PHY11L A4 E204

E204: TORQUE: SECOND CONDITION OF EQUILIBRIUM FRISNEDI, Nadine T.

OBJECTIVE Torque, which is developed by Archimedes, is the ability of force to change the rotational motion of a particle and an influence to change the rotational motion of an object. For a body to be in equilibrium the sum of all the torques acting on it, clockwise and counter clockwise, should be zero. Equilibrium implies a state of balance. Its second condition states that the net torque acting on the body should be zero for angular acceleration to be zero. The purpose of this experiment is to study the principles of torque through the application of Newton’s second condition of equilibrium. The students were tasked with obtaining the weight and forces of certain apparatuses through the analysis of equilibrium so as to practice and understand more clearly the significance of torque in the process. In order to evaluate their findings, the students were prompted to compare their gathered data with actual values through the computation of the percent differences. This relationship between torque and equilibrium is the main background of the experiment which was conducted. By the end of the experiment, it is expected for students to know the second condition of equilibrium. They will learn how the second condition affects an object or a body. They will also learn how to apply the second condition in computing the unknown data in the experiment. Through this experiment, the students will gain more knowledge and appreciation about the concepts on torque and how different is first condition of equilibrium to the second one. Students will also appreciate the concept of second

condition of equilibrium and how it is important in studying Physics. MATERIALS AND METHODS The performed experiment used the following materials and equipment which are: two pieces of weight pans, a model balance, a protractor, a meter stick, a spring balance, set of weights and an electronic weighing scale.

Figure 1. The materials and equipment used in the experiment.

Before conducting the experiment, the table should be made stable, stationary and leveled. The model balance, meter stick, and the weight pans were the main materials for this experiment. The model balance was set-up based on the figure given on the laboratory manual wherein the axis of rotation passes through the center of gravity of the beam. Since there is a missing nail on the beam, in which there is hole left on it to be used later on the experiment, it is necessary to put a piece of paper, it can be rolled so that it fits on the last empty hole on the right side of the beam. Adjust the amount of that piece of paper until the beam is balanced.

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The L1 is the distance between the pan, P1 and the axis of rotation, L2 on the other hand is the distance between the pan, P2 to the axis of rotation. L1 and L2 was measured using the meter stick.

Figure 2. Set-up for the determination of the weight of the pans.

After the set-up is done, a 10-gram weight is considered to be the W1, was placed on the pan, P1. The pans are to be placed on the beam in which the hooks or nails can be used to hang the pans. The pan, P1 is positioned on the right side of the beam while the pan, P2 is positioned on the left side of the beam. The two pans were placed on the beam while it is made sure that the beam becomes horizontally orientated or be seen as balanced in both of its sides which states that the system is in equilibrium. It is not necessary that the pans must be placed on the hooks or the nails, they can be hanged on top of the body of the beam inwardly to make the pans become balanced.

Figure 4. Measuring L1 using the meter stick.

Figure 5. Measuring L2 using the meter stick.

After that, the 10g weight was taken off from the pan, P1. A weight of 5g, which is considered the W2, was placed on P2. The two pans were placed again on the beam for the system to be in equilibrium.

Figure 3. P1 with a W1 and P2 in equilibrium.

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The angle of inclination of the spring balance was measured using the protractor, which is less than 90 degrees in which the beam was kept in horizontal position. The reading of the spring balance was recorded and was marked as the F(Measured).

Figure 6. P1 and P1 with W2 in equilibrium.

The L3 this time is also the distance between the pan, P1 and the axis of rotation, L4 on the other hand this time is the distance between the pan, P2 to the axis of rotation. Using the meter stick, L3 and L4 were measured. The previous procedure was repeated for the second and the third trial however the amount of weights placed on the pans were different. After conducting all the trials, the mass of P1 and P2 was computed for each trial. For the second part of the experiment, a weight of 50g is considered as W1, was placed on P1 which is at the left side of the beam. The spring balanced was also placed on the left side of the beam in a manner that will make the beam balanced.

Figure 8. Determining the angle of inclination of the spring balance.

