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Conservation of total energy E This is a Formal report. Include the following sections: Abstract, theory, experimental setup, results, analysis, conclusion d d The idea: For a CONSERVATIVE SYSTEM, the total energy E is CONSTANT. In this experiment a glider, mass m = 0.190 kg, will float on an air track, and so there is zero friction. We will release the glider from rest at the top of the air track (position zero in the figure). The air track ramp is 2 meters long. If we assume there is no significant air drag, then the total energy anywhere along the air track will be constant (until it hits the bottom of course). Recall, near the surface of the earth: (1) PEgravity = mgh where the height h is maximum at the top of the ramp (x = 0), and h is ZERO at the bottom of the ramp (x = 2 meters) (2) h = (2-x)∙sin(ϴ) = (2-x)∙0.025 Here x is in meters (3) KE = ½ mv2 (4) total energy E = KE + PE • • • • Obtain speed data at various positions x down the ramp. For each position x calculate the height h using h = (2-x) )∙0.025 Calculate the PE and the KE, and total energy E. The units for energy is Joules (kg∙m2/s2 = Joules) PLOT on a SINGLE plot: KE, PE and E, all on the vertical (y) axis. Plot x (meters) on the horizontal (x) axis. Discuss the plot. Does it show conservation of enery E? Lab D – The spring constant “K” of a spring, and simple harmonic motion Abstract The purpose of this assignment is to study the estimate the value of the spring constant “k” of a string. The estimate is obtained after performing and analysing the results of two different experiments, involving said string. Theory and Experimental Setup A spring is an elastic object which can store mechanical energy. If a string is stretched or compressed, it will react to this deformation by applying a force. It can be observed that, under certain conditions, this force is proportional to the displacement of the spring. This behaviour is summed up by Hooke’s Law: 𝐹 = −𝑘𝑥 𝑁 The term 𝑘 in Hooke’s Law is called spring constant and is measured in [𝑚]. When the constant 𝑘 is “large”, the spring is “strong” or “rigid”; when the constant 𝑘 is “small” the spring is “weak” and “easy” to yield. To obtain an estimate of the spring constant 𝑘 it is possible to proceed in two simple, but different ways. The first experiment to measure the spring constant consists of applying a known and constant force to the spring and measuring the displacement of the spring. The constant force is exerted by a known mass subject to the gravitational acceleration. It is easily possible to repeat the experiment and change the applied mass to obtain multiple measurements thus reducing the measurement error after an appropriate analysis of the results. In the second experiment, the spring constant is measured indirectly by measuring the period of oscillation of the harmonic motion produced by letting a mass oscillate up and down. The oscillation period is a function of the mass attached to the spring (the spring mass is not considered) and of the spring coefficient. On the other hand, it can be demonstrated that the period of the oscillation is (with a good approximation) independent on the amplitude of the oscillation, so it is not needed to make the mass oscillate with a known amplitude to determine the spring constant. From Hook’s Law and the definition of velocity and position it can be written: 𝐹 = −𝑘𝑥 = 𝑚𝑎𝑦 = 𝑚 𝑑𝑣𝑦 𝑑2 𝑦 =𝑚 2 𝑑𝑡 𝑑𝑡 The solution to this differential equation is the following: 𝑦 = 𝑦𝑚𝑎𝑥 cos 𝜔𝑡 Where 𝑦𝑚𝑎𝑥 is the maximum distance y from equilibrium (or the amplitude of the oscillations) and 𝜔 is the angular velocity: 𝜔= 2𝜋 𝑇 It can be easily proved that the given sine wave is a solution to the differential equation by substitution. It can also be demonstrated the following: 𝑘 𝜔=√ 𝑚 → 𝑇 = 2𝜋√ 𝑚 𝑘 Where T is the natural period of oscillation. Finally, by rewriting the last equation: 𝑚= 𝑘 2 𝑇 4𝜋 2 This equation will be used to estimate the value of the spring coefficient. Results Analysis of Experiment 1 (a) – (b) Graph plot with regression line and regression equation: Spring Force vs Spring Displacement 1.6 1.4 Force [N] 1.2 F = 3.5905 x – 0.6963 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Position [m] From the linear trendline it is determined that the spring coefficient estimated value is: 𝑘 = 3.59 𝑁 𝑚 (c) – (d) 𝑁 Using the regression tool of Excel it can be obtained the slope standard error: 𝜎 = 0.020 𝑚 The computed 95% confidence interval for the spring coefficient is: 𝑘1 = 3.59 ± 0.047 𝑁 𝑚 → 𝑘 ∈ [3.544; 3.637] 𝑁 𝑚 Analysis of experiment 2 (a) – (b) Graph plot with regression line and regression equation: Mass [kg] Mass vs Squared Period 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 m = 0.0917 T2 – 0.0545 0 0.2 0.4 0.6 0.8 1 Squared Period 1.2 1.4 1.6 1.8 [s2] The slope of the equation is 0.917. From it the spring coefficient can be calculated using the following formula: 𝑘 = slope ⋅ 4𝜋 2 = 0.917 ⋅ 4𝜋 2 = 3.62 𝑁 𝑚 (c) – (d) The slope standard error is 0.000912. The slope 95% confidence interval is the following: slope = 0.0917 ± 0.0021 → slope ∈ [0.0896; 0.0938] Thus, knowing that the spring coefficient uncertainty is proportional to the slope uncertainty, with a factor equal to 4𝜋 2 , it follows: 𝑘2 = 3.620 ± 0.083 𝑁 𝑚 → k ∈ [3.537; 3.703] 𝑁 𝑚 (e) It is very easy to determine that the value estimated with the first experiment does fall into the uncertainty interval obtained with the second experiment (which has a bigger uncertainty interval). This strongly suggests that the two computed values: 𝑘1 and 𝑘2 do agree with each other. Conclusions The results obtained when estimating the spring coefficient can be considered satisfying. Two completely different methods (based on different physical principles) have been used to estimate the spring coefficient, and the results were consistent with each other. It is interesting to note that the measurement uncertainty obtained with the first experiment is smaller compared to the one obtained with the second experiment. This is probably due to the fact that it is difficult to reach a good trade-off of the number of oscillations to measure and to perform dynamic measurements of time. In fact, using too few periods would greatly increase the measurement errors. On the other hand, it is not recommended to let the oscillations last too much because the period would change due to unmodeled friction. x(cm) x/100(m) comp 170 140 120 90 60 40 END h(m) 0,01 v(m/s) PE(J) 35,2 31,5 28,8 26,3 19,8 15,4 KE(J) E(J) COMP x (cm) x (m) h (m) = Too =(2-)x0.025 ² .) +0,025 >PE (J) KE(J) E (J)
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