Physics 141
Forces and Vector -- 1
FORCES AND VECTORS
GOAL: To study the additions of vectors in the and
context of forces in equilibrium
INTRODUCTION: If an object is not accelerating
(3)
then the forces acting on the object must add up to
zero. If this condition holds, one says the object is in Much of this experiment, involves experimentally
equilibrium. Experimentally, the absence of adding two forces ( and ) and measuring the
acceleration is the "proof" that the sum of the forces is
zero. Because people are better at seeing that an force ( ) required to counteract this sum and keep
object is not moving than it is at judging constant the system at rest.
motion, one can make this experimental test easier to
perform by starting the system at rest, i.e., NOT Q i. How is the force, , from a suspended ( non-
MOVING. Then one applies something like: "A body accelerating) object related to the mass, m, of
at rest tends to stay at rest unless acted on by a non- the object?
zero net force."
(4)
A force has both a magnitude and a direction,
so forces must be added as vectors. In most cases one
adds vectors by resolving each vector into
perpendicular components ( say horizontal and
PROCEDURE:
vertical, or x and y), next add the components and then
combine the components.
I Tug of war VS the TA.
The class is divided in half and each group
pulls on an end of a rope, This is basically a tug of
war, but the group pull hard but establish an
equilibrium. The TA then attempts to push the center
of the rope sideways while the two groups of students
try to prevent it.
TENSION IN A VERTICAL PLANE
Figure 1 Vector addition and
equilibrium.
To add forces and (at angles " and $
respectively) that add together to get the resulting in
force (with associated angle, (), one first does:
(1)
Then
Figure 2 The pulley framework mounted
(2) on the counter.
Physics 141
Forces and Vector -- 2
There is a set of equipment for studying the THE FORCE TABLE
vector nature of forces consisting of a framework on
which two spring balances and a pulley are mounted as
shown in Fig. 2. This setup will be used for parts II, II
and IV. The position of the pulley can be adjusted up
or down to change the configuration of the system, and
the position of the suspended mass can be adjusted by
moving the binder clip. The angles " and $ can be
found by measuring Ax , Ay , Bx , and By.
II. First, test the system without any suspended
mass or mass hanger.
Q ii. What relationship do you expect between the
readings of the two spring balances in this
arrangement? Change the position of the
pulley and test your hypothesis.
III. Forces in a Vertical Plane Figure 3 The force table.
Try several (3) different masses on the system for each
of the following three configurations of the suspension
point for the mass: The remainder of this experiment will be done
1. Point below the bottom of the pulley. using a force table, see Fig. 3. This is an elevated
2. Point above the bottom of the pulley. metal disk about 40 cm in diameter. Angles in degrees
3. Point at same level as the bottom of the are ruled near the rim of the disk and pulleys can be
pulley. attached to the rim. Masses are suspended by strings
that run over pulley and thus exerts a force on the
Find each individual forces at the point where the mass center ring. If the center ring (while not touching the
is suspended ( at the binder clip) and compare to the center pin) is stationary then the sum of these forces
expected values. from the suspended masses is zero.
The magnitude of the force associated with a
IV. A Different Set of Axes suspended mass is its weight ( NOT the length
The obvious choice of axes, when working in a of the string).
vertical plane is for the x-axis to be horizontal (left and The direction of this force on the center ring
right) and the y-axis to be vertical ( up and down). is outward from the center toward the pulley
There are cases such as problems dealing with incline and can be read from the protractor scale on
planes where this choice may complicate the the edge of the force table.
mathematical operations rather than simplify them.
Consider the following configuration on the
framework. One string is about 30° from the NOTE: In the following
horizontal and the other string is at a right angle to the procedures, each force is written as
first. Set this up. Use the corner of a piece of paper or an angle and an added mass. This
a book as guide to setting the right angle. mass must be added to the mass of
the mass hanger before calculating
Experimentally this resolves the force the force.
associated with the suspended mass into two
perpendicular components. Identify the new axes.
Compare the experimental results with the calculated
values for a few suspended masses. Relate this new V. 1-D Equilibrium ( do very quickly)
choice of axis to the inclined plane problem, the First, test the general idea that two equal but
normal force and the force down the plane. opposite forces cancel each other and leave the system
Physics 141
Forces and Vector -- 3
in equilibrium. Do this on the force table with only Mass #1 Angle Mass #2 Angle
two pulleys and two strings. grams #1 grams #2
Place 200 grams on one of the mass holders,
200 175° 50 265°
with the associated pulley at 0°.
200 30° 50 120°
200 0° 50 90°
Q iii. How much mass do you expect you will need
200 351° 50 81°
to put on the other mass holder to bring the
system into equilibrium, and at what angle?