The distance of the pan, P1 from the axis of rotation is measured as the L1 and the distance of the spring balance from the axis of rotation was measured and was marked as L2. The force exerted by the spring balance was measured using the second condition of equilibrium. The procedures were repeated for the second trial but the spring balance was placed at the right side of the beam. For the third part of the experiment, the second hole on the beam was used as the axis of rotation. The weight 50g was, which is again considered as the W1, was placed on the pan, P1. The location of pan was adjusted for the system until equilibrium is achieved. The distance of P1 from the axis of rotation was measured as the L1. The distance from the axis of rotation to Wb, which is the previous axis of rotation from the first two parts of the experiment, was measured and marked as the L2.

Figure 7. Set-up for determining the force needed for equilibrium

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The previous procedure was repeated for the second and the third trial however the amount of weights placed on the pans were different. For the second trial, the W1=60g while on the third trial, the W1=70g were used. After this the weight of the beam was computed using the second condition of equilibrium. OBSERVATIONS AND RESULTS

Figure 9. Set-up for determining the weight of the beam.

In the first part of the experiment, the L1, L2, L3, L4, P1(Computed), and P2(Computed) were needed. The weights were already given. The L1 and L2 was measured by the distance of the P1 with W1 of P2 to the axis of rotation. The L3 and L4 was measured by the distance of the P1 and P2 with W2 to the axis of rotation. The P1(Computed) and P2(Computed) were computed using the elimination method of two equations from the procedures. The average weight of pans, were obtained by getting the average of P1 and P2 in the three trials. The percent difference for both of the pans were computed in which the actual value of the pans was the first variable while the average experimental weight was the second variable.

Table 1. Determining the Weight of the Pan Actual Value of pan 1, P1 = 24.8g Actual Value of pan 2, P2 = 24.8g

Figure 10. Measuring L1 using the meter stick.

TRIAL 1

2

3

W1= 10g W2= 5g W1= 15g W2= 25g W1= 30g W2= 20g

L1

L2

L3

L4

17.7 cm

24.7 cm

21.1 cm

17.7 cm

10.2 cm

16.3 cm

19.9 cm

10.1 cm

10.1 cm

22.1 cm

18.2 cm

10.1 cm

Figure 11. Measuring L2 using the meter stick.

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TRIAL

1

2 3

P1

P2

(COMPUTED)

(COMPUTED)

25.59g

25.5g

W1 = 10g W2 = 5g

25.5 + 25.39 + 25.17 3 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃2 = 25.35𝑔

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃2 =

Percent Difference |𝐸𝑉1 − 𝐸𝑉2 | 𝐸𝑉 + 𝐸𝑉2 ( 1 ) 2 |24.8 − 25.41| % 𝑑𝑖𝑓𝑓 𝑃1 = 24.8 + 25.41 ( ) 2 %𝑑𝑖𝑓𝑓 𝑃1 = 2.43% % 𝑑𝑖𝑓𝑓 𝑃1 =

W1 = 15g W2 = 25g

25.58g

25.39g

W1 = 30g W2 = 20g

25.06g

25.17g

Average Weight of P1 = 25.41g Average Weight of P2 = 25.35g

|𝐸𝑉1 − 𝐸𝑉2 | 𝐸𝑉 + 𝐸𝑉2 ( 1 ) 2 |24.8 − 25.35| % 𝑑𝑖𝑓𝑓 𝑃2 = 24.8 + 25.35 ( ) 2 %𝑑𝑖𝑓𝑓 𝑃2 = 2.21% % 𝑑𝑖𝑓𝑓 𝑃2 =

Percent Difference for P1 = 2.43% Percent Difference for P2 = 2.21% Sample Computations: Getting the P1(Computed) and P2(Computed) for the first trial: (𝑃1 + 𝑊1 )𝐿1 = (𝑃2 𝐿2 ) (𝑃2 + 𝑊2 )𝐿4 = (𝑃1 𝐿3 ) 𝑃2 =

