Experiment 3: Conditions for EquilibriumLaboratory Report

Josemaria Castro, Maria Rosalina Coleto, Cherish Cruz, Krystal Grace Cruz, Jose Luis de Guzman

Department of Math and PhysicsCollege of Science, University of Santo Tomas

España, Manila Philippines

Abstract

1. Introduction

2. Theory

3. Methodology

The Conditions for Equilibrium experiment is composed of four activities each with different procedures and materials used. On the first activity which is the determination of Equilibrant Force, three pans of the force table were weighed and were labeled as A, B, and C, respectively. Pans A and B were hung at 30o and 200o marks on the force table, respectively. A 100g weight was placed on pan A and 150g weight on pan B. These were recorded as TA and TB. The two tensions in the strings were balanced by placing weight on pan C and adjusting its position in the force table. To know, if the tensions are balanced, the pin must be exactly at the center of the ring. The magnitude and position of the equilibrant were recorded and the theoretical equilibrant of the two tensions were computed by component method. Percent error was also computed using the values obtained by component method as the accepted value for magnitude as well as for direction. The next activity was about the First condition for Equilibrium. A cylinder of unknown weight was suspended by means

of two strings using the force board. A spring scale was attached to one of the strings. The string was pulled horizontally until the pin on the force board was exactly at the middle of the ring. The reading on the spring scale was recorded as T1. The angle that the other string made with the horizontal was measured and recorded as θ. A free body diagram of the ring was drawn. The tension T2 in the other string and the weight of the cylinder was computed, and the weight of the cylinder was obtained for the accepted value and the percent error was obtained. For the third activity which was Locating the Center of Gravity, no percent error was obtained. In this activity, a circle with a diameter of 10 cm and a square of side 10 cm was cut out from a cardboard. Their weights were obtained separately and recorded as WC

and WS, respectively. The center of gravity of the composite figure was determined by balancing method and plumb line method. The position of the center of gravity was specified using the leftmost side of the square as the y-axis and the bottom of the square as the x-axis. Computations were done using this formula:

x=xcw c+x sws

w y=

ycwc+ ysw sw

Lastly, the Second Condition for Equilibrium activity, an aluminum bar was the main

Θ = 32.0o

T3 = 200g

material. The center of gravity of the aluminum bar was located by balancing it on a pencil or other knife edge. The position of the center of the gravity was marked. The cylinder used in the previous activity was hanged 5.0 cm from one end of the bar. The aluminum bar was supported using the force board and by means of a spring scale on end and a string on the other end until the bar assumes a horizontal position. By this procedure, the forces acting on the bar were now balanced. A free body diagram of the bar was drawn. The second condition for equilibrium was used to determine the weight of the bar and the tension in the string. The theoretical weight of the cylinder was used in the computation and the weight of the bar obtained from the electronic gram balance for the accepted value. Percent error was computed.

4. Results and Discussion

Equilibrant forces

Tensions Magnitude (g)

Position (o)

TA 82.90 40o

TB 195.4 300o

Experimental Equilibrant

192.17 145

Theoretical Equilibrant

201.24 144.09o

% Error 4.51%Table 1 Data Summary for Equilibrant Forces Experiment

Table 1 shows the result of our first experiment. The two forces, A and B, that are acting was equalized by a force acting relatively opposite the direction of the two forces. This force, called the equilibrant force would make the sum of the three forces together equal to zero. In our case, the tensions mounted by A and B, with 82.90 grams at a 40o angle and 195.4 grams at a 300o

angle, respectively, required a theoretical equilibrant force of 201.24 grams at an angle of

144.09o (with all angles based on the positive horizontal axis). Experimentally, we obtained the equilibrant forces to be at 192.17 grams at roughly 145o. Relative to the magnitude, our data was off 4.51% from the theoretical equilibrant.

First Condition of Equilibrium

T1 (g) 320

Θ (o) 32.0o

T2 (g) 377

Experimental wt. (g) 200

Theoretical wt. (g) 200

% Error 0%

Table 2 Data Summary for First Condition of Equilibrium Experiment

Table 2 shows a summary of our results. T1

which was at an angle of 0o relative to the positive horizontal axis recorded a pull of 200 grams of force using the spring scale. The force of T2 which was the tension exerted by the string pulled at a recorded angle of 32.0o relative to the negative horizontal axis was solved and recorded to be 377 grams. Using the calculated values, the experimental weight of the metal cylinder used was recorded to be 200 grams. The theoretical weight obtained from the weighing scale was recorded to also be 200 grams (values expressed in 3 significant figures). Therefore, the % error in this experiment is 0%.

Locating the Center of Gravity

Method Center of Gravity

10cm10cm

7.94g 6.16g

Point of reference

220g

5cm 34.5cm

19.75cm

39.5cm

fgwc

X-coordinate Y-coordinate

Plumb line Method (cm)

9.50 5.00

Computation (cm)

9.37 5.00

Weight of square

7.94g Weight of

Circle

6.16g

Table 3 Data Summary for Locating the Center of Gravity Experiment

The plumb line method made use of a string that should theoretically consistently pass through a single point in an object if it would be made to hang in different positions. That point in the object where the string spawned from different positions pass is said to be the center of gravity of the object. This was measured from a single point of reference by ruler and was found to be 9.50 cm in the x-coordinate and 5.00 cm in the y-coordinate (based on the point of reference shown in figure 1). The computed value of the center of gravity of the combined objects was at 9.37cm in the x-axis and 5.00cm in the y-axis based on the same point of reference. The data confirms the concept of the plumb line method regardless of the negligible difference in the computation for the x-axis wherein the plumb line method is therefore limited by the measuring instrument used, i.e., the accuracy of the ruler.

Second Condition of Equilibrium

Reading of Spring Scale

220g

Wt. of Cylinder 202g

Tension in the String 119g

Experimental wt. of Bar

87.14g

Theoretical wt. of Bar 79.73g

% Error 9.29%

Table 4 Data Summary for Second Condition of Equilibrium Experiment

The result of the fourth experiment is summarized above in table 4. The reading of the spring scale was recorded at 220 grams which is the pull at that end of the aluminum bar with a 202 grams cylinder hanging on. The tension of the string was calculated to be at 119 grams while the experimental weight of the bar was calculated to be at 87.19g. The aluminum bar was recorded to actually be 79.73 grams resulting into a 9.29% error in our data. The error in the data could have come from inaccurately placing the cylinder 5cm from the leftmost side of the aluminum bar.

5. Conclusion

6. Applications

1.) State the first condition for equilibrium. If a body is in equilibrium, are there no forces acting on it?

Figure 1 Illustration of Object Used

The first condition of equilibrium is when a body at rest or moving with uniform velocity has zero acceleration.

4.) Devise a way by which you could determine your center of gravity.

The center of gravity (CG) is the center of an object's weight distribution, where the force of gravity can be considered to act. It is the point in any object about which it is in perfect balance no matter how it is turned or rotated around that point. A person could recline on a piece of lumber for about the person’s height and size. The person must execute two directions: parallel and perpendicular to the board and would move from one side to the other. To determine the center, the point where the body is in equilibrium in each axis would be it. The two axes will meet at a point and that certain point will be the center of gravity.

5.) In general, the women’s centers of gravity tend to be lower than men’s. Can you explain why?

Based on some research, it says that the lower body of women is generally heavier in comparison to their whole body, as opposed to the lower body of men. This would make the center of gravity of women slightly lower, because of their wider hips and heavier bone structure in the lower abdominal part of the skeleton. Men in general have wider shoulders as compared to women.

7. References