Torque Lab

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Analyzing the Effect of Mass and Position on Rotational Torque at Translational and Rotational Equilibrium Mahika Tyagi Partners: Neha Polavurapu and Kanussh Jain February 7 th , 2022 Dr. Crowe AP Physics C, Academy of Science

ABSTRACT This experiment aimed to explore the relationship between position ( x ), mass ( m ), and rotational torque ( τ ) at equilibrium through the analysis of a meter stick-fulcrum and meter stick-force sensor apparatus. Through a series of successive components, the relationship between center of mass and center of gravity were determined to be equal to one another. This claim was then substantiated through a series of experiments and successive analyses using percent difference and percent error. This claim was then used to calculate the torques of various positions of acting forces on the meter stick-fulcrum apparatus and the meter stick-force sensor apparatus. The purpose of part one was to determine the center of mass of the meter stick at equilibrium with no additional mass. The purpose of part two was to analyze the relationship between force and distance of the acting force from the center of equilibrium and the rotational torque. In order to model the relationship between position, mass, and torque, a mathematical model was derived and created. The model was then explored to be viable through a series of calculated percent differences. The purpose of part three was to determine if the center of mass remained constant even with the addition of various increments of imbalanced forces, and the purpose of part four was to determine if the rotational torques of a system at equilibrium were impacted by various points of rotational origin. Using the derived mathematical model and the calculated percent differences in part two ( D 2 ), it was concluded that the product of mass and distance from the origins were equal on opposing sides at horizontal equilibrium. Using the calculated percent error in part three ( E 3 ) , it was determined that the center of mass remains the center of gravity at equilibrium even as the mass of the system is changed in imbalanced increments. Using the calculated percent differences in part four ( D 4 , D E , D 0 , D m2 ) it was determined that the sum of the forces oriented upwards and downwards are equal at translational, horizontal equilibrium, and that the sum of the clockwise and counterclockwise torques (or torques oriented in opposing rotational directions) are equal in rotational equilibrium. The measures of accuracy

and precision of the data displayed a high congruence of data and, thus, it was concluded that there was minimal impact of the experimental errors in the validity of the results; however, there were a few systematic and random errors present throughout the experiment that could have impacted the collection of data. Some systematic errors include the inaccuracy of the tick marks on the wooden meter sticks, the lack of equilibrium of the meter stick during set-up and data collection, and the imperfect leveling of the scale before collecting the values of mass. Some random errors include the estimation of the hundredth centimeter (reliance on resolution), the inconsistency in density of mass of the wooden meter stick, random gusts of wind present during the set-up of the apparatus and warping of the balanced surface. To fix these potential sources of error, future experiments utilizing this procedure could be one in controlled, automated environment where measure of position and mass is more precise and accurate, and the risk of warping of a balanced surface is eliminated.

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MATERIALS AND METHODS In order to model the relationship between rotational torque and equilibrium, an experimental design was created in order to allow for the manipulation of mass ( m ) and position ( x ) using a fulcrum and meter stick apparatus, modeling a series of shifts in the center of gravity using metal blocks ( b ). In order to accurately represent the relationship between torque and equilibrium, the procedure of this experiment occurred in a series of four successive parts that enabled the use of assumptions, substantiated by the preceding sections, to be used in successive parts of the lab. In each part, blocks of different metallic materials and mass were utilized to alter the mass of the system when necessary. To replicate a model of horizontal equilibrium, a wooden meter stick 1 and fulcrum apparatus was used to allow for alterations in the position of equilibrium. In this experiment, the mass of the system and positions of the blocks and meter sticks were altered, and the position of the fulcrum and metal blocks at equilibrium were recorded. To set up the fulcrum and meter stick apparatus used in parts one through three, a metal fulcrum 2 was placed upon a flat surface (in this case, it was a table). Next, a metal clamp 3 containing a set screw was placed around the center of a wooden meter stick, ensuring that the screw was facing downwards to aid in measuring the position of the clamp and setting the meter stick at horizontal equilibrium (when the screw is facing upwards, it acts against the downward force placed on the meter stick by gravity and creates more difficulty when setting the meter stick at horizontal equilibrium). The clamp was then placed onto the center of the fulcrum using its jutting wedge, and the set screw of the central clamp was tightened to the point where the position of the meter stick was adjustable in small increments. The first component of this experiment was to observe the position of the meter stick at equilibrium with no additional mass attached to the system ( Figure 1 ). Thus, following the set-up of the apparatus, the 1 Sergeant Welch Meter stick 2 Sergeant Welch Metal Fulcrum 3 Sergeant Welch Hook and Set Screw Clamp