𝑃2 =

𝐿1 (𝑊2 𝐿4 + 𝑊1 𝐿3 ) 𝐿2 𝐿3 − 𝐿1 𝐿4

17.7((5)(17.7) + (10)(21.1)) ((24.7)(21.1) − (17.7)(17.7)) 𝑃2 = 25.5𝑔 𝑃1 =

(𝑃2 + 𝑊2 )𝐿4 𝐿3

(25.5 + 5)17.7 𝑃1 = 21.1 𝑃1 = 25.59𝑔

Average weight of pans: 25.59 + 25.58 + 25.06 3 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃1 = 25.41𝑔

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃1 =

For the second part of the experiment, the L1 and L2 was measured using the same procedure in the first part of the experiment by replacing the P 2 with the spring balance. The angle of inclination was measures using the protractor. The force F by the spring balance on the beam was computed using the second condition of equilibrium. The percent difference for both trials were computed in which the F(MEASURED) was the first variable and the F(COMPUTED) as the second variable. Table 2. Determining the Force Needed to be in Equilibrium TRIAL 1 2

L1 17.7 cm 17.7 cm

L2

W1+P1

7.3cm

74.8g

15.1 cm

74.8g

F

F

(COMPUTED)

(MEASURED)

1

224.18g

240g

6.82%

2

108.38g

90g

18.53%

TRIAL

%Diff

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Sample Computations:

TRIAL

wB (COMPUTED)

Getting the F(COMPUTED) for the first trial:

1

144.48g

Given: L1 = 17.7cm, L2 = 7.3cm, W1 = 50g, P1 =24.8g, θ=54°

2

144.04g

3

144.15g

wB (MEASURED)

137g

𝑊1 + 𝑃1 = 50𝑔 + 24.8𝑔 = 74.8𝑔 𝐹(𝐶𝑂𝑀𝑃𝑈𝑇𝐸𝐷)

Average Weight WB = Percent Difference =

(𝑃1 + 𝑊1 )𝐿1 = sin 𝜃 𝐿2

𝐹(𝐶𝑂𝑀𝑃𝑈𝑇𝐸𝐷) =

144.22g 5.14%

Sample Computations:

(74.8)17.7 (sin 54°)7.3

Getting the WB (COMPUTED) for the first trial:

𝐹(𝐶𝑂𝑀𝑃𝑈𝑇𝐸𝐷) = 224.18𝑔

Given: L1 = 14.1cm, L2 = 7.3cm, W1 = 50g, P1 =24.8g

Percent Difference for the first trial: F(MEASURED) = 240g |𝐸𝑉1 − 𝐸𝑉2 | % 𝑑𝑖𝑓𝑓 = 𝐸𝑉 + 𝐸𝑉2 ( 1 ) 2 |240 − 224.18| % 𝑑𝑖𝑓𝑓 = 240 + 224.18 ( ) 2 %𝑑𝑖𝑓𝑓 = 6.82%

𝑊1 + 𝑃1 = 50𝑔 + 24.8𝑔 = 74.8𝑔 𝑊𝐵 =

(𝑃1 + 𝑊1 )𝐿1 𝐿2

𝑊𝐵 =

(74.8)14.1 7.3

𝑊𝐵 = 144.48𝑔 For the third part of the experiment, the L1 and L2 was measured by getting the distance between P1 and Wb from the axis of rotation respectively. The weight if the beam was computed using the given formula. The average weight of pans, were obtained by getting the average of P 1 and P2 in the three trials. The percent difference for both trials were computed in which the WB(MEASURED) was the first variable and the average of the WB(COMPUTED) as the second variable.

Average weight of the beam: 144.48 + 144.04 + 144.15 3 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑊𝐵 = 144.22𝑔

