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In a prototype of a BMMD, six different springs are available for testing with the device. The six springs are very similar, but with different restoring constant k. A student decides to test the device with all springs to estimate the value of his body mass. Thus, he measured 4 oscillation periods for each spring and calculated the average of the periods to obtain six pairs of data (k,7), as shown in Table 1 and Figure 1. Table 1. Results of the average oscillation period for each spring of known constant in a BMMD. Spring constant k ± 0,001 kg/s² Oscillation period T ± 0,001 s 455,000 2,684 480,000 2,613 515,000 2,522 590,000 2,357 712,000 2,145 900,000 1,955 Answer the following questions based on the information provided and the operating fundamentals of the BMMD. A. Plot T as a function of k. Justify why it is not a linear relationship and explain why it has a decreasing or increasing trend. Figure 1 Graph of Period of oscillation (T) as a function of spring constant (k). Period of oscillation (T) as a function of spring constant (k) 2,700 2,684

In a prototype of a BMMD, six different springs are available for testing with the device. The six springs are very similar, but with different restoring constant k. A student decides to test the device with all springs to estimate the value of his body mass. Thus, he measured 4 oscillation periods for each spring and calculated the average of the periods to obtain six pairs of data (k,7), as shown in Table 1 and Figure 1. Table 1. Results of the average oscillation period for each spring of known constant in a BMMD. Spring constant k ± 0,001 kg/s² Oscillation period T ± 0,001 s 455,000 2,684 480,000 2,613 515,000 2,522 590,000 2,357 712,000 2,145 900,000 1,955 Answer the following questions based on the information provided and the operating fundamentals of the BMMD. A. Plot T as a function of k. Justify why it is not a linear relationship and explain why it has a decreasing or increasing trend. Figure 1 Graph of Period of oscillation (T) as a function of spring constant (k). Period of oscillation (T) as a function of spring constant (k) 2,700 2,684

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
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Problem In a prototype of a BMMD, six different springs are available for testing with the device. The six springs are very similar, but with different restoring constant k. A student decides to test the device with all springs to estimate the value of his body mass. Thus, he measured 4 oscillation periods for each spring and calculated the average of the periods to obtain six pairs of data (k,T), as shown in Table 1 and Figure 1. Table 1. Results of the average oscillation period for each spring of known constant in a BMMD. Spring constant k ± 0,001 kg/s² Oscillation period T ± 0,001 s 455,000 2,684 480,000 2,613 515,000 2,522 590,000 2,357 712,000 2,145 900,000 1,955 Answer the following questions based on the information provided and the operating fundamentals of the BMMD. A. Plot T as a function of k. Justify why it is not a linear relationship and explain why it has a decreasing or increasing trend. Figure 1 Graph of Period of oscillation (T) as a function of spring constant (k). Period of oscillation (T) as a function of spring constant (k) 2,700 2,684 2,613 2,600 2,522 2,500 2,400 2,357 2,300 2,200 2,100 2,000 1,900 Period of oscillation (T) 455,000 480,000 515,000 Spring constant (k) 530,000 2,145 712,000 1,955 900,000
Transcribed Image Text:Problem In a prototype of a BMMD, six different springs are available for testing with the device. The six springs are very similar, but with different restoring constant k. A student decides to test the device with all springs to estimate the value of his body mass. Thus, he measured 4 oscillation periods for each spring and calculated the average of the periods to obtain six pairs of data (k,T), as shown in Table 1 and Figure 1. Table 1. Results of the average oscillation period for each spring of known constant in a BMMD. Spring constant k ± 0,001 kg/s² Oscillation period T ± 0,001 s 455,000 2,684 480,000 2,613 515,000 2,522 590,000 2,357 712,000 2,145 900,000 1,955 Answer the following questions based on the information provided and the operating fundamentals of the BMMD. A. Plot T as a function of k. Justify why it is not a linear relationship and explain why it has a decreasing or increasing trend. Figure 1 Graph of Period of oscillation (T) as a function of spring constant (k). Period of oscillation (T) as a function of spring constant (k) 2,700 2,684 2,613 2,600 2,522 2,500 2,400 2,357 2,300 2,200 2,100 2,000 1,900 Period of oscillation (T) 455,000 480,000 515,000 Spring constant (k) 530,000 2,145 712,000 1,955 900,000
The displayed graph shows that there is a decreasing trend in the period as the spring constant increases. It should be remembered that these results are the product of tests on the same object, i.e. the same mass. This is because the spring constant k represents the ease with which the spring can be elongated. In other words, according to Sears et al. (2018), in Simple Harmonic Motion, the restoring force Fx is directly proportional to the displacement x generated by the mass with respect to equilibrium. Thus, the constant of proportionality between Fx and x is the force constant k, which will have opposite signs, as this represents the desire of the spring to return to its equilibrium state. This is defined by Hooke's law, described by the formula F = kx Now, in simple harmonic motion the period T can be defined as m T = 2π₁1 =2 TT-√ √ k where the mass will exert a force on the spring, which will determine the maximum displacement that this will have according to its constant k. As k is dividing inside the root, it implies that the greater its value, the less displacement it will allow in the spring, causing the period to be smaller, since the period is the time in which a cycle is completed. The greater the force of restitution with which the spring acts, the faster it will return to the point of equilibrium, and the further it moves away from it, the faster it will return, passing through the point of equilibrium at a higher speed and completing the cycle in less time. B. Sometimes it is necessary to reduce the relationship between two variables to a linear relationship of the form y=mx+b, to do this, first, compute the first column as the reciprocal of k and the second column as T². Make a scatter plot of the result (do not forget to place title, labeling of axes and their respective units) and find the equation for the resulting linear function by making a linear fit. Write down the value of the slope and the intercept by placing the respective units that both have.
Transcribed Image Text:The displayed graph shows that there is a decreasing trend in the period as the spring constant increases. It should be remembered that these results are the product of tests on the same object, i.e. the same mass. This is because the spring constant k represents the ease with which the spring can be elongated. In other words, according to Sears et al. (2018), in Simple Harmonic Motion, the restoring force Fx is directly proportional to the displacement x generated by the mass with respect to equilibrium. Thus, the constant of proportionality between Fx and x is the force constant k, which will have opposite signs, as this represents the desire of the spring to return to its equilibrium state. This is defined by Hooke's law, described by the formula F = kx Now, in simple harmonic motion the period T can be defined as m T = 2π₁1 =2 TT-√ √ k where the mass will exert a force on the spring, which will determine the maximum displacement that this will have according to its constant k. As k is dividing inside the root, it implies that the greater its value, the less displacement it will allow in the spring, causing the period to be smaller, since the period is the time in which a cycle is completed. The greater the force of restitution with which the spring acts, the faster it will return to the point of equilibrium, and the further it moves away from it, the faster it will return, passing through the point of equilibrium at a higher speed and completing the cycle in less time. B. Sometimes it is necessary to reduce the relationship between two variables to a linear relationship of the form y=mx+b, to do this, first, compute the first column as the reciprocal of k and the second column as T². Make a scatter plot of the result (do not forget to place title, labeling of axes and their respective units) and find the equation for the resulting linear function by making a linear fit. Write down the value of the slope and the intercept by placing the respective units that both have.
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