meter stick’s horizontal position was adjusted until it reached an equilibrium position, ensuring that no persons or external objects were leaning on the table during adjustment to prevent any warping of the balanced surface of the table. The position of the central clamp at equilibrium (termed x E ) was recorded by looking at the position of the central notch of the clamp on the meter stick. In this experiment, the position of the meter stick was both altered and observed for a total of one trial. The second component of this experiment was used to observe the effect of position and mass on the equilibrium of the meter stick apparatus. Three combinations of a pair of different metal blocks 4 were created to be observed. Each metal block was attached to a hooked clamp, and the masses of the clamp and block systems (termed m 1 and m 2 ) were measured using a scale 5 ( Figure 2 ). Before measuring the mass of each block-clamp system, the scale was leveled by adjusting the feet located on its bottom surface so that the leveling display on the back of the scale was centered. When measuring the masses of the varying block clamp systems, the scale was first tared, and each block and clamp was on the metal surface. The mass of each system was then read and recorded in grams and the material of the block in the system was noted. To conduct this component of the experiment, the block clamp systems of the two blocks were placed onto the meter stick (already at horizontal equilibrium), ensuring that the set screw and hook were oriented downwards, according to the combination determined for that trial. The apparatus was then adjusted by shifting the block and clamp systems until the meter stick reached horizontal equilibrium. The positions of the block-clamp systems (termed x 1 and x 2 ) were recorded by reading the notches on the meter stick through the jutting wedge of the clamps. In each trial, the combination of additional masses was varied (by varying the type of block attached to the meter stick) and the position of the block-clamp systems were observed in a total of three trials. 4 Sergeant Welch Specific Gravity Set Blocks (4/ST) 5 Mettler Toledo PL1502E

The third component of this experiment was utilized to observe the relationship between the center of mass of a system and equilibrium. To conduct this experiment, three metal blocks of various masses were utilized to alter the mass of the fulcrum and meter stick apparatus. To determine how much mass was added to the system from each block, a hooked clamp was attached to each of the blocks, and the mass of each block-clamp system was measured using a scale (ensuring that the scale was leveled and tared before each measurement) and the material of each block in the block-clamp system was recorded in each trial. Additionally, the mass of the meter stick (termed m m )–without the central clamp attached–was measured and recorded, leveling and taring the scale before use. Before conducting the experiment, the central clamp was attached roughly around the center of the meter stick, with the set screw facing down, and the meter stick was placed onto the fulcrum using the wedge of the clamp. The meter stick was then adjusted until it reached horizontal equilibrium (ensuring no persons or external objects were leaning on the balanced surface to avoid warping), and the position of the central clamp ( x E ) was recorded ( Figure 3 ). To start the experiment, each block-clamp system was attached to the meter stick (with the set screw and hook oriented downwards) and the position was kept constant throughout each trial. The position of the block-clamp system ( x 1 ) on the meter stick was recorded. The meter stick was then adjusted using the central clamp until the apparatus reached horizontal equilibrium. The position of the central clamp ( x 2 ) was recorded at equilibrium in each trial. In this component, the increment of additional mass was varied and the position of the fulcrum at equilibrium was observed for a total of three trials. The fourth component of this experiment was utilized to observe the relationship between position, mass, and varying centers of rotation at equilibrium. A hanging meter stick fulcrum apparatus was utilized in this experiment to allow for analysis of varying origins of rotation, and two blocks of different material and mass were chosen to use for the alteration of the mass of the system. To set up the apparatus, a ring stand was placed onto a balanced surface (a table). A right-angled clamp was then

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attached to the top of the ring stand, and a Vernier GoDirect force sensor 6 was placed perpendicular to the clamp, making sure that the hook on the sensor was oriented downwards. The force sensor was then paired to a recording device 7 via Bluetooth by opening the Vernier Graphical Analysis application 8 , clicking the ‘New Experiment’ button, and then clicking the ‘Devices’ button. To pair the device, the ID of the force sensor was selected and connected to the recording device. The force sensor was then zeroed by pressing the ‘Force’ button in Vernier Graphical Analysis, and then pressing the ‘Zero’ option in the dropdown menu. Next, a central clamp containing a hook was placed onto a meter stick (with the set screw facing downwards and hook oriented upwards) and the mass of the meter stick and clamp system ( m mc ) was measured using a scale (ensuring that the scale was leveled and tared before measurement). Both the mass of the block-clamp system and the corresponding material of the block were recorded. The meter stick was then attached to the force sensor via the hook on the clamp, and the position of the clamp was adjusted until the meter stick reached horizontal equilibrium. The position of the central clamp at equilibrium ( x E ) was recorded. Next, the two chosen metal blocks of differing mass were attached to hooked clamps, and the mass of the resulting block and clamp systems ( m 1 and m 2 ) were measured using a scale (ensuring that the scale was leveled and tared before use). The blocks were then attached to the meter stick, ensuring that the set screw and hook were oriented downwards. To conduct the experiment, the positions of the blocks were adjusted until the meter stick reached horizontal equilibrium in each trail. The positions of each block and clamp system ( x 1 and x 2 ) were recorded. Once the meter stick reached horizontal equilibrium, the tension force of the system was recorded by pressing the ‘Collect’ button on the graphical analysis software, which recorded the tension force of the system for an automatic period of two seconds. The resulting tension force generated by the software read and recorded (termed F ). This 6 Vernier GoDirect Force and Acceleration Sensor 7 iPhone 13 Pro 8 Version 5.14.0

procedure was repeated in each trial for a total of three trials, changing the initial position of the blocks each time and observing the positions of the blocks at equilibrium. Figure 1: Meter stick and Fulcrum apparatus set up for Part One Figure 2: Meter Stick, Fulcrum, and Metal Block apparatus set up for Part Two Figure 3: Meter stick, Fulcrum, and Metal Block apparatus set up for Part Three

