Spring-Mass Lab - Spring-Mass Lab PDF

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Spring-Mass Lab...

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Kachi Anyanwu PHYS.1030L-813 Properties of a Vertical Spring-Mass System 9 November 2017 Ngan Trinh, Kristen Geraghty Sanjanee Waniganeththi Objective: The periodic motion of a mass hanging from a spring is studied as the hanging mass, spring constant, and amplitude are varied in relation to its period.

Background and Predictions When a spring is stretched by adding a hanging mass as in Figure 1, there are two forces acting on the spring-mass system. The first is gravity, since the hanging mass is hanging. The second is the force exerted by the spring to prevent the mass from free falling. This force known as the restoring force. According to Hooke’s Law, the restoring force of a spring can be modeled through if the spring constant and the distance the spring is stretched are known. (1) F R=−kx where FR is the restoring force of the spring, k is the spring constant, and x is the distance the spring is stretched. At equilibrium, force of gravity acting on the spring-mass system is equal the restoring force because there is no net force. In this case, (2) mg= kx , where m is the hanging mass, g is gravitational acceleration, k is the spring constant, and x is the distance the spring is stretched. Since, the restorative force of the spring is proportional to the distance it is stretched or compressed, the spring-mass system oscillates in simple harmonic motion where the restorative force always works opposite to its displacement to restore equilibrium. For example, if the spring is compressed, it acts to lengthen the spring. Likewise, if the spring is stretched, the restoring force acts to compress the spring. This causes the spring to oscillate until there is no longer a net force acting on the spring as a result and equilibrium is reached. The distance that the spring is compressed or stretched is known as the amplitude. The simple harmonic motion of the spring can be predicted using the equation (3)

√

m where T is the period, m is the hanging mass, and k is the spring constant. k This demonstrates that the period of the spring in simple harmonic motion is dependent on the hanging mass and the spring constant, but independent of the amplitude. Based on the Equation 3 above, we expected the period to be independent of amplitude, to increase as mass increased, and decrease as the spring constant increased. T =2 π

Figure 1. Spring-Mass System at Equilibrium

Apparatus and Procedure Figure 2. Apparatus Setup

List of Materials:  Various hanging masses  Two springs  Hook for spring attachment  Position sensor  Meter stick Procedure Part 1. Measuring the Spring Constant. Given two different springs, we first hung the lighter spring from the hook. We measure the position of the spring at rest with no added mass using the last ring as a reference. The change in position was then measured using the last ring as a reference point after a hanging mass of 30 g was added. We repeated the measurements with a hanging mass of 50 g attached to the lighter spring. Then, we repeated the entire procedure using a heavier spring and hanging masses of 50 g and 70 g. Part 2. Period vs. Amplitude. We attached the lighter spring to the hook with a hanging mass of 75 g and placed the position sensor on the ground directly under the spring. We measured the initial position of the spring with a meter stick. After compressing the spring 2.0 cm, we released it. The change in position caused by the oscillation of the spring at an amplitude of 2.0 cm was analyzed using the program Data Studio. We recorded the period of oscillation and repeat the procedure with an amplitude of 4.0 cm, 6.0 cm, and 8.0 cm.

Part 3. Period vs. Hanging Mass. We attached the lighter spring to the hook with an initial hanging mass of 35 g and placed the position sensor on the ground directly under the spring. After compressing the spring 2.0 cm, we released it. The change in position caused by the oscillation of the spring at an amplitude of 2.0 cm was analyzed using the program Data Studio. We recorded the period of oscillation and repeat the procedure with a hanging mass of 40 g, 45 g, 50 g, 55 g, and 60 g. Part 4. Period vs. Spring Constant. We attached the heavier spring to the hook with an initial hanging mass of 55 g and placed the position sensor on the ground directly under the spring. After compressing the spring 2.0 cm, we released it. The change in position caused by the oscillation of the spring at an amplitude of 2.0 cm was analyzed using the program Data Studio. We recorded the period of oscillation and repeat the procedure with a hanging mass of 60 g, 65 g, 70 g, 75 g, 80 g. Results and Analysis Results Figure 3. Spring Constants Lighter Spring Heavier Spring Mass, g 30 50

