Forces
According to Newton's Second Law
F = m a,
if there is a net force acting on an object, then the object will accelerate.
In this experiment we will consider a situation of static equilibrium,
i.e. a case in which nothing is moving. If an object is not moving, then it is
certainly not accelerating. And then using Newton's Second Law we can further
conclude that there is no net force on the object. No net force does not mean
there are no forces on the object, but rather that the effects of all of the
individual forces cancel.
Forces have both a magnitude and a direction. The magnitude tells one the size
of the push or pull supplied by the force, while the direction tells which way
(e.g. along what angle) the object is being pushed or pulled. Forces are thus represented
mathematically by vectors. A quantity must have certain properties in order for
it to be considered a vector. For instance, there are rules expressing vectors
in terms of components and rules for adding vectors.
Expressing a vector in terms of its components is also known as "resolving" it.
Let us consider resolving a force F into components. Find the angle
between the x axis and F as shown below. There are two conventions for
measuring angles:

To measure the angle starting from the positive x
axis, proceeding in a counterclockwise direction and ending on the vector of
interest. This approach results in angles between 0 and 360 degrees.

To measure the angle between the x axis (positive or negative, whichever
is closer) and the vector of interest. This approach yields angles between 0
and 90 degrees, and one must further keep track of which quadrant the vector
lies in.
We'll adopt the second method in the following discussion. Another thing to
keep in mind when dealing with angles is to keep track of whether one is
working with degrees or radians.
We can think of F and its x and y components (F_{x} and F_{y})
as the sides of a right triangle, with F
as the hypoteneuse and the components as the legs. Using the trigonometric
relation that
cos θ =
adjacent / hypoteneuse
we can determine that with the angle as drawn above, F_{x}, the x
component of F is, up to a sign, given by
F_{x} = F cos θ
where F is the magnitude of F. Similarly, up to a sign the y
component of F is given by
F_{y} = F sin θ
the magnitude of F times sin theta. The signs can be determined by which quadrant one is in
Quadrant 
Sign of x
component

Sign of y
component 
I 
+

+

II 
—

+

III 
—

—

IV 
+

—

Part I. Resolving the force vector into components.

Tie three strings to washer. Two of the strings should be of a length such that
if the washer is in the center of the force table, the strings can pass over a
pulley at the edge of the table and hang several centimeters below table level.

Attach the two longer strings to hangers, and we want the hangers to hang
several centimeters below the pulleys and several centimeters above the lab
table as well.

The length of the third string should be slightly longer than the radius of the
force table. We will be attaching the third string to the Force Sensor.

Put one pulley at 360 and the second at 90. Place the longer strings over the
pulleys and suspend hangers from them. Add unequal masses (at least 100 g) to
the hangers and determine the weight of each (multiply the mass in kilograms by g=9.8 m/s^{2},
don't forget the mass of the hanger).
 Click on the Tare button on the side of the Force Sensor when nothing is pulling on the Force Sensor.
What does the Taring process do?
Attach the hook of the Force Sensor to a loop at the end of your shorter
string. Hold the Force Sensor so that the strings attached to the washer are horizontal
(not vertical!)
and so that the washer is over the center of the force table. The Force Sensor
should be aligned with the string attached to it.
Hold it there as
steadily as possible while one partner clicks on Record in Data
Studio 
you only need a few seconds. Also note the angle of the string attached to the
Force Sensor (the angles are marked on the Force table). Some groups attach
the Force Sensor to a ringstand; make sure the Force Sensor is pointed
along the same direction as the string pulling on it. 
Record the mean magnitude of the force. The Force Sensor reading may be negative. The
negative implies a direction; however, we are obtaining the direction of the
force by noting the angle. Since we want only the magnitude, take the
absolute value of the Force Sensor reading.

We will first want a coordinate system; we will take the x direction to point
in the radial direction from the center of the table toward the 360 degree mark
and the y direction to point in the radial direction from the center of the
table to the 90 degree.
Let us draw the three forces in this coordinate system.
(Sometimes when picturing vectors the lengths of the lines are meant to be
proportional to the magnitudes of the vectors. This is not true in the
picture above.) The way we have set it up and the way we chose the coordinate system
F_{1} has only a positive x component and F_{2}
has only a positive y component.

The way vectors add is that one adds their components. So the x component of
the net force (F_{1}+F_{2} + F_{3}) will in this case
be
F_{1} + F_{3x}
and the corresponding y component will be
F_{2} + F_{3y}
For a vector to be zero, all of its components must be zero. Thus the two
expressions above should yield zero. Verify this.

Magnitude 
Angle 
Quadrant 
F_{x} 
F_{y} 
Force 1 





Force 2 





Force 3 





Sum 
XXX 
XXX 
XXX 



Change one of the masses (make a change of at least 50 grams) and repeat the
measurement and calculations.

Magnitude 
Angle 
Quadrant 
F_{x} 
F_{y} 
Force 1 





Force 2 





Force 3 





Sum 
XXX 
XXX 
XXX 


Are the sums of the components acceptably close to zero?
Part II. Adding vectors.

Add a fourth string to your setup and place it at angle between 30 and 60
degrees (do not choose 45 degrees!) and attach it to a hanger. Put some mass on
the hanger. Our goal now is to replace two of the
tensions (the force exerted by a string is referred to as tension) by a single
tension that serves the same purpose in balancing the washer. First measure F_{3}
(the Force Sensor force) as in Part II (obtain the magnitude and the
direction).

Now add the vectors representing F_{1} and F_{4} by first
resolving F_{4}
into x and y components and adding the components.

Adding the components of F_{1} and F_{4}, you now have the
components of the sum, which we will call G.

We now need to find the magnitude and direction of G. If G_{x} and G_{y}
are the components, then the magnitude of G is given by
G = (G_{x}^{2} + G_{y}^{2})^{1/2}
(Pythagorean theorem) and the direction (angle) by
θ =
Arctan (G_{y}/G_{x}).

Make the replacement you just determined.
 Let the F_{2} string (the spectactor string) alone
 Remove the fourth string
 change weight hanging on string 1 to correspond to the magnitude of G,
 and change its angle to the direction of G. (As a check,
the direction of G should be between F_{1} and F_{4}.)

Then remeasure F_{3} (with the Force Sensor). It should be essentially unchanged. In your report
you should give all the vectors before, show your work to determine the
"resultant" (the sum) and show your measurements (all the vectors) after.
Before  After 

Magnitude 
Angle 
Quadrant 
F_{x} 
F_{y} 
Force 1 





Force 4 





Force 2 





Force 3
(before) 







Magnitude 
Angle 
Quadrant 
F_{x} 
F_{y} 
Force G 





Force 2 





Force 3
(after) 







Repeat the set up with four strings again, but this time change all of the
angles. Strings 1 and 2 should no longer be at 360 and 90. Add vectors 1 and 4
as before. (This time you will have to resolve both vectors into components
before adding.)

Then repeat the above measurements and calculations.
Before  After 

Magnitude 
Angle 
Quadrant 
F_{x} 
F_{y} 
Force 1 





Force 4 





Force 2 





Force 3
(before) 







Magnitude 
Angle 
Quadrant 
F_{x} 
F_{y} 
Force G 





Force 2 





Force 3
(after) 






Did Force 3 change substantially when you replaced Forces 1 and 4 with Force G?
Are any differences acceptable? Explain.