DATA Table 1: Position of central clamp of meter stick at horizontal equilibrium with no additional mass (Part 1) Figure 4: Meter stick, Ring Stand, and Force Sensor apparatus set up for Part Four

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Trial # x E (cm) 1 50.11 Table 2: Position of central clamp of meter stick at horizontal equilibrium with no additional mass (Part 2) Trial # x E (cm) 1 50.11 Table 3: Measurements of position and mass of block-clamp systems at horizontal equilibrium (Part 2) Table 4: Position of central clamp of meter stick at horizontal equilibrium with no additional mass (Part 3) Trial # x E (cm) 1 50.21 Table 5: Measurements of position of block-clamp system and central clamp at horizontal equilibrium and mass of block-clamp system and meter stick without additional mass (Part 3) Table 6: Measurements of position of central clamp of meter stick at horizontal equilibrium with no additional mass, along with the mass of the meter stick with the central clamp attached (Part 4) Trial # b 1 b 2 m 1 (g) m 2 (g) x 1 (cm) x 2 (cm) 1 Aluminum Brass 103.19 280.36 98.68 32.59 2 Lead Brass 361.90 280.36 15.99 93.32 3 Lead Steel 361.90 265.14 17.65 94.54 Trial # b 1 m 1 (g) m 2 (g) x 1 (cm) x 2 (cm) 1 Aluminum 103.29 82.34 95.00 75.15 2 Brass 287.22 82.34 95.00 85.01 3 Steel 362.49 82.34 95.00 86.72 Trial # x e (cm) m mc (g) 1 50.16 98.38

Table 7: Measurements of position and mass of block-clamp systems and Force of Tension of apparatus (Part 4) Trial # b 1 b 2 m 1 (g) m 2 (g) x 1 (cm) x 2 (cm) F (N) 1 Aluminum Brass 103.27 286.95 6.08 66.09 4.83 2 Aluminum Brass 103.27 286.95 9.98 64.56 4.82 3 Aluminum Brass 103.27 286.95 18.36 61.74 4.82 ANALYSIS A series of assumptions were made, using an established physical model, that allowed an easily observable model of the relationship between position ( x ), mass ( m ), and rotational torque at equilibrium. To establish observational references for position and mass, a particle model was utilized for the block and clamp systems, only–where one point on each block-clamp system was followed throughout data collection, assuming that all other points on the cart would behave the same. Furthermore, it was assumed that the density and weight distribution of the wooden meter stick remained equal throughout the object. Additionally, it was assumed that there was minimal to no external pressures acting on the balanced surface on which the apparatuses were assembled, thus producing a negligible effect on the horizontal equilibrium of the meter stick. It was assumed that the only forces acting upon the meter stick was the force of gravitational field strength, the normal force of the fulcrum, the force produced by the mass of the blocks, and the tension force of the force sensor. Furthermore, it was assumed that the force exerted by the

gravitational field strength remained constant throughout the experiment. Moreover, in components one through three, it was assumed that the force of the gravitational field strength was equal to the normal force applied by the fulcrum. Thus, external forces such as gusts of wind and drafts, were considered negligible. In this experiment, analyses of each component were performed consecutively, referencing results from previous components to substantiate reasoning for following analysis procedures. The first component of this experiment allowed for the analysis of equilibrium with no external forces acting on the fulcrum and wooden meter stick apparatus. The equilibrium position was found to be roughly in the center of the meter stick, with the force of mass acting upon the meter stick on each side of the fulcrum concluded (within reason) to have been roughly equal. No further analysis was performed in this component. In the second component of the experiment, the recorded masses of the block-clamp systems and their associated positions, along with the positions of the central clamp, were converted into standard units (grams to kilograms and centimeters to meters, respectively) using the associated conversion equations ( Equation 1 and 2, respectively). Resulting values were recorded and used in further analysis ( Table _). m ( g ) × 1 ( g ) 1000 ( kg ) (1) Where m represents the mass of the object/system in reference. x ( cm ) × 1 ( cm ) 100 ( m ) (2) Where x represents the position of the object/system in reference.

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Table 8: Mass and Position of Block-Clamp Systems in Standard Units. The objective of the analysis in the second component was to create a model of the relationship between position and mass at equilibrium. In order to do so, a mathematical model was created using conclusion of equal force of mass from part one. It was hypothesized that the product of the force of mass, or weight 9 , of the first block- clamp system and the absolute value of the change in displacement of the system ( x 1 ) from the position of equilibrium ( x E ) would be equal to the product of the weight of the second block-clamp system and its distance from the position of equilibrium, shown in Figure 5 and modeled Equation 3 . | x E − x 1 | ×m 1 g = | x E − x 2 | ×m 2 g (3) To derive the relationship between position, mass, and torque at equilibrium, the change in displacement ( ∆ x 1 and ∆ x 2 , respectively) of each block-clamp system was denoted as l 1 and l 2 , respectively ( Equation 4 ). l 1 m 1 g = l 1 m 1 g (4) Utilizing the established constructs of Newton’s second law, the product of the mass of the blocks and the gravitational field constant of each system was termed as F 1 and F 2 , respectively ( Equation 5 ) using standard units of Newtons (N). 9 The gravitational field strength of 9.8, or g , was multiplied by mass in order to factor in gravity into the model, shown in Equation 1 . Without the influence of gravitational force in this experiment, any position of the meter stick would have been at equilibrium. Trial # b 1 b 2 m 1 (kg) m 2 (kg) x 1 (m) x 2 (m) 1 Aluminum Brass 0.10319 0.2803 6 0.9868 0.3259 2 Lead Brass 0.36190 0.2803 6 0.1599 0.9332 3 Lead Steel 0.36190 0.3619 0 0.1765 0.9454