If necessary set up on the force table for a few of these
configurations to help visualize the systems. (This
Try it and test your hypothesis. This cancellation of
series of measurements is more about a concept than
forces, such that the system remains in equilibrium, is
about numbers!)
central to the rest of this experiment.
VII. Adding Perpendicular Forces
VI. Resolving a Force into Components
Consider adding two perpendicular forces, for
Forces can be resolved experimentally into
example, one in the x direction and one in the y
perpendicular components using a force table. First,
direction.
set one pulley at 0° and a second pulley at 90°. Next,
Q v. How, in words, do you expect the angle of the
set the third pulley at the angle of the force given in
the table below and add the required mass to the third resultant force to vary with the ratio of the x-
mass hanger. Finally, balance the forces by adding the directed force to y directed force? ( increase
right amount of mass to the first and second mass or decrease?)
hangers. Q vi. And how mathematically does the angle
depend on this ratio? Test these hypotheses,
Q iv. What would you need to do if the component using all three pulleys and mass hangers.
of the force in the 0°-direction was negative?
Experimentally determine the third force required to
keep the system in equilibrium if the following pairs of
Resolve the following forces ( added masses and forces ( added masses) are applied
angles):
Mass #3 Angle Mass #1 Angle Mass #2 Angle
grams #3 grams #1 grams #2
200 225° 200 0° 50 90°
200 200° 200 0° 100 90°
200 190° 200 0° 150 90°
200 164° 200 0° 200 90°
Consider displaying your data in a table with enough 200 0° 300 90°
columns to include various calculated values. 200 0° 400 90°
A i. Compare the force required to maintain
VII. Choice of Axis when Adding Forces equilibrium with the calculated sum of the two forces.
Adding vectors at right angles is relatively simple (Consider displaying your data in a large table that
because, with a clever choice of coordinates, the forces includes various calculated values of interest including
to be added are already resolved. Consider the the ratio of the two applied forces) Is there any
similarities of adding the following pairs of forces symmetry amongst these situations?
(added pairs of masses):
Physics 141
Forces and Vector -- 4
VIII. Adding Non-Perpendicular Forces Use the binder clips rather than knots when
Compared to adding perpendicular forces, adding two suspending the mass from the string.
forces that are not perpendicular is more complex in The horizontal force table:
terms of the mathematical operations that are required, Level the tables
but on the force table the procedure is identical. Use fishing line
Remember that the mass of the mass hangers
Q vii. As the angle between the two applied forces must be included in the calculations.
increases, what do you expect the When the forces are nearly balanced, tapping
counteracting force to vary? the center ring may help in finding the true
equilibrium of the system by overcoming the
Test your hypothesis with the following pairs of forces small friction in the system.
(added masses): Check that the strings moved on the center
ring to positions that are appropriate for the
Mass #1 Angle Mass #2 Angle angles to the pulleys.
grams #1 grams #2
ANALYSIS:
200 0° 200 15°
A ii. How accurate are your measurements? What
200 0° 200 30°
is the major cause of your inaccuracy?
200 0° 200 45°
200 0° 200 60°
A iii. For the force table system, draw free-body
200 0° 200 90°
diagrams for the center ring and the three suspended
200 0° 200 120°
masses. How are the various forces in these diagrams
200 0° 200 150°
related? Is there something special about their
200 0° 200 180°
relationship if the system is in equilibrium? How is
the free-body diagram for the center ring different
when the suspended mass balance and the system is in
equilibrium?
EQUIPMENT:
Force table with 3 pulleys
A iv. For the system consisting of the framework
Frame with two spring balances
and spring scales and suspended masses, draw the free-
and a pulley
body diagram for the suspended mass.
Mass set and hangers
Binder clips
A v. How does the concept of tension enter into the
Meterstick & ruler
analysis of these experiments?
In room
Incline Plane Demo
A vi. Consider the following two methods of
Large Hemp Rope
analyzing your results:
Scissors
1. Vectorially adding the two applied forces
String and fishing line
and the measured counterbalancing force.
Levels
Because the system is in equilibrium, one can
compare this sum of three forces to zero.
EQUIPMENT HINTS:
2. Calculate the sum of the two applied forces
and compare it to the measured
The pulley frame work:
counterbalancing force.
Check the zero of the spring balances. Ask for
help if they are off.
How does one calculate the relative difference
Check the calibration of the spring balances.
from an expected value of zero?
To determine the angles in this system,
measure various horizontal and vertical
distances ( Ax , Ay, Bx and By in the figure) and
use a bit of trigonometry.