Spring Amplitude, cm Constant, N/m Mass, g 4.3 6.8 6.9 7.1 Average k = 6.95

Table 2. Period vs. Amplitude Amplitude, cm Period, s 2.0 0.6474 4.0 0.6474 6.0 0.6474 8.0 0.6474 Table 3. Period vs. Hanging Mass Mass, g Period, s 35 0.4731 40 0.4980 45 0.5225 50 0.5478 55 0.5810 60 0.5976

Amplitude, cm 50 70

Spring Constant, N/m 4.4 11.1 7.1 9.7 Average k = 10.4

Table 4. Period vs. Spring Constant Mass, g Period, s 55 0.5478 60 0.5810 65 0.6225 70 0.6474 75 0.6640 80 0.6723 Amplitude Period

± 0.01 cm ± 0.0001 s

Figure 3. Period vs. Mass with Different Spring Constants 0.7000 f(x) = − 0 x² + 0.03 x − 0.44

Period, s

0.6500

Lighter Spring Power (Lighter Spring) Heavier Spring Polynomial (Heavier Spring)

0.6000 f(x) = 0.1 x^0.45 0.5500 0.5000 0.4500 30

40

50 Mass, g

Sample Calculations 1. Spring Constant mg k= x 0.030 kg ×9.8 ¿

m s2

0. 043 m N 6.8 ¿ m 2. Average Spring Constant (k −k ) k= 2 1 2

60

70

80

1 N ¿ ( 7.1−6.8) m 2 N ¿ 7.0 m Analysis The results of Part 1 shown in Table 1, follow the general trend where the amplitude increases with increased hanging mass. When comparing the spring constants using the 50 g hanging mass, the data demonstrates how the amplitude decreases as the spring constants increase. In Table 1, the period remained 0.6474 s despite the change in amplitude. Therefore, the data supports that the period is independent of amplitude. As the hanging mass increased in Part 3 and Part 4 (Tables 3 and 4) the period increased. Comparing the data for 55 g and 60 g hanging mass, the data follows the trend where an increased spring constant decreased the period of the spring-mass system in simple harmonic motion. Figure 3 shows how the change in the period of the spring-mass system decreases as more weight is added. When comparing 55 g and 60 g specifically, the heavier spring has a shorter period for the same added hanging mass. Discussion The results of the experiment are generally reliable; the data follows the trends which are generally accepted. The data collected in Part 3 demonstrates how the amplitude of the springmass system is independent of the period. As the spring constant increased, the amplitude decreased in Part 2 and the period decreased as expected in Part 3 and 4. Likewise, the period is directly related to the mass of the object and the spring constant. However, this relationship is a square root function that is poorly modeled by the best fit lines. While the data from the heavier spring has a distinct curve similar to that of a square root function, the data from the lighter spring does not. But, the data from lighter spring in theory it does still support the general model because when fit to a power function, the power is 0.44; the theoretical value should be 0.5 for square root. Without the ability to fit the data collected to a square root function, there is no real way to calculate the exact percent error associated with the data collected in this experiment. The biggest source of error in this experiment was the alignment. In theory, the position of the spring-mass system should be a simple sine graph. But throughout the course of the experiment, the position of the spring-mass system showed signs of interference; although the spring clearly oscillates, there were sometimes multiple crests to the sinusoidal waves. This could also be a result of the hanging masses used; as more weight was added or the amplitude was increased, the restorative force of the spring increased resulting in a less controlled oscillation and possible deformation of the spring. The springs that were used were also damaged which could have affected the oscillation of the spring-mass system. The interference could be minimized if the hanging mass was reduced, the spring contant was increased, the amplitude decreased, a mechanism which could align the spring, or new springs were used. Conclusion As the hanging mass of a spring-mass system in simple harmonic motion is increased, its period also increases; the period is independent of the amplitude....