F 1 l 1 = F 2 l 2 (5) τ 1 = τ 2 (6) The product of the force of mass and distance from the position of equilibrium was noted as torque (denoted as ? 1 and ? 2 respectively) ( Equation 6 ). The collected values of data from each trial were plugged into Equation 1 and the resulting values of torque in the block-clamp systems were recorded. Once the torque values were attained, the percent difference between the two values of torque was calculated to test if the torques of both objects were equal at horizontal equilibrium. The percent difference was calculated by taking the absolute value of the difference between the two torque values calculated in each trial, and then dividing by the average value of the torques. To convert the calculated into a percent, the resulting value was then multiplied by a factor of 100 ( Equation 7 ). If the value of D 2 was less than, equal to, or slightly greater than one percent, the difference between each datum was considered insignificant and the hypothesis of the relationship between torques at horizontal equilibrium would be viable. Values of torque and the associated percent differences were recorded. D 2 = | τ 1 − τ 2 | ( τ 1 + τ 2 2 ) × 100 (7) Figure 5: Diagram of derivation of mathematical equation modeling the relationship between position and mass at equilibrium.

| x 2 − x 1 | ×m 1 g = | x 2 − x F | ×m 2 g (8) In part three, the rudimentary equation for torque was utilized in order to determine the position of the center of mass of the wooden meter stick at horizontal equilibrium ( x F ), using the assumption that the torque of one side of the fulcrum would equal the torque of the other side. Since the mass of the opposing side of the meter stick was unknown, the total mass of the meter stick without the clamp was utilized as m 2 , or the weight of the opposing side of the block-clamp system, to calculate the center of mass of the meter stick; thus, the position of the block-clamp apparatus ( x 1 ), the position of the central clamp at equilibrium ( x 2 ), the addition of mass from the block-clamp system ( m 1 ), and the mass of the meter stick ( m 2 ) were substituted into Equation 8 and the position of the center of mass of the meter stick ( x F ) was solved for. Once the values for x F were calculated, the percent error for value was calculated ( Equation 9 ) by finding the absolute value of the difference between the calculated x F values and the expected position of the center of mass, then dividing by the expected value of the center of mass and multiplying by a factor of 100. Percent error ( E 3 ) was utilized in this component as the theoretical center of mass of the wooden meter stick was previously measured ( x E ); thus, the values of x F were compared to the expected value of center of mass in each trial. E 3 = | x F − x E | x E × 100 (9) Figure 6: Diagram of derivation of solution to find the center of mass of the meter stick given m1, m2, and xE.

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In the fourth component, all collected data of mass and position measurements were converted into standard units using Equation 1 and Equation 2 ( Table 9 ). The objective of analysis in part four was to analyze the rotational torques of different points of force at various points of referential origins at equilibrium, known as translational equilibrium. According to the previously established physical model, it was assumed that the points of force acting upon the meter stick were at the center of the block-clamp systems and at the central clamp of the meter stick. Three origins of reference were chosen for rotational torque to be analyzed: Around the center of mass of the wooden meter stick ( x E ), the 0-cm mark of the meter stick ( x 0 ), and around the position of the m 2 block-clamp system ( x 2 ). Table 9: Mass and Position of Block-Clamp Systems in Standard Units. Trial # b 1 and b 2 m 1 and m 2 (kg) x 1 and x 2 (m) m s (kg) 1 Aluminum 0.10327 0.0608 0.4886 Brass 0.28695 0.6609 2 Aluminum 0.10327 0.0998 0.4886 Brass 0.28695 0.6456 3 Aluminum 0.10327 0.1836 0.4886 Brass 0.28695 0.6174

When calculating the various points of force ( Figure 7 ), it was assumed that the meter stick was in translational, horizontal equilibrium. It was derived that the sum of the forces acting in the upwards and downwards direction would equate to zero at equilibrium. Thus, the values of the forces at the center of mass of the system were derived to be equal to one another in order for the meter stick to have been suspended in translational equilibrium; therefore, the sum of the forces acting upwards on the meter stick apparatus Σ F U , was equal to the sum of the forces acting downwards on the meter stick apparatus, Σ F D . The sum of the forces in the upwards direction was equated to the force reading of the sensor in each trial, F T ( Equation 10 ). Σ F U = F T (10) To calculate the sum of the forces in the downward direction, the total mass of the system ( m s ) was calculated ( Equation 11 ) and multiplied by the gravitational field strength constant ( Equation 12 ). m s = m 1 + m 2 + m mc (11) Where m 1 is the mass of the first block-clamp system, m 2 is the mass of the second block-clamp system, and m mc is the mass of the meter stick and the central clamp. Σ F D = m s g (12) The percent difference ( D 2 ) between the forces oriented upwards and the forces oriented downwards was calculated ( Equation 12 ) in order to measure the accuracy of the data collection. If the value of D 4 was less than, equal to, or slightly greater than one percent, the difference between each datum was considered insignificant and the postulated relationship between upwards and downward force at translational equilibrium would hold true. The rotational equilibrium of the meter stick apparatus was then analyzed around various frames of reference. Using the established assumptions derived from the previous components of the experiment, it was assumed that the sum of the torques in the opposing rotational directions of motion were equal to zero at rotational equilibrium. Thus, the torques in opposing directions would be equal to one another. In

