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Unformatted text preview: NAME LAB PARTNERS Station Number Addition of Vectors: ExperimentZ
Translational Equilibrium INTRODUCTION For many physical quantities, the direction of a quantity is just as important as its mag
nitude. Quantities that require speciﬁcation of both magnitude and direction are called vectors.
Scalars, on the other hand, have magnitude only. Mass, length, and temperature are examples of
scalar quantities. Displacement, velocity, and force are examples of vectors. The object of this
experiment is to learn to add and subtract vectors by several methods. The vectors with which
you will work are force vectors, but the methods you will learn apply to any vector. You will
also learn about translational equilibrium. THEORY One method of adding vectors is to add them graphically. A vector is represented by a
straight line with an arrowhead. The length of the straight line is proportional to the magnitude
of the vector, and the direction of the vector is indicated by the arrowhead. A simple example will serve to illustrate graphical vector addition. Suppose you walk 300
meters due east and then walk 400 meters northeast. (The direction northeast is 450 north of
east.) Figure 2.1 shows one way to add these two displacement vectors. Vector A represents the
300meter displacement, and vector B represents the 400meter displacement to the northeast.
Vector R represents the resultant (or sum) of vectors A and B. Measurements made with a ruler
and a protractor give a magnitude of 648 meters and an angle of 260 north of east. The method of
addition shown in Figure 2.1 is often called the triangle or polygon method of vector addition. If
more than two vectors are added, Figure 2.2 shows the resultant. The resultant R joins the tail of
the ﬁrst vector to the arrowhead of the last vector. This process can be used for adding many
vectors. N N
C
R
R
B B
E E
Figure 2.1 Graphical addition Figure 2.2 Polygon method
of two vectors A and B. The of vector addition scale is 1/4 inch = 100 m The two vectors A and B shown in Figure 2.1 can be added graphically in yet another way.
Figure 2.3 illustrates this technique, which is called the parallelogram method of vector addition.
The tails of the two vectors are drawn at the origin so that the vectors
form two sides of a parallelogram. The diagonal of the parallelogram
is the resultant vector R. 3. Graphical methods of vector addition suffer from the obvious
difﬁculty that accuracy is limited by the accuracy of the measuring
devices and by the care taken to construct the drawings. This limited
accuracy makes it necessary to develop an analytical method for
adding vectors. This analytical method is based on trigonometry and
geometry, and relies on ﬁnding the x and ycomponents of each
vector. To add vectors analytically, ﬁrst ﬁnd the x and ycom
ponents of each vector, deﬁned by Figure 2.3 Parallelogram
method of vector addition AX = A cos 6 (1)
and
Ay = A sin 6. (2) Here A is the magnitude of the vector A, and 6 is the angle
between vector A and the +x axis. Figure 2.4 shows the construction
of the x and ycomponents of vector A. Since xcomponents and y
components of all vectors are now parallel to the x or y axis,
respectively, they can be added algebraically so that the resultant x
and ycomponents are given by Figure 2.4 x and y— R = A + B + C + , . . (3) components of vector
X X X X
and
Ry=Ay+By+Cy+o, (4) where RX and Ry are the x and ycomponents of the resultant vector. By standard convention,
xcomponents to the right are positive, and xcomponents to the left are negative. Similarly,
upward ycomponents are positive, and downward ycomponents are negative. Since x
components are perpendicular to the ycomponents by deﬁnition, the Pythagorean theorem is
used to ﬁnd the magnitude R of the vector R so that R = [sz + Ry2]1/2 5 (5)
and the angle between the vector R and the +x axis is given by 6 = tan'1 (Ry /Rx). (6) The addition of force vectors in this experiment makes use of the concept of translational
equilibrium. If several forces act on an object, they may be replaced by a single resultant force
that is the vector sum of all forces acting of the object. Translational equilibrium occurs when the
resultant force on an object is zero. EXPERIMENT NO. 2 1. Your instructor will assign you three masses. m1, m2, and m3. and
the angles at which to apply these masses on the force table. Top
and side views of the force table are shown in Figures 2.5 and 2.6,
respectively. The force F provided by each mass is given F = mg,
where g is the acceleration due to gravity. Record the values for
the three masses (in kg) and the angles at which they are applied
on the force table. Record all angles as measured counterclockwise
with respect to the +x axis. Figure 2.5 Top view
m1 = at an angle of of force table m2 = at an angle of
m3 = at an angle of Calculate the magnitude of the forces F1, F2, and F3 caused by
each of the masses. Fi ure 2.6 Side View
F = F = 1: = g
1 —— 2 m 3 _ of force table 2. Select any two of the three forces and determine the magnitude and direction of the resultant
vector using the parallelogram method. Show your construction on the graph paper provided.
Be sure to record the scale you used on the drawing. Record the direction and the magnitude of
the resultant vector and the equilibrant vector. The equilibrant vector is equal in magnitude but
oppositely directed to the resultant vector. The vector sum of the resultant vector and equilibrant vector is zero. Magnitude of resultant = Magnitude of equilibrant = Direction of resultant = Direction of equilibrant = 3. Apply the two forces you selected and the equilibrant to the small ring on the force table. A
peg in the center of the table keeps the ring from moving very far until equilibrium is attained.
Be sure the connecting strings are kept in the grooves of the pulleys. Adjustments should be
tested by removing the peg and pulling the ring slightly to the side and releasing it. If equili
brium has been established, the ring will return to the center of the force table. Record the
magnitude and direction of the equilibrant vector, as determined using the force table. Magnitude of equilibrant = Direction of equilibrant = 4. Using the three forces from Part 1, determine the direction and magnitude of the resultant
using the polygon method of vector addition. Show your diagram on the graph paper
provided. State the scale. Magnitude of resultant = Direction of resultant = 5. Check the measurement of Part 4 on the force table as you did in Part 3. Record the magnitude
and direction of the equilibrant vector. Magnitude of equilibrant = Direction of equilibrant = How should the resultant from Part 4 and the equilibrant from Part 5 be related? How are
yours related? 6. Using the three forces from Part I, calculate the magnitude and direction of the resultant
vector analytically. Use Equations (1) — (4) to complete the table. Show your calculations in
the space below the table. F1 Resultant X 10 Use Equations (5) and (6) to calculate the magnitude and direction of the resultant. Magnitude of resultant = Direction of resultant = QUESTIONS 1. Draw three vectors that originate at the origin and lie in the ﬁrst quadrant of an xy coordinate
system. Explain why the resultant of your vectors can or cannot be zero. 2. Vector S has a magnitude of 8 units and makes an angle of 46 degrees with respect to the +x
axis. Vector T has a magnitude of 6 units and makes an angle of 36 degrees measured clockwise
with respect to the negative x axis. Sketch the two vectors and calculate analytically the magnitude and direction of the resultant vector. What are the magnitude and direction of the
equilibrant vector? 11 3. Two vectors, A and B, are drawn on an xy coordinate system as shown. Vector A has a
magnitude of 8 units, and vector B has a magnitude of 6 units. Find the X and ycomponents
of vectors A and B in the xy system. Compute the magnitude of the resultant in the xy
coordinate system. A second coordinate system, the x'y' system, is rotated 37° with respect
to the xy system as shown. Find the x' and y' components of A and B in the x'y' system.
Compute the magnitude of the resultant in the x'y' coordinate system. (Hint: Calculate the angle that A and B made with respect to the x'—y' axes.) 12 ...
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 Fall '10
 LOWELLWOOD

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