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑊𝐵 =

Percent Difference Actual Value of 𝑊𝐵 = 137g % 𝑑𝑖𝑓𝑓 =

Table 3. Determining the Weight of the Beam TRIAL

L1

L2

W1+P1

1

14.1cm

7.3cm

74.8g

2

12.4cm

7.3cm

84.8g

3

11.1cm

7.3cm

94.8g

% 𝑑𝑖𝑓𝑓 =

|𝐸𝑉1 − 𝐸𝑉2 | 𝐸𝑉 + 𝐸𝑉2 ( 1 ) 2

|137 − 144.22| 137 + 144.22 ( ) 2

%𝑑𝑖𝑓𝑓 = 5.14%

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DISCUSSION & CONCLUSION Torque is a measure of how much a force is acting on an object causes that object to rotate. It is also called as the moment of force. On the experiment the model balance serves as the axis of rotation. Our data in the first table shows that we have determined the weight of the pan. The concept of torque and the second condition of equilibrium was used in this part. Based on our data, as the weight increases the P1 and the P2 also increases. It shows that as the force applied increases, torque also increases. Therefore, torque is directly proportional with the force applied on the object and is also dependent on the perpendicular distance of the applied force to the axis of rotation. Since the summation of torque in the body must be equal to 0, the clockwise torques must be equal to the counterclockwise torques, the P1 should be equal to P2 so that the beam will not rotate. The % difference that we got for the P1 and P2 are 2.41% and 2.35% respectively, which is small. Our data on the second table shows that we have determined the force exerted by the spring balance to the beam. The force needed for the system to be in equilibrium is greater when the angle is greater than zero but less than 90 degrees. As the angle is reaching zero, the system is reaching equilibrium. For this part of the experiment, we got a high percent difference. Maybe it is due to the wrong measurements of the distances and the angles and we assumed that the beam is balanced already. On the third activity we need to use the second hole in the beam as the axis of rotation. The data from the third table shows that we have determined the weight of the beam. The weight was computed using the concept of the second condition of equilibrium. The weight we have come up is quite close to the actual value and this means that we did the experiment properly.

In the three parts of experiment which finds the weight of the pan, force exerted and weight of the beam respectively, we noticed how torque is affected by the forces acting on the system and their radial distance from the axis and also, how the rotational equilibrium is applied. We have come up to the conclusion when second condition of equilibrium is satisfied, there is no angular acceleration and body will not be moving and will be in rotational equilibrium. Since all the parts of the experiment have been applied by the second condition of equilibrium and we have analyzed each part to get the unknown data, the first objective of second condition of equilibrium was fulfilled. Since we have applied the second condition of equilibrium, we have known its importance for solving the unknown and learning how to use it. This fulfills the second objective of the experiment. This makes our experiment successful since all the objectives were achieved The possible sources of error for this experiment are wrong judgement of balancing the beam and inaccurate measurement of the distances of the object to the axis of rotation. We might say that the beam is already balance but maybe it is not really balanced totally. If the beam is not balanced, it will affect our data since we applied the second condition of equilibrium. For the measurement of the distances, since they are all measured manually, there is a tendency to approximate the measurement since the object we are measuring are not stable. We could recommend to the students in the future that will also conduct this experiment is that they make sure that the beam is totally balanced and make sure that when measuring distances, it should be done properly and accurately. Also, doing a sub-trial per added weight is recommended to verify the measurements so that the data to be used will be of the least error.

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ACKNOWLEDGMENT & REFERENCE I would like to thank my groupmates for being so cooperative upon doing the experiment. Although it was a lot of pressure doing two experiments in one period they kept cool and relaxed even if time is really limited. I appreciate all of their efforts since without their initiative in doing the tasks assigned to them, our experiment will have a great chance of failure. I would also like to thank our professor, Prof. Ricardo F. De Leon, Jr. for guiding all throughout the experiment and for pointing out the things we should remember in conducting the experiment. I would also like to thank him for giving us additional points for our performance in this experiment. I would like to thank my friends, Vivi, Elijah, and Alvin for giving me ideas on how I should properly layout my report and how my ideas should flow. They were very nice when I ask for their help. I also would like to acknowledge the lab assistants for reminding us how to properly handle the materials and equipment they are lending us so that it would be easy for us to set it up and we won’t damage those materials. Lastly, I would like to thank my family for their never ending support and encouragement for me in my studies as I pursue my degree in Mapúa. They have been so understanding and I am so blessed to have them. Reference: Calderon, Jose C., (2000) College Physics Laboratory Manual, Mapúa Institute of Technology, Manila: Department of Physics.

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