Experiment I: Vectors

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Goals

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- Practice resolution of vectors into their components and addition
of vectors
- Visualize these processes on a force table

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** **One of the most important
concepts in physics is the concept of vector quantities, such as velocity,
acceleration and force. It is essential that you understand the concept of
vector, the addition of vectors, and the resolution of a vector into its
components. This lab is designed to give you exercises in performing these processes
and help you visualize them on a force table. The diagram below illustrates the
resolution of a vector **F** into *x*- and *y*-component relative to a given choice of *x*-*y* coordinate system:

F_{x}
= F cosq

F_{y}
= F sinq

tanq = F_{y} / F_{x}

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There are two common methods
for finding the resultant of many vector quantities: the first is the graphic
method (parallelogram or tail-to-tip, see your textbook for details); the
mathematically precise method is by adding components in which each vector is
first resolved into its *x*- and *y*-components, the total *x*- and *y*-components are then obtained from the algebraic sum of all the *x*- and *y*-components:

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Since we are dealing with algebraic
sums, it is important to keep track of the signs of each component.

In this lab we will use the
weight of objects (one type of force) as a representative vector to study the
addition of vectors. We will make use of the condition for static equilibrium:
the vector sum of all the forces acting on a body at rest is zero. We will
calculate the resultant of a number of forces, **F _{R}** =

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*Equipment***: **Force table, pulley, weights
and hangers, ruler, polar graph paper

** Setup:** Figure 1-2 shows a schematic
drawing of the experimental setup. Weight (force of gravity) always points
downward. We use a pulley mounted on the edge of the force table to turn it
into a horizontal force with the same magnitude. The table is marked in 1

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For each part of the
experiment draw a vector diagram *to scale*
on a piece of polar graph paper.

1. (a) Hang a 0.20 kg mass at 110^{o}. Calculate the weight
of the mass.

Hang a second mass so that
the ring is in equilibrium. What is the direction (in degrees) and magnitude of
this balancing force?

(b) Calculate the *x*- and
*y*-components of the force at 110^{o}
in Part 1(a) for a *x-y* coordinate
system with the positive *x*-axis at 0^{o}
and the *y*-axis at 90^{o}.
Replace this force by its *x*- and *y*-components on the force table while
leaving the balancing force unchanged. Is the ring still in equilibrium?

(c) Calculate the *x*- and *y*-components of the force at 110^{o} in Part 1(a) for a *x-y* coordinate system with the positive *x*-axis at 70^{o}. At what angle
on the force table is the *y*-axis now?
Replace the original force at 110^{o} by these new *x*- and *y*-components on
the force table while leaving the balancing force unchanged. Do the components
of a vector depend on your choice of the coordinate system?

2. Mount a pulley at 20^{o} and suspend
a 0.98 N weight from the string; mount another pulley at 140^{o} and
suspend a 1.96 N weight from it. Draw a vector diagram to scale and find the
resultant force graphically by the parallelogram method. Determine both the
magnitude and direction of the resultant.

For a more accurate result,
find the resultant from the components with the *x*-axis at 0^{o}. Calculate both the magnitude and direction
of the resultant. Set up the *negative*
of the resultant force (equal in magnitude but 180^{o} away in
direction), in addition to the two forces already on the force table. Is the
ring in equilibrium?

3. Set up the pulleys at 20^{o} and 140^{o}
as in Part 2, suing the same weights. Mount a third pulley at 220^{o}
and suspend 1.47 N over it. Calculate the resultant of the three forces using
components with the *x*-axis at 0^{o}.
Set up the negative of the resultant and check for equilibrium.

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In the context of the
results you have obtained, discuss the following:

1.
Are
the components of a vector unique?

2.
Are
the components of a vector equivalent in their effect to the force itself?

3.
Does
the order in which you add two or more vectors affect the resultant of the
vectors?

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Briefly discuss whether you
have accomplished the goals listed at the beginning.

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