Experiment 4: Conditions for Equilibrium Laboratory Report Nathaniel Baluyut, Kate Baroro, Camille Bausa, Patricia Benito, Jamiellie Chua Department of Speech-Language Pathology College of Rehabilitation Sciences, University of Sto. Tomas España, Manila Philippines
Abstract The condition of a system when neither its state of motion nor its internal energy tends to change with time is called equilibrium [1]. Any object in equilibrium is acted upon by a force (or a number of forces) called the resultant which is balanced by its equilibrant force. In this experiment, the members of the group tackled on the conditions for equilibrium, equilibrant force and center of gravity.
Since a body is in equilibrium, there are forces that cancel out to remain stable. The equilibrant force is a force that is equal to the resultant force in magnitude but opposite in direction. Relatively, there are factors that affect equilibrium. An object’s state of equilibrium is affected by the center of gravity. The center of gravity of an object is located at the point from where all the weight of the object is considered concentrated.
1. Introduction In Physics, equilibrium is defined as the state of balance. A body that is in uniform motion is said to be in dynamic equilibrium while a body at rest is assumed to be in static equilibrium. In order to know whether a body is in equilibrium, two conditions must be met. The first condition for equilibrium states that an object is in equilibrium if the resultant of the forces acting upon it is equal to zero. The second condition for equilibrium states that for a body to be in equilibrium, all clockwise torques are equal to all counter clockwise torques. Torque is the quantity that measures how effectively a force causes acceleration. It is defined as the product of the force and the lever arm. [2]
The objectives of this experiment were to: 1) determine the equilibrant force using the force table and component method 2) determine the unknown forces using the first and second conditions for equilibrium 3) locate the center of gravity of a composite body and 4) demonstrate rotational equilibrium. 2. Theory When an object is balanced by a force, the balancing force is said to be the equilibrant force. In Activity 1, the concept of equilibrant forces is made use of. Given the conditions of the experiment, the equilibrant force was computed using the component method. Specifically, substituting variables and values, the
equations for finding the coordinates of the resultant force are x TA cos 30° TB cos 200°
y TA sin 30° TB sin 200°
∑
∑
The magnitude of the equilibrant is found ∑ using the equation: √ ∑ | | and the resultant’s position (or angle) is ∑ calculated using the formula: |∑ | Since the equilibrant force is opposite in direction to the resultant, the angle computed from the equation is then added to 180° to find the angle of the equilibrant.
Since sin0° = 0 and sin 270° = -1, the equation for finding the weight of the cylinder becomes Wc = T2 sinϴ2 where Wc is the experimental weight. Using an electronic gram balance, the theoretical weight of the cylinder was found. Together with the experimental weight, the %error was computed following the formula: |
.
Using the first condition for equilibrium, unknown values were found in Activity 2 of the experiment. Based from the formula, ∑ , this equation was derived: T1 cos0° + T2 cosϴ2 + Wc cos270° = 0 where T1 is the force or tension represented in the spring balance, T2, an unknown quantity, is the tension produced in pulling the string, and Wc is the weight of the cylinder and an unknown quantity. Since cos0° is equal to 1 and cos270° is equal to 0, the final derived equation for finding the value of T2 is: T2 = Deriving from the formula, ∑ , the equation for the weight of the cylinder was found: T1 sin0° + T2 sinϴ2 + Wc sin270° = 0
|
The center of gravity may be obtained using the balancing method and the plumb line method. In balancing method, the object is placed on one finger while the person balances it. The point at which the finger points to is the center of gravity. In plumb line method, the object is suspended at one edge with the slightest contact possible. A vertical line is then drawn from the point of contact. The process is repeated once again and the point of intersection of the two lines is the center of gravity. In Activity 3, whereby the goal is to locate the center of gravity of a square and circle attached together whose diameter are both 10cm, the following formulas were used:
̅
and ̅
where xc and yc are the coordinates of the CG of the circle, xs and ys are the coordinates of the CG of the square, ̅ and ̅ are the coordinates of the center of gravity of the composite figure. Wc and Ws are the weight of the circle and square, respectively.
In Activity 4, in order to find the weight of the aluminium bar, the concept of the second condition of equilibrium was used. In symbols, ∑ . Deriving from this formula, the equation for finding the weight of the bar is,
where Tss is the tension measured by the spring scale, lss is the lever arm of the spring scale, Wc is the weight of the cylinder, lc is the lever arm of the cylinder and lb is the lever arm of the bar. 3. Methodology The materials used in the experiment were force table and accessories, force board, cylinder of unknown weight, spring scale, electronic gram balance, card board, aluminum bar, unknown weight, and protractor. In Activity 1, three pans of a force table were weighed individually and labeled Pan A, B, and C. Pan A was positioned at 30° mark while Pan B was at 200° mark on the force table. A 100-gram and 150-gram weight was added to Pan A and Pan B, respectively. The weight of the pan with the weight added is the tension acting on the string and these tensions were recorded as TA and TB. The tension in the strings were balanced by adding a weight on Pan C and/or moving its position along the force table. The tension was balanced when the pin was exactly at the center of the ring. The magnitude or position of Pan C or the equilibrant was recorded as the experimental
equilibrant. The theoretical equilibrant was determined using the component method. In Activity 2, a cylinder of unknown weight was hung on the force board using two strings. A spring scale was attached to one end of one string and the other string was pulled to balance the forces until the pin on the board was exactly at the middle of the ring and the angle made was recorded as Ɵ. The reading on the spring scale was recorded as T1. Using T1, the tension T2 in the tension in the other string and the weight of the cylinder was determined. The cylinder was weighed for the theoretical value. In Activity 3, a circle with a diameter of 10cm and a square with a side of 10cm were cut from a cardboard. Using electronic gram balance, the weight of the square and the circle was determined and recorded as WS and WC, respectively. Points on the x and y axes were plotted. Using the balancing method, the center of gravity must be determined by balancing the cardboard on one finger and estimate its location along the x and y axes. Another method to know the center of gravity is the plumb line method wherein the figure was held with the slightest contact and two vertical lines were traced, the intersection of the lines would determine the center of gravity. Obtaining the points, the experimental results were compared with the actual computation of center of gravity. In Activity 4, the weight of an aluminum bar was to be determined by placing a cylinder with a mass of 70 grams 0.05 m from the left end of the bar (LC) and a spring scale at the right end of the bar. The spring scale was hung on the other end to
balance the bar and acquire a horizontal position. The second condition for equilibrium was used to determine the weight of the bar. The bar was weighed using the electronic gram balance and was compared with the theoretical weight of the bar. 4. Results and Discussions Table 1 shows the magnitude, position, experimental equilibrant, theoretical equilibrant of the weights in the pan and its resulting percentage error acquired in Activity 1. As the table presents, the pans A and B along their respective weights of 100g and 150g produced a resultant force equal to the equilibrant. Through measuring and component method, two values, theoretical and experimental, were produced with a %error of 15.62% for the magnitude and 4.74% for the position. Probable errors committed by the group were systematic and gross errors. It was possible that the computation and the variation in measurements led to the %error. Table 1. Magnitude, Position, Experimental Equilibrant, Theoretical Equilibrant and Resulting Percentage Error acquired in Activity 1. Tensions Magnitude Positions (N) (o) TA 1.512 30 TB 1.965 200 Experimental 0.459 163 Equilibrant Theoretical 0.544 171.11 Equilibrant % error 15.62% 4.74%
Table 2 shows the magnitude, position, experimental weight, theoretical weight of the strings and its resulting percentage error acquired in Activity 2. Applying the first condition for equilibrium, the value for the tension upon the second string is 3.46 N. This means that three forces caused the ring to be in the center (or in equilibrium): a 3.33N force in line with the +x-axis, a force of 3.46N at an angle of 164 relative to the x-axis° and a cylinder whose theoretical weight is 0.98 N. Experimentally, the weight of the cylinder is 0.95N which produces a %error of 3.06%. Table 2. Magnitude, Position, Experimental Weight, Theoretical Weight and Percentage Error acquired in Activity 2. T1 (N) 3.33 Free Body Diagram of ring ϴ(o) 164 T2 (N)
3.46
Experimental 0.95 Weight (N) Theoretical 0.98 Weight (N) % error 3.06%
Table 3 shows the data gathered in Activity 3.Three methods were used to find the center of gravity of the composite figure (or “squircle”): plumb line method, balancing method, and by computation. Upon finding the weight of each figure, the formula for the computation was followed to yield the coordinates for the center of gravity. As the results show, both the plumb line and balancing method yielded results close to the ones by computation.
Table 3. X-coordinates and ycoordinates of the center of gravity of the “Squircle” body using the Plumb line Method, Balancing Method and Computation Method in Activity 3. Method Center of Gravity x-coordinate y-coordinate Plumb line 9.2cm 5.7cm Method Balancing 9.5cm 6.9cm Method Computation 9.42cm 5cm In Table 4, the reading scale, weight of the cylinder, experimental and theoretical weight of the bar, and % error were determined. As shown, theoretical and experimental weight are precisely calculated relatively to each other. With that, the percent error had been computed as 4.04%. The said % error were possibly brought about by miscalculations, mistakes in measuring and in obtaining the mass of the regularly shaped bar as well as its weight. Table 4. Reading of the Spring Scale, Weight of the Cylinder, Experimental and Theoretical Weight of the Bar and %Error Reading of Spring Scale (N) Weight of the Cylinder (N) Experimental Weight of the Bar (N) Theoretical Weight of the Bar (N) % Error
6.86 19.73 0.882 0.8477 4.04%
5. Conclusions Equilibrium is achieved only when two conditions are satisfied. The first condition for equilibrium states that the
resultant of the forces acting upon a body is zero. Meanwhile, the second condition for equilibrium states that the clockwise moment and the counter-clockwise moment are equal to each other. In the first activity, the equilibrant was determined using a force table and three (3) strings placed in different positions with different weights on them. The component method was used to determine the theoretical equilibrant. In the second activity, the weight was determined using methods similar to Activity 1.With great care and focus, both activities were performed with significant precision. It is therefore concluded that there is a force responsible for equilibrium: the equilibrant force. In the third activity, the center of gravity, the point where the weight of a body is concentrated, was determined using 3 methods: Plumb Line Method, Balancing Method, and Computation Method. All methods were proven to be effectively precise.
6. Applications 1) State the first condition for equilibrium. If a body is in equilibrium, are there no forces acting on it? The first condition for equilibrium states that a body is in equilibrium if the resultant of the forces acting on the body is zero. A body which is in equilibrium does not mean that there are no forces acting upon it but rather a number of forces that counter each other producing zero resultant.
2) The Russell traction system below is used for a fractured femur. Identify the force needed to immobilize the femur. What will supply this force?
4) The human body is a mechanical system designed by God and consists mostly of levers. Give examples of these “human levers.”
forearm arm ankle (tendon of Achilles) leg (lower)
7. References [1]Britannica. (n.d.). Equilibrium. Retrieved
January 19, 2013, from Britannica Encyclopedia: http://www.britannica.com/EBchecked/topic /190745/equilibrium
3) What happens to the center of gravity of a person under the following situations: (a) His upper right extremity is amputated. (b) He carries all his books using the right arm only. Theoretically, the center of mass of an average adult man is at the second sacral vertebra. (a) If a man’s upper right extremity is amputated, his CG would move to the left since his weight is now more concentrated on the left side of the body. (b) If a man is carrying all his books using the right arm, his center of gravity shifts to the right where the weight becomes more concentrated.
[2]Padua, A. L., & Crisostomo, R. M. (2007). Practical and Explorational Physics Modular Approach. Quezon City: Vibal Publishing House.
