_Lab Report 3_ Static and Kinetic Friction

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Toronto Metropolitan University *

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Jan 9, 2024

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Verifying the Coefficients of Kinetic and Static Friction PCS 120

Introduction The objective of this experiment is to verify the equations for kinetic and static friction and confirm that the coefficients of friction are constant through investigating both scenarios on a carpeted surface at varying mass configurations resulting in changes to the force of friction present. The core data collection of the investigation will be recorded using the Vernier Force Sensor which will be equipped with the software Vernier Graphical Analysis which will graph the force in Newtons over a time interval in seconds. Theory Friction is an external contact force which exerts resistance when two objects/surfaces make contact and the frictional force always opposes the relative motion. There are two types of friction forces, static friction is the frictional force which keeps an object from slipping, the static friction force will point in the opposite direction preventing motion (Knight, 2016). When objects are at equilibrium the frictional force is completely in balance with the tension force ( f s = T) . Typically there are three scenarios which determine the motion of the object relative to the maximum possible static friction force, as evident in Figure 1, if an object remains at rest the friction force is less than the friction maximum ( f s < f s max ), but if the object begins to slip this happens when the friction force has reached the friction maximum ( f s = f s max ) and the friction maximum will never be less than the frictional force. The formula for static friction states that f s max is proportional to the magnitude of the normal force, f s max = μ s n (Knight, 2016). The static friction force formula can be rearranged for the coefficient of static friction which varies in magnitude depending on the materials of the object and surface in contact, the formula μ s = can be used to calculate and confirm whether the 𝑓 𝑠 ?𝑎𝑥 ? coefficient is constant or not. However, in cases where an object starts to slide and set into full motion the static friction force is replaced by the kinetic friction force as shown in Figure 1 , when in comparison the kinetic friction force is nearly constant magnitude (Knight, 2016). As always the kinetic frictional force will be in the opposite direction of the sliding object and it is important to notice that the size of the kinetic friction force is less than the maximum static friction as μ s > μ k better explaining as to why it is easier to keep an object in motion than to start moving it (Knight, 2016). Likewise, the kinetic friction formula is f k = μ k n and can be rearranged to verify the coefficient of kinetic friction, μ k = . 𝑓 𝑘 ? 1

Procedure This tutorial helps to perform and collect data to verify the equations of static and kinetic friction and confirm the coefficients of friction, by calculating the amount of frictional force exerted on an object pulled across a carpeted surface at 6 different mass configurations. Ensure to have the Vernier Graphical Analysis software downloaded on a computer along with a Vernier Force sensor, along with a wooden block attached to a string placed on a carpeted surface with a 500 g ± 1% mass and two 1 kg ± 1% masses. Make sure the Vernier Graphical Analysis software is launched and connect the force sensor and set the range to ± 10N on the sensor. The force sensor is calibrated by performing a two-point calibration using a 0 g mass and a 500 g mass the. Click the Sensor Meter button in the bottom right and click calibrate. Assure the force sensor is held vertically and enter the known value as 0 and click “keep”. In the second part of the calibration the 500 g mass hangs from the sensor, the second known value, 4.91 is entered and clicking keep saves the data. Click “apply” to complet the force sensor calibration. Once calibrated, measure the force of gravity acting on the wooden block and record that value for calculations. Place the wooden block with the attached string at the top of the carpeted surface, hook on the force sensor to the string, press collect on the Graphical Analysis software and pull the block horizontally shortly after the data collection has started, run and collect data for 4 trials for the same wooden block (0 g mass). The Graphical Analysis software will produce a graph which measures the frictional force in Newtons over seconds. The highest point before the constant horizontal line on the graph is considered to be the f s max, take note of this value, as this portion of the graph will account for the static friction. On the same graph the horizontal/constant section after the peak on the graph is selected and referred to as the kinetic friction, use the “graphing tools” options in the bottom left to find the statistics of the selected portions. There are a total of 6 mass configurations possible each of the configurations will include the wooden box hence the calibration indicating 0 g = wooden box. The configurations include (1) 0 g, just the wooden box, (2) 500 g + wooden box, (3) 1 kg + wooden box, (4) 2 kg + wooden box, (5) 500 g + 1 kg + wooden box and (6) 500 g + 2 kg + wooden box. Therefore, the whole experiment must be repeated 6 times for each mass configuration, and for each configuration collect 4 trial runs and l make sure the wooden box is placed at the same starting position each time. At the end of data collection make sure to access the File menu in the top right corner to save and export as a gambl so that the file can be opened on the software, but also export the data as a CVS so that it can later downloaded as an excel spreadsheet, where necessary analysis can rake place to calculate and conclude results in terms of the magnitudes of the coefficients. 2

