PHYSICS EXPERIMENTS — 131
18-1
Experiment 18
Equilibrium Under the Influence of Nonconcurrent Forces
In this experiment you study the forces on an
extended object in equilibrium on a force table.
When an object is in equilibrium, there are two
criteria that must be satisfied. The first is that the
resultant of all the forces must be zero. The second
condition is that the resultant of all the torques
must be zero. You will set up a system in which a
laminar sheet of plastic is in equilibrium when
pulled by strings in different directions in the case
that the string tensions are nonconcurrent. (See
Figure 1.) Nonconcurrent forces are those whose
lines of action do not all intersect at one point.
of the torque. Assign a positive sign to a torque
causing a counterclockwise rotation and a negative
sign to a torque causing a clockwise rotation.
Procedure.
• Attach strings through the holes in the corners of
the thin plastic plate irregular shape. Set the plate
on a force table with the threads from it’s corners
hanging over pulleys. Make sure that the pulleys
are turned so that they are parallel to the strings
passing over them. This will prevent the strings
from slipping off the pulleys and minimizes the
pulley resistance.
Preliminaries.
A force is a vector, with magnitude and direction.
Graphically, a vector is represented by an arrow; the
direction of the arrow gives the direction of the
vector and the length of the arrow represents the
magnitude of the vector to some suitable scale.
Forces must be added as vectors using the tip to tail
construction to determine the vector sum (or
resultant). The sum of forces holding an object in
equilibrium is zero.
Each force acting on an extended object exerts a
torque. The torque (in two dimensions) has
magnitude and sign. The magnitude of the torque is a
measure of the tendency of the force to cause a
rotation of the object while the sign of the torque
indicates the direction of the rotation (clockwise or
counterclockwise).
The torque is determined from the force vector as
well as where the force is applied to the object. The
construction for determining the value of the torque
is shown in Figure 1. The steps are:
i. choose a point for the axis (indicated by the
circle on Figure 1). For an object in equilibrium, any
choice works.
ii. draw the line of action (LOA) through each
force vector.
iii. draw the lever arm for each force (indicated
by x in Figure 1) from the axis point perpendicular to
the line of action for that force.
iv. for each force, multiply the force magnitude
by the lever arm distance to determine the magnitude
• If left on their own the four forces along the
strings will try to arrange themselves to be
concurrent. In order to avoid this make sure that the
placement of the pulleys is such that the forces are
not concurrent. This may be quickly checked by
observing the intersections of the lines of action. If
all the lines of action pass through a 4 cm circle
(about the size of a 1 g slotted mass), then they are
nearly concurrent and must be readjusted to create a
nonconcurrent state. One way to assure
nonconcurrency is to have two strings parallel or to
have two strings cross.
• Select weights to hang from the strings which
make the laminar come to equilibrium, i.e. float
smoothly on the table and return to its position
when gently displaced. The weights should be large
enough so that the plate is not resting heavily on the
surface of the table. You may find it helpful to
place a small mass on the plate to stabilize it,
reducing the likelihood of its flipping over.
• Once equilibrium is established, place a sheet of
paper under the laminar on the table and trace the
plate and strings. Label the strings / forces clearly.
• Record the amount of suspended mass for each
string with its proper string. Construct x-y axes on
the tracing through the axis point. Measure and
record the angle each force makes with respect to
18-2
PHYSICS EXPERIMENTS — 131
the x axis. (Do not record the angles on the force
table as they do not apply in this experiment.)
• Construct the lever arm for each force with a
straight edge according to the construction
described above and shown in Figure 1. Record
each lever arm length. For ease of measurement,
choose the axis point so that it is not closer than 1
cm to any of the lines of action. The axis point does
not have to lie within the body of the plastic sheet.
• Draw a vector map to scale showing the four
string tension force vectors added tip to tail.
• Compute the algebraic (including signs) sum of
the four torques.
Questions (Answer clearly and completely).
1. rDoes the system show Newton’s First Law,
∑ Fi = 0 ? How do you know by examination of the
i
• Compute the torque for each force by multiplying
the force times its lever arm.
map showing the graphical tip to tail addition of the
force vectors?
• Assign a positive sign to a torque causing a
counterclockwise rotation and a negative sign to a
torque causing a clockwise rotation. To determine
if a torque is clockwise or counterclockwise, place a
pencil point on the axis. Place your finger at the
intersection of the moment arm and LOA and push
the paper in the direction the force acts. The
direction the paper rotates will show you the sense
of the torque.
2. Does your analysis show that the system does not
tend to rotate? How do you know?
3. Do the lines of action depend on the choice of
axis point? Does the size of each lever arm depend on
the choice of axis point? Does the sign of each torque
depend on the choice of axis point?
LOA
F1
Line of Action
x3
F2
x4
x1
x2
F3
LOA
LOA
F4
Figure 1. Free Body Diagram of Laminar with Lever Arm Construction