC as a function of If the maximum tension allowed ineach cable is 5 kN, determine the shortest lengths of cablesAB and AC that can be used for the lift. The center ofgravity of the container is located at G.
u. A
CB
1.5 m 1.5 m
G
F
θ θ
■3–15.
The spring has a stiffness of and an unstretchedlength of 200 mm. Determine the force in cables BC and BDwhen the spring is held in the position shown.
k = 800 N>m
A Bk 800 N/m
D
500 mm 400 mm
400 mm
300 mm
C
SOLUTIONThe Force in The Spring: sehcterts gnirps ehTApplying Eq. 3–2, we have
A 4-kg sphere rests on the smooth parabolic surface.Determine the normal force it exerts on the surface and themass of block B needed to hold it in the equilibriumposition shown.
mB
B
A
y
x0.4 m
0.4 m
60
y 2.5x2
3–31.
If the bucket weighs 50 lb, determine the tension developedin each of the wires.
A
B
E
C
D4
3
5
30
30
SOLUTIONEquations of Equilibrium: First, we will apply the equations of equilibrium along the x and y axes to the free-body diagram of joint E shown in Fig. a.
(1)
(2)
Solving Eqs. (1) and (2), yields
Ans.
Using the result and applying the equations of equilibrium to thefree-body diagram of joint B shown in Fig. b,
Force Vectors: We can express each of the forces on the free-body diagram shown inFig. a in Cartesian vector form as
Equations of Equilibrium: Equilibrium requires
Equating the i, j, and k components yields
(1)
(2)
(3)
Solving Eqs. (1) through (3) yields
Ans.Ans.Ans.FAE = 2354 lb = 2.35 kip
FAC = 538 lbFAB = 808 lb
-
67
FAB -
67
FAC + FAE - 1200 = 0
-
37
FAB -
27
FAC + 500 = 0
27
FAB -
37
FAC = 0
¢27
FAB -
37
FAC≤ i + ¢- 37
FAB -
27
FAC + 500≤j + ¢- 67
FAB -
67
FAC + FAE - 1200≤k = 0
¢27
FAB i -
37
FAB j -
67
FAB k≤ + ¢- 37
FAC i -
27
FAC j -
67
FAC k≤ + (500j - 1200k) + FAE k = 0
gF = 0; FAB + FAC + FAD + FAE = 0
FAE = FAE k
FAD = FAD C (0 - 0)i + (12.5 - 0)j + (-30 - 0)k
2(0 - 0)2+ (12.5 - 0)2
+ (-30 - 0)2S = {500j - 1200k} lb
FAC = FAC C (-15 - 0)i + (-10 - 0)j + (-30 - 0)k
2(-15 - 0)2+ (-10 - 0)2
+ (-30 - 0)2S = -
37
FAC i -
27
FAC j -
67
FAC k
FAB = FAB C (10 - 0)i + (-15 - 0)j + (-30 - 0)k
2(10 - 0)2+ (-15 - 0)2
+ (-30 - 0)2S =
27
FAB i -
37
FAB j -
67
FAB k
3–61.
If cable AD is tightened by a turnbuckle and develops atension of 1300 lb, determine the tension developed incables AB and AC and the force developed along theantenna tower AE at point A.