Simple Harmonic Motion and Oscillations on an Incline
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1 Eric Dammer Dr. Fauerbach Physics 14 November 2001 Purpose Simple Harmonic Motion and Oscillations on an Incline The force constants of two springs are found experimentally. Taking angular frequency ω = 2πf, to give T=2π/ω; since ω = (k/m), T = 2π (m/k) and experimental period measurements are thus used to find theoretical spring constants which are compared with experimental values. Mass of the springs is neglected. Materials and Apparatus: Procedure Massive dynamics cart with inclinable track Two springs one strong and another medium Base-supported support rod Mass set Stopwatch Mass balance Meter stick The cart-track apparatus is set up with an incline determined trigonometrically between twenty to thirty degrees such that the end supporting the spring in turn attached to the cart is raised and this point is the zero position. Data for the displacement of both springs tested are given for varying forces in addition to that imparted across the spring by the weight of the cart alone (precisely 500.0g). See Table 1. In table 2, data for experimental period measurements of both springs are taken with varying incline but controlled mass. Finally, in table 3, mass is varied while the incline is returned to the initial incline angle of degrees [h=51, l=113 cm; sin -1 (h/l)=26.83 ].
2 Dammer, 2 Data & Observations Table 1: Equilibrium and Displacement on an Inclined Spring-Cart Apparatus Stiff Spring at incline Medium Spring at incline Added Mass Equilibrium Position Shift Position (from Equilib.) Addtl Force mg sin θ Added Mass Equilibrium Position Shift Addtl Force Position (from Equilib.) mg sin θ 0g g g g g g g g g g g g Table 2: Period of the Spring-Cart Apparatus in Motion with Statistical Analysis times in sec Trial at at Medium Stiff Medium Stiff Spring Spring Spring Spring One Two Three Four Five Average Std. Deviation Std. Error Table 3