this experiment, rotational movement in the downwards direction (towards the left) was termed as clockwise rotation, whereas rotational movement in the upwards direction (towards the right) was termed as counterclockwise rotation. At each point, the force acting upon the system was calculated using Newton’s second law ( Equation 13) in order to calculate the torques of the objects at that position. F j = m j g (13) Where F j represents the force of an object, j , with mass m j, and g represents the gravitational acceleration of 9.8 m/s 2 . The variable m j was substituted for m mc when solving for the downwards force of the central clamp, m 1 when solving for the force exerted by the first block-clamp system, and m 2 when solving for the force exerted by the second block-clamp system, denoted as F mc , F 1 , and F 2 , respectively. The tension force was equal to the recorded force on the force sensor in each trial ( F T ). The torque values of the forces at various points about an origin were calculated using the general torque equation ( Equation 14 ). τ = l F (14) Where τ represents torque about some origin, l represents the distance of the position of force acting on the system from the origin and F represents the acting force of the object on the system at that position. To determine if the meter stick apparatus was at a true state of rotational equilibrium, the torques of each object acting upon the meter stick (shown in Figure 7 ) were calculated and analyzed about the three varying origins of rotation ( Equation 15). τ zj = F j | x z − x j | (15) Where τ zj represents the torque of an object, j, around origin, z, F j represents the force of the object at its position, x j , and x z represents the position of the origin of rotation. To calculate the torques of each force about the center of mass, x E was plugged into the x z value, and the corresponding force ( F 1 , F 2 , F mc , and F T ) and position values ( x 1, x 2 , and x E 10 , respectively) of the objects were plugged into the 10 Both F mc and F T have the position x E

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corresponding variables to calculate the resulting torques of each object ( τ E 1 , τ E 2 , τ Emc , and τ T , respectively). To calculate the torques of each force about 0-cm mark, x 0 was plugged into the x z value, and the corresponding force ( F 1 , F 2 , F mc , and F T ) and position values ( x 1, x 2 , and x E , respectively) of the objects were plugged into the corresponding variables to calculate the resulting torques of each object ( τ 01 , τ 02 , τ 0 mc , and τ 0 T , respectively). To calculate the torques of each force about the position of mass, m 2 , x 2 was plugged into the x z value, and the corresponding force ( F 1 , F 2 , F mc , and F T ) and position values ( x 1, x 2 , and x E , respectively) of the objects were plugged into the corresponding variables to calculate the resulting torques of each object ( τ 21 , τ 22 , τ 2 mc , and τ 2 T , respectively). To determine if the torques acting in the opposing directions were equal to one another, the sum of the clockwise and counterclockwise torques were calculated about all three origins in each trial. In order to determine what forces acted in a clockwise or in a counterclockwise motion, each force was looked at in isolation ( Figure 7 ) about the origin of reference. If the force, acting by itself, would rotate the meter stick in a clockwise direction about the origin of reference, it was deemed clockwise. If the isolated force would have rotated the meter stick in a counterclockwise direction about the origin of reference, it was deemed a counterclockwise force. To calculate the sum of the clockwise torques about the center of mass ( Σ τ cwE ¿ , the force that appeared to be acting in a clockwise motion when looking at Figure 7 is F 2 . Thus, the sum of the clockwise torques around the center of mass is equal to the torque of m 2 ( Equation 16 ). The force that appeared to act in a counterclockwise motion is F 1 . Thus, the sum of the counterclockwise torques about the center of mass ( Σ τ ccwE ) was equal to the torque of the sensor ( Equation 17 ). Σ τ cwE = τ E 2 (16)

The F T and F mc have no effect on the rotation of the meter stick about the center of mass as it shares the same pivot position as the center of mass and, thus, has no clockwise or counterclockwise rotational direction. To calculate the sum of the clockwise torques about the 0-cm mark ( Σ τ cw 0 ¿ , the forces that appeared to be acting in a clockwise motion when looking at Figure 7 were F 1 , F 2 , and F mc . Thus, the sum of the clockwise torques around the center of mass can be calculated by adding the torques of these forces ( Equation 18 ). Where τ 01 was the torque of block one about the designated origin, τ 02 was the torque of block two about the designated origin, and τ 0 mc was the downward torque of the meter stick at its center of mass around the designated origin. The force that appeared to act in a counterclockwise motion was F T . Thus, the counterclockwise torque about the center of mass ( τ ccw 0 ) was equal to the torque of the sensor ( Equation 19 ). The force that appeared to act in a counterclockwise motion was F T . Thus, the sum of the counterclockwise torque about the center of mass ( Σ τ ccw 0 ) was equal to the torque of the tension force ( F T ) ( Equation 20 ). To calculate the sum of the clockwise torques about the position of m 2 mark ( Σ τ cw 2 ¿ , the forces that appeared to be acting in a clockwise motion when looking at Figure 7 was F T . Thus, the sum of the clockwise torques around the position of block two was equal to the torque of the tension force ( F T ) ( Equation 21 ). Σ τ ccwE = τ E 1 (17) Σ τ cw 0 = τ 01 + τ 02 + τ 0 mc (18) Σ τ ccw 0 = τ 0 T (19) Σ τ ccw 0 = τ 0 T (20)

The forces that appeared to act in a counterclockwise motion around the origin reference of block two’s position were F mc and F 1 . Thus, the sum of the counterclockwise torques about the position of m 2 ( Σ τ ccw 2 ) was equal to the sum of the torques of the gravitational force of the meter stick ( τ 2 mc ) and the weight force of block one ( τ 21 ¿ ( Equation 22 ). The torque of block two was assumed to have no rotational effect as it was located on the rotational point of the origin. At each origin of rotation, the percent difference between the clockwise and counterclockwise torques were calculated ( Equation 23 ) in each trial in order to determine the accuracy of data collected. Where Σ τ cwz represents the sum of the clockwise torques at one of the three origins, generally termed z , and Σ τ ccwz represents the sum of the counterclockwise torques at origin z . Figure 7: Diagram of rotational origins and positions of force using in the derivation of clockwise and counterclockwise torque Σ τ cw 2 = τ 2 T (21) Σ τ ccw 2 = τ 2 mc + τ 21 (22) D z = | Στ cwz − Σ τ ccwz | ( Στ cwz + Σ τ ccwz 2 ) × 100 (23)

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RESULTS Table 10: Position of central clamp of meter stick at horizontal equilibrium with no additional mass (Part 1) Trial # x E (cm) 1 50.11 Table 11: Calculated torque’s of block-clamp systems and associated percent difference (Part 2) Table 12: Calculated center of mass of meter stick and associated percent error (Part 3) Table 13: Calculated sum of upward and downward forces and associated percent difference (Part 4) Trial # b 1 b 2 ? 1 (N ∙ m) ? 2 (N ∙ m) D 2 ( % ) 1 Aluminum Brass 0.491 0.492 0.102 2 Lead Brass 1.210 1.212 0.155 3 Lead Steel 1.151 1.154 0.280 Trial # x f (cm) E 3 ( %) 1 50.24 0.060 2 50.12 0.179 3 50.27 0.119

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Trial # ? E1 (N ∙ m) ? E2 (N ∙ m) ? Emc (N ∙ m) ? ET (N ∙ m) Σ ? CWE (N ∙ m) Σ ? CCWE (N ∙ m) D E ( % ) 1 0.446 0.448 0.00 0.00 0.448 0.446 0.416 2 0.407 0.405 0.00 0.00 0.405 0.407 0.418 3 0.322 0.326 0.00 0.00 0.326 0.322 1.177 Table 14: Calculated torques of objects and sum of clockwise and counterclockwise torques around the center of mass along with the associated percent difference Table 15: Calculated torques of objects and sum of clockwise and counterclockwise torques around the 0-cm mark along with the associated percent difference Trial # ? 01 (N ∙ m) ? 02 (N ∙ m) ? 0mc (N ∙ m) ? 0T (N ∙ m) Σ ? CW0 (N ∙ m) Σ ? CCW0 (N ∙ m) D 0 ( % ) 1 0.0615 1.859 0.484 2.423 2.404 2.423 0.790 2 0.101 1.815 0.484 2.418 2.400 2.418 0.731 3 0.186 1.736 0.484 2.418 2.406 2.418 0.502 Table 16: Calculated torques of objects and sum of clockwise and counterclockwise torques around the position of m 2 along with the associated percent difference Trial # b 1 and b 2 Σ F U (N) Σ F D (N) D 4 ( % ) 1 Aluminum 4.83 4.788 0.868 Brass 2 Aluminum 4.82 4.788 0.660 Brass 3 Aluminum 4.82 4.788 0.660 Brass

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Trial # ? 21 (N ∙ m) ? 22 (N ∙ m) ? 2mc (N ∙ m) ? 2T (N ∙ m) Σ ? CW2 (N ∙ m) Σ ? CCW2 (N ∙ m) Dm 2 ( % ) 1 0.607 0 0.154 0.769 0.761 0.769 1.112 2 0.552 0 0.139 0.694 0.694 0.691 0.415 3 0.439 0 0.112 0.558 0.558 0.551 1.350 DISCUSSION The purpose of this experiment was to analyze the relationship between position, mass, and rotational torque at moments of horizontal and rotational equilibrium. Using a meter stick and fulcrum apparatus, moments of horizontal equilibrium were modeled, allowing for the variation of mass and position through the addition of metal blocks to the system via hooked clamps. Using a hanging meter stick and force sensor apparatus, rotational equilibrium was modeled through the addition of an upwards oriented force of tension, allowing for various orientations of torque around different origins of rotation to be used in analysis of the relationship between rotational equilibrium and torque. The procedure of this experiment was split into four components in order to allow for the use of theoretical provisions (substantiated by preceding parts of the lab) to create accurate experimental and mathematical models in each part. Utilizing assumptions from the previously established physical model, subsequent mathematical models were derived through successive analysis in order to characterize a relationship between torque and horizontal and rotational equilibrium. The first component of this experiment was used to determine if the theoretical center of mass of the wooden meter stick was equal to the equilibrium point of the apparatus with no additional mass

