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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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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 3. P1 with a W1 and P2 in equilibrium.

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 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.

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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 7. Set-up for determining the force needed for

equilibrium

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 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.

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

Figure 10. Measuring L1 using the meter stick.

Figure 11. Measuring L2 using the meter stick.

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

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

TRIAL L1 L2 L3 L4

1 W1= 10g

17.7

cm

24.7

cm

21.1

cm

17.7

cm W2= 5g

2 W1= 15g

10.2

cm

16.3

cm

19.9

cm

10.1

cm W2= 25g

3 W1= 30g

10.1

cm

22.1

cm

18.2

cm

10.1

cm W2= 20g

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TRIAL P1

(COMPUTED)

P2

(COMPUTED)

1

W1 = 10g 25.59g 25.5g

W2 = 5g

2 W1 = 15g

25.58g 25.39g W2 = 25g

3

W1 = 30g 25.06g 25.17g W2 = 20g

Average Weight of P1 = 25.41g

Average Weight of P2 = 25.35g

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 = (𝑃2𝐿2)

(𝑃2 + 𝑊2)𝐿4 = (𝑃1𝐿3)

𝑃2 =𝐿1(𝑊2𝐿4 + 𝑊1𝐿3)

𝐿2𝐿3 − 𝐿1𝐿4

𝑃2 =17.7((5)(17.7) + (10)(21.1))

((24.7)(21.1) − (17.7)(17.7))

𝑃2 = 25.5𝑔

𝑃1 =(𝑃2 + 𝑊2)𝐿4

𝐿3

𝑃1 =(25.5 + 5)17.7

21.1

𝑃1 = 25.59𝑔

Average weight of pans:

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃1 = 25.59 + 25.58 + 25.06

3

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃1 = 25.41𝑔

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃2 = 25.5 + 25.39 + 25.17

3

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃2 = 25.35𝑔

Percent Difference

% 𝑑𝑖𝑓𝑓 𝑃1 =|𝐸𝑉1 − 𝐸𝑉2|

(𝐸𝑉1 + 𝐸𝑉2

2)

% 𝑑𝑖𝑓𝑓 𝑃1 = |24.8 − 25.41|

(24.8 + 25.41

2)

%𝑑𝑖𝑓𝑓 𝑃1 = 2.43%

% 𝑑𝑖𝑓𝑓 𝑃2 =|𝐸𝑉1 − 𝐸𝑉2|

(𝐸𝑉1 + 𝐸𝑉2

2)

% 𝑑𝑖𝑓𝑓 𝑃2 = |24.8 − 25.35|

(24.8 + 25.35

2)

%𝑑𝑖𝑓𝑓 𝑃2 = 2.21%

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 P2

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 L1 L2 W1+P1

1 17.7

cm 7.3cm 74.8g

2 17.7

cm 15.1 cm 74.8g

TRIAL F

(COMPUTED)

F

(MEASURED) %Diff

1 224.18g 240g 6.82%

2 108.38g 90g 18.53%

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Sample Computations:

Getting the F(COMPUTED) for the first trial:

Given: L1 = 17.7cm, L2 = 7.3cm, W1 = 50g,

P1 =24.8g, θ=54°

𝑊1 + 𝑃1 = 50𝑔 + 24.8𝑔 = 74.8𝑔

𝐹(𝐶𝑂𝑀𝑃𝑈𝑇𝐸𝐷) =(𝑃1 + 𝑊1)𝐿1

sin 𝜃 𝐿2

𝐹(𝐶𝑂𝑀𝑃𝑈𝑇𝐸𝐷) =(74.8)17.7

(sin 54°)7.3

𝐹(𝐶𝑂𝑀𝑃𝑈𝑇𝐸𝐷) = 224.18𝑔

Percent Difference for the first trial:

F(MEASURED) = 240g

% 𝑑𝑖𝑓𝑓 =|𝐸𝑉1 − 𝐸𝑉2|

(𝐸𝑉1 + 𝐸𝑉2

2)

% 𝑑𝑖𝑓𝑓 = |240 − 224.18|

(240 + 224.18

2)

%𝑑𝑖𝑓𝑓 = 6.82%

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 P1 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.

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

TRIAL wB (COMPUTED) wB (MEASURED)

1 144.48g

137g

2 144.04g

3 144.15g

Average Weight WB =

Percent Difference =

144.22g 5.14%

Sample Computations:

Getting the WB (COMPUTED) for the first trial:

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

P1 =24.8g

𝑊1 + 𝑃1 = 50𝑔 + 24.8𝑔 = 74.8𝑔

𝑊𝐵 =(𝑃1 + 𝑊1)𝐿1

𝐿2

𝑊𝐵 =(74.8)14.1

7.3

𝑊𝐵 = 144.48𝑔

Average weight of the beam:

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑊𝐵 = 144.48 + 144.04 + 144.15

3

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑊𝐵 = 144.22𝑔

Percent Difference

Actual Value of 𝑊𝐵 = 137g

% 𝑑𝑖𝑓𝑓 =|𝐸𝑉1 − 𝐸𝑉2|

(𝐸𝑉1 + 𝐸𝑉2

2)

% 𝑑𝑖𝑓𝑓 = |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.