Rigid Object in Equilibrium: Torques
Timothy De Guzman

Purpose:
A static equilibrium condition is first studied experimentally and then theoretically. A meter stick, with small holes drilled at specific locations, is the rigid body and five forces are applied in order to produce a static condition that satisfies given constraints. Three of these forces have their magnitudes, directions, and points of application specified. The point of application the fourth force is variable and both the magnitude and direction of the fifth force are variable. These variable quantities are to be determined both experimentally and theoretically, and the results compared.
Theory:
The conditions for a rigid body to be in equilibrium are that the sum of the external forces acting on a rigid body must be equal to zero,


F = 0 ,
(1) known as the First Condition of Equilibrium; and that the sum of the external torques acting on a rigid body must be equal to zero,
 

= 0 ,
(2) known as the Second Condition of Equilibrium. When the forces are confined to the x-y plane, (1) and
(2) become

F x
= 0 ,

F y
= 0 ,
  z
= 0 ,
(3) where
F x 1 and
F y 2 are the forces in the x
3 and y
4 directions and
 z 5 is the torque in the z 6 direction (i.e., "around" the z-axis).
In this experiment the forces acting on the rigid body are coplanar. According to (3), then, there exist three independent equations that relate the quantities m
1 7, m
2 8, m
3 9, m
4 10, x
11,

12, and m
5 13. (Refer to Figure 1.) Therefore, specification of all but three of the quantities allows the unspecified quantities to be uniquely determined. These variables, or unknowns, are chosen to be

15, and m
5 16. x
14,
When the conditions for static equilibrium are applied to any system, the location and orientation of the axes around which the torques are calculated are arbitrary. It is advantageous, then, to locate the origin at a position that will reduce the number of terms in the torque equation.

Sketch:
Part I:
Part II:
Part III:

Data:
Part 1:
Center of Gravity: 49.53 cm / 1000 = .0495 m
Part 2:
M
1
= m
1
+ m clip1
= .0218 kg + .1 kg = .122 kg
M
2
= m
2
+ m clip2
= .0214 kg + .05 kg = .0714 kg
M
3
= m
3
+ m clip3
= .0214 kg + .2kg = .221 kg d
1
= .2 m d
2
= .1 m d
3
= .140m
F
1
= M
1 g = .1218 kg x 9.8 m/s 2 = 1.19 N
F
2
= M
2 g = .0714 kg x 9.8 m/s 2 = .699 N
F
3
= M
3 g = .2214 kg x 9.8 m/s 2 = 2.17 N
Part 3:
L = .0205 m
M = .121 kg m total
= m + m clip
= .2 kg + .0214 kg = .221 kg d = .0112 m
Analysis:
Part 2: d
3
= (M
1 d
1
+ M
2 d
2
)/(M
3
) =
τ ccw
= F
1 d
1
+ F
2 d
2
(.122kg x .2m + .0714kg x .1m)/ (.221 kg) = .142 m
= 1.19 N x .2 m + .699 N x .1 m = .308 N
τ cw
= F
3 d
3
= 2.17 N x .140 m = .304 N

Conclusion:
In all of our calculations in this lab, our results were always within less than 1 percent difference with an average percent difference of 0.416%. This clearly indicates that our method of experimentation was quite accurate. Another reason for this level of accuracy is that there is very little room for error in order to get the ruler to balance without tipping towards either side. Nevertheless, there was still error. This can be attributed to the fact that, at times, even though the ruler is perfectly balanced and is not tipping on either side, the ruler rests at a slight angle - even though it is not moving at all. This could be due to the inconsistency of the density of the wood that the ruler is made of; which is another reason why in step 1 of the procedure the ruler did not quite balance exactly at o.500 m on the meter stick.
When comparing our calculations, we did not compare net torque to our calculated torques, since our net torques were assumed to be zero. Instead, we compared our calculations to the supposed length/s or at times mass/es to determine the difference between our calculations and the actual or experimental results. This was the basis of our calculations for the percent difference in this lab.
Using torque, we were able to complete this lab and all of the calculations it required -everything from determining the center of gravity of a system of masses to the conditions for rotational equilibrium of a rigid bar (i.e. ruler).