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attached. It was expected that the equilibrium point would be at exactly half of the total length of the meter stick, positioned at 50.00 centimeters, where the mass of the meter stick would be equal on each side; therefore, the weight of mass of the meter stick on each side would be equal and the system would be in horizontal equilibrium. However, when observing the actual value of the central position of the meter stick at equilibrium in part one, it was noticed that the point of equilibrium was not exactly at 50.00, although close to the theoretical value (50.11 centimeters). This difference may have been due to a few experimental errors associated with the assumptions of our physical model. The second component of the experiment was used to model and observe the effect of unequal mass additions and unequal position measurements on the static horizontal equilibrium of the meter stick apparatus, keeping the initial center of mass position constant. A mathematical equation was developed in order to model the relationship between the center of mass of the apparatus, the position of the blocks, and the mass of the block-clamp systems. The hypothesis of the model ( Equation 8 ) was that the proportion of length and mass of one block-clamp system was equal to the proportion of length and mass of the opposing block-clamp system about the origin of the center of mass ( Figure 5 ). The force of gravitational field strength had the same effect on both sides of the apparatus; therefore, the factor g was not considered to have a large impact on the difference between torques of each block-clamp system. It was noticed in Table 11 that the torques on each opposing side of the center of mass ( Figure 5 ) were relatively equal to one another. This was further substantiated by the calculation of a percent difference between the two torque values in each trial: when looking at the values of D 2 (Table 11 ), the differences were all less than 1%, indicating a highly similar value of torque on each side of the apparatus. Thus, it was concluded that the product of the force and position of the force were equal to one another, and therefore the proportion of mass and length were equal to one another, in horizontal static equilibrium.

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The third component of the experiment was utilized to determine if the center of mass of the meter stick apparatus would remain the same as the initial center of mass (with no additional mass added) at equilibrium, even with imbalanced additions of various increments of mass. It was hypothesized that the center of mass of the meter stick would not remain the same as the initial center of mass measured ( Table 4) when the blocks would be added to the meter stick; however, once data had been collected ( Table 5 ) and the values of the center of mass, x f , were calculated (Table 12 ) it was noticed that the center of mass calculated in each trial was nearly equal to the initial center of mass at horizontal equilibrium. This was further tested using a percent error calculation for each trial, where the initial center of mass measured (50.21 cm) was used as the standard value of comparison. Looking at the resulting percent errors ( Table 12 ) it can be seen that the error is less than 1%, indicating that there was minimal error in the calculation of the center of mass; therefore, it can be concluded that the center of mass of the meter stick does not change even with the addition of mass and rotational imbalance. This viability of the constant center of mass allows for the establishment of the use of the center of mass of the meter stick in calculations of torque to be used in the fourth component of this experiment. The fourth component of this experiment was used to analyze how rotational torque of a system at equilibrium is impacted when calculated from different origins of rotation. Before calculating the torque of the various forces acting upon the system, however, analysis was performed on the apparatus in order to determine if the meter stick was actually in translational equilibrium throughout data collection. To determine if the meter stick was in translational equilibrium, the net force oriented upwards and downwards were assumed to equal zero. Thus, it was assumed that the sum of the forces acting in an upwards direction on the apparatus was equal to the sum of the forces acting in a downwards orientation on the whole apparatus ( Table 13 ). When looking at the calculated values of the sums of the forces, it was noticed that the upwards and downwards facing forces were almost at the same value. To substantiate this,

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the percent difference between the sums was calculated ( Table 13 ) and used as a test for accuracy and comparison between the two groups. When looking at the values of D 4 , the percent differences were all less than 1% in each trial, indicating that the data accurately reflected the claim that the meter stick was at translational equilibrium; therefore, it was assumed that the meter stick was at both translational, horizontal equilibrium and rotational equilibrium (as the motion of the meter stick was static and unmoving during collection of data). It was hypothesized that at rotational equilibrium, the sum of the torques acting in opposing directions (clockwise and counterclockwise) would equate to zero as the apparatus was not rotating in any direction at the time of data collection; thus, it was derived that the sum of the clockwise torques were equal to the sum of the counterclockwise torques. Thus, the total clockwise and counterclockwise torques were calculated in each trial, and associated percent differences were calculated in order to substantiate the claim that the meter stick was in rotational equilibrium. The clockwise and counterclockwise torques were calculated about three rotational origins to determine if there was a significant impact on the rotational equilibrium of the meter stick ( Tables 14, 15, and 16 ). When looking at the percent differences between clockwise and counterclockwise torques about the center of mass ( D E ), it can be seen that the values are all less than or around 1% ( Table 14 ), indicating that the difference between the clockwise and counterclockwise torques is minimal. This trend continues in the percent differences of the clockwise and counterclockwise torques about the 0-cm mark and around the position of block two. Looking at the percent difference between the clockwise and counterclockwise torques about the 0-cm mark ( D 0 ), it can be seen that the values are all less than 0.8%, indicating that the difference between the opposing-oriented torques is minimal. When looking at the percent difference between the clockwise and counterclockwise torques about the position of m 2 ( D m2 ) in Table 16, it is noticed that all the values are less than or around 1%, indicating that there is a minimal difference

