FORCE TABLE
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:
1. 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.
2. 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.
2012
y
quadrant
I
II
x
Fx
Fy
F
III
IV
We can think of F and its x and y components (Fx and Fy) as the sides of a right triangle, with F as
the hypotenuse and the components as the legs. Using the trigonometric relation that
cos θ = adjacent / hypotenuse
we can determine that with the angle as drawn above, Fx, the x component of F is, up to a sign,
given by
Fx = |F| cos θ
where |F| is the magnitude of F. Similarly, up to a sign the y component of F is given by
Fy = |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
+
—
-2-
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.
washer
string
•
•
•
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/s2, don't
forget the mass of the hanger).
2
3
1
m2
m1
•
•
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.
-3-
•
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.
y (90°)
x (360°)
Let us draw the three forces in this coordinate system.
F2
F1
F3
(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 F1 has only a positive x
component and F2 has only a positive y component.
•
The way vectors add is that one adds their components. So the x component of the net
force (F1+F2 + F3) will in this case be
F1 + F3x
and the corresponding y component will be
F2 + F3y
-4-
For a vector to be zero, all of its components must be zero. Thus the two expressions
above should yield zero. Verify this.
Force 1
Force 2
Force 3
Sum
•
Angle
Quadrant
XXX
XXX
XXX
Fx
Fy
Change one of the masses (make a change of at least 50 grams) and repeat the
measurement and calculations.
Force 1
Force 2
Force 3
Sum
•
Magnitude
Magnitude
Angle
Quadrant
XXX
XXX
XXX
Fx
Fy
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 F3 (the Force Sensor force) as in Part II (obtain the magnitude and the
direction).
Now add the vectors representing F1 and F4 by first resolving F4 into x and y components
and adding the components.
Adding the components of F1 and F4, 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 Gx and Gy are the
components, then the magnitude of G is given by
|G| = (Gx2 + Gy2)1/2
(Pythagorean theorem) and the direction (angle) by
θ = Arctan (Gy/Gx).
-5-
•
•
Make the replacement you just determined.
o Let the F2 string (the spectactor string) alone
o Remove the fourth string
o change weight hanging on string 1 to correspond to the magnitude of G,
o and change its angle to the direction of G. (As a check, the direction of G should
be between F1 and F4.)
Then remeasure F3 (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
Quad
-rant
Fx
Fy
Magnitude Angle
Force 1
Force G
Force 4
Force 2
Force 2
Force 3
(after)
Quad
-rant
Fx
Force 3
(before)
•
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
Quad
-rant
Fx
Fy
Magnitude Angle
Force 1
Force G
Force 4
Force 2
Force 2
Force 3
(after)
Quad
-rant
Fx
Fy
Force 3
(before)
•
Did Force 3 change substantially when you replaced Forces 1 and 4 with Force G? Are any
differences acceptable? Explain.
-6-
Fy