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Results and Calculation Once all the data has been collected, the mass of the wooden block must be calculated. The value given from the sensor was 3.40407N. Using the formula F G = mg, the weight of block can be calculated as follows: F G = mg 3.40307 = m(9.81) m = 0.347 kg Since the force of friction is calculated using the formula μ mg, the normal force for each mass configuration must be calculated. Since each trial was conducted on a horizontal surface, the normal force will equal mg. Below are the normal force values of each mass configuration used in the experiment. Description of setup Mass (kg) Normal force (N) Wooden block 0.347 3.40407 WB + 500g 0.847 ± 0.005 8.30907 ± 0.04905 WB+ 1kg 1.347 ± 0.01 13.21407 ± 0.0981 WB + 1.5 kg 1.847 ± 0.015 18.11907 ± 0.14715 WB + 2kg 2.347 ± 0.02 23.02407 ± 0.1962 WB + 2.5kg 2.847 ± 0.025 27.92907 ± 0.24525 Table 1: Mass configurations with their respective mass and weight. The maximum value of friction will equal the force applied on it right before movement occurs, signaling the force has overcome the friction force. Using the maximum force value, the static friction coefficient can be calculated using this equation: F f MAX = F App . The value for the applied force will be the average of each run’s maximum force. A sample calculation for the first configuration consisting of just the wooden block will be demonstrated, and a graph representing the values from each trial will be shown below To calculate the error for the normal force, the error given for the masses are taken into consideration. Since the normal force is dependent on the mass of an object, the uncertainty is factored in. This is represented by taking the average of each uncertainty value and dividing by 6. This is used for the horizontal error bar in each graph. As for the force, the standard deviation was calculated from all the values used for the force. Static friction only used the maximum values, and kinetic used all values while the object is in constant motion. 3

F f MAX = F APP Normal Force (N) Average Maximum Force (N) μ s MAX mg = F App 3.40407 0.8110033 μ s (0.347)(9.81)= 0.811003261 8.30907 ± 0.04905 2.1838211 μ s MAX = 0.25244522 13.21407 ± 0.0981 3.4048355 18.11907 ± 0.14715 4.3400784 Table 2: Maximum static 23.02407 ± 0.1962 5.3866912 friction forces. 27.92907 ± 0.24525 6.3782777 The horizontal error bars represent the error associated with the weighted masses. There are vertical error bars ( 0.24525 ) from the standard deviation of the maximum forces. The slope represents the static friction coefficient since it is the relationship between how much force is needed to overcome the normal force of an object Figure 2: Maximum force applied versus normal force. According to the graph, the static friction coefficient is 0.222. Another method of determining the coefficient is to calculate the μ s MAX value of each weight and taking the average of each. Following this method μ s MAX = 0.2398. Once the maximum static friction force has been overcome, the kinetic friction coefficient takes over. In order to calculate the kinetic friction coefficient, a similar process of creating a Force vs Normal Force graph should be created. Data collected from the time the object only had kinetic friction acting upon it was used to calculate the average force. These values represent the frictional force, and will also be compared to the normal force of each object. 4

F k = F APP Normal Force (N) Average Maximum Force (N) μ k mg = F App 3.40407 0.6369628 μ (0.347)(9.81)= 0.6369628 8.30907 ± 0.04905 1.7728028 μ k = 0.18 13.21407 ± 0.0981 2.7779519 18.11907 ± 0.14715 3.8463557 Table 3: Kinetic friction 23.02407 ± 0.1962 4.85313 forces. 27.92907 ± 0.24525 5.9204822 The horizontal error bars are the error associated with the weighted masses identical to the static friction values. There are vertical error bars ( 0.24525 ) from the standard deviation of the maximum forces. The slope represents the kinetic friction coefficient since it is the relationship between how much force is needed to overcome the normal force of an object The value given from the graph shows 0.2123, but calculating the coefficients individually and taking the average gives a value of 0.2046. Discussion and Conclusion The static friction coefficient is always going to have a higher value than the kinetic friction, and this follows the results achieved through this experiment. Both ways of determining these values (graphically and mathematically) result in two acceptable values for the friction coefficients of a soft surface. Since the material tested is a carpet, the friction coefficients are expected to be lower than a material such as rubber ( μ s = 1.15). Although the lab is successful at deepening the understanding of friction and how to derive its elements, there could be some improvements to ensure the experiment is more accurate and in depth. One factor that hindered the experiment was how many mass configurations were used. Since the results relied heavily on the graphical representation of the data taken, it is advised to increase the sample size of masses used. This leads to less outliers in the event of a poorly 5

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executed experiment, and shows proof of a continuous trend of the relationship between friction and the normal force. Similarly, having more trials per mass configuration would also reduce the likelihood of outliers. More trials shows how each individual mass reacts to the same forces. Same ideology as adding more masses where quantity in data leads to consistent results. Finally, it would be ideal to have a device that is able to pull the objects at a consistent force. Human error impacted this experiment since there is an uneven force applied over the entire experiment. When the object starts moving, the human body will not maintain a consistent pulling force, thus affecting the sensor. 6