FORCES AND TORQUES IN EQUILIBRIUM
Purpose The purpose of this lab is to study a system in which the forces and torques
are in equilibrium.
Pre-Lab Assignment
1.
Read this laboratory handout in entirety.
2.
Refer to Figure 2, and use the following values for the forces, angles, and
moment arms, to determine whether the forces and torques are in equilibrium.
Note that the mass of non-current force apparatus = 1.3 kg, so the weight of the
apparatus is W = mg = (1.3 kg) (9.8 m/s2) = 12.74 N
T1
T2
T3
T4
=
=
=
=
5N
1.5N
1.2N
9.7N
θ1 = 72˚
θ2 = 50˚
θ3 = 81˚
θ4 = 80˚
θw = 70˚
x1
x2
x3
x4
xcog
=
=
=
=
=
.05m
.15m
.25m
.40m
.30m
Please note that this problem is not the same as the experiment you will be doing.
Theory
An object will remain at rest if and only if:
1.
All external forces on the object balance (so the net force is zero).
2.
All external torques on the object balance (so the net torque is zero].
Description of Apparatus
In this experiment an object is suspended at rest with up to five forces acting on it; the
object remains at rest. The magnitudes and directions of the forces are measured,
and calculations are done to verify that the forces balance and the torques balance, so
the system is in equilibrium.
To do this, we will use a “non-concurrent force apparatus;” the forces on the object
usually don’t all pass through the same point. The apparatus has four protractors
mounted on it to measure the directions of the applied forces (see Figure 1.) Forces
are applied to hooks attached to the center of the protractor bases.
Figure 1. The non-concurrent force apparatus.
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Calculations of Forces and Torques
Figure 2. This example of several forces acting on the apparatus is used in the pre-lab
assignment. Note that x1, x2, x3, x4 and xcog are the moment arms of the y components of the
force about the origin.
To analyze the forces it is convenient to let the x-axis be parallel to the apparatus, and
the y-axis perpendicular to the apparatus, with the origin at the left end, as shown in
Figure 2. For calculation of torques, an axis or pivot point of rotation is assumed at
the left end of the apparatus, with the rotation axis perpendicular to the paper. Then
the x-components of the forces will be parallel to the moment arms, causing no torque.
Thus, only the y-components of the forces will exert a torque about this pivot point,
and the distances x1, x2, x3, x4, and xcog shown in Figure 2 are the moment arms of the
y-components of the forces.
In Figure 2, four forces T1 to T4 are shown acting on the apparatus. Also notice the
weight force acting at the center of gravity is the fifth external force. For each force,
the angle between the force and the apparatus is shown. Components of the forces
are obtained using the sin or cosine of the angle as appropriate; for example, the x and
y components of T1 are: T1x = T1 cosθ1 and T1y = T1 sinθ1.
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Equipment
Non-concurrent force apparatus,
2 table clamps
3 aluminum rods (1.2 m.)
6 aluminum right angle clamps
2-20 Newton spring scales, 2-50 gram mass holders
about 4000 grams of assorted masses
2 pulleys with attached rods
string (30 lb. strength).
Figure3. The apparatus set up with forces applied to each of the four hooks of the apparatus.
Procedure
1. The non-concurrent force apparatus is shown in Figure 1. Measure the distance
from the left end to the center of each protractor (x1 to x4).
2. Measure the mass of the non-concurrent force apparatus.
3. Locate its center of gravity of balancing. To do this, rest the apparatus on a pen or
rod and mark the center of gravity (cog). Measure the distance from the left end to
the center of gravity. (xcog).
4. Set both of the 20 Newton scales to zero when no load is applied. (Note, each scale
should read zero when it is held by the ring and suspended with no load.)
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5. Set up the apparatus as shown in Figure 3. Use scissors and the 30 pound string
provided, and attach the string securely wherever indicated. The concurrent force
apparatus should be at least 10° from the horizontal. Each of the hanging weights
should be in the range from 400 to 1500 grams. The spring tensions should be
between 10 and 20 Newtons.
6. . Read each of the four protractors to measure the smaller angle (< 90°) between
each force (T1 to T4) and the concurrent force apparatus. Draw a full page Figure to
scale showing the apparatus, the four forces, and the angles measured.
7. Record the force in Newtons for T1 and T4 (the scale readings.) Record the values of
the suspended masses causing the forces T2 and T3, and convert these to Newtons,
using W = mg.
8. Note that the direction of force T2 is the same as the direction of the weight of the
concurrent force apparatus.
Calculations
1. Write down the equations that describe the equilibrium of forces in the x direction,
the y direction, and the equilibrium of all torques acting about an axis through the
origin (as defined earlier).
2. Calculate the x and y components of each of the five forces.
3. Using your measured values of the forces (magnitudes and directions), calculate
the total force in the – x direction and the total force in the + x direction. Find the
percent difference between them, and compare this to the expected value.
4. Find the total force in the – y direction, and the total force in the + y direction. Find
the percent difference between them, and compare this to the expected value.
5. Using the left end of the apparatus as the axis of rotation, and the measured
moment arms, find the total counter clockwise torque and the total clockwise
torque. Find the percent difference between them, and compare this to the
expected value.
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