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between the torques of opposing orientation. Thus, it can be concluded that the meter stick was in rotational equilibrium, and that the opposing torques at equilibrium are equal to one another. In this experiment, the calculations of percent difference and percent error were used as a measure of accuracy and precision, respectively. When looking at the culmination of the percent differences calculated in each trial, the values remained relatively low (at a value less than or around the value of 1%), indicating that the data collected was accurate and had minimal error systematically skewing the collection of data throughout the experiment. Furthermore, the low values of percent error calculated in part three of this experiment indicate that the precision of the data was in line with the accuracy of the data to some extent (corroborated by the percent error values that were less than 1%). However, there were experimental errors, both systematic and random, that may have impacted the data collection and results of this experiment. One systematic error that may have directionally skewed the data was that the tick marks on the meter stick did not start at the exact edge of the stick; therefore, the position measurements collected would have been skewed to a greater measurement of position than the actual position of each point. Another systematic error present throughout the experiment was that the initial position of equilibrium was not recorded when the meter stick was perfectly horizontal, and therefore would have impacted the trials measuring the positions of mass increments at equilibrium (the skew could have been systematically greater or less than the actual values of position). Additionally, the scale used to measure mass may have not been perfectly level and thus, the mass recorded could have been directionally skewed (either greater than or less than) from the actual values of the mass. Another systematic error that potentially skewed the data, particularly parts one and three of data collection and analysis, was the assumption that the density of the wooden meter stick was constant throughout; however, the mass distribution of the wooden material is not always continuous and, therefore, the measured center of equilibrium of the meter stick in part one and calculations of the center of mass of the

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meter stick in part three may have been skewed (either greater than or less than the actual value of position and mass). Another systematic error present in the experiment was due to the particle model used to observe the mass of the block-clamp systems. It was assumed that the mass of the system was distributed and behaved the same throughout the system and thus, the measured and calculated influence of the force of mass may have been greater or less than the actual value throughout the experiment. Random errors were also present throughout this experiment and may have had an impact on the precision of measurement. One random error that was heavily present throughout this experiment was the reliance on resolution, or estimation of a final significant digit, as a form of precision. Since the meter stick only measured up to 0.1 centimeters, a resolution of the following digit measurement was made in each measurement of position in order to get a precise recording of the measuring of the blocks. This reliance on resolution may have caused random skewing of the data (some data sets would have been under or overestimated), resulting in an imprecise measurement. Another random error was that the protruding edge of the clamp that was supposed to fit in the fulcrum was not fully snug and had wiggle room, which may have affected the true equilibrium of the meter stick. Additionally, the presence of random gusts of wind and drafts may have influenced the equilibrium when setting up the apparatus and during the collection of data, randomly skewing true measurements of position at equilibrium. Another source of random error was the warping of the balanced surface due to external forces (such as leaning, placing down objects in the middle of data collection). The warping of the balanced surface would skew the actual equilibrium, altering the observed equilibrium to a value greater or less than the true position. Compared to previous experiments that aimed to accurately analyze the position of objects, this experiment was considerably less accurate. In a previous experiment utilizing a cart and track system to model collisions, the positions of each object were tracked using a motion sensor that was connected to a Vernier Graphical Analysis recording software. In this experiment, the collection of raw position was

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more accurate as it was automated and did not rely on resolution for precision; however, both experiments shared similar sources of errors regarding the effects of external forces on the validity of the results. In both experiments, the measure of mass may have been skewed due to a mistaken leveling of the scale. Additionally, the speeds of the carts were heavily influenced by the horizontal equilibrium of the balanced surface. Thus, if the surface was not balanced, the results were heavily skewed (either greater than or less than the actual values) throughout the process of data collection. In order to improve the precision and accuracy of the data, while avoiding any errors that may occur throughout the conduction of the procedure, various measures could be implemented in the experimental design to ensure that the collection of position and mass of the block-clamp systems are as accurate as possible. One main improvement that must be made in this experiment is the implementation of an automated and more precise measurement of position of the points of force throughout this experiment. This would eliminate the need to use resolution for precision throughout the experiment, and simultaneously create results with more validity and use. Additionally, a metal meter stick could be implemented in this experiment in order to rid of the inconsistent density of mass within wooden meter sticks, effectively increasing the accuracy and precision of the analysis calculations used to determine the center of mass of the meter stick. Furthermore, an improvement regarding the measure of horizontal equilibrium could be made in order to determine if the meter stick is in true horizontal equilibrium before data collection ensues. Using a structure to evaluate the horizontal tilt of the meter stick would allow for greater accuracy when setting up the apparatus and conducting data collection. Additionally, it would be ideal if this experiment were performed in an automated and isolated system, where there would be a nominal occurrence of warping of the balanced surface in order to ensure that the meter stick is in true static and translational equilibrium.

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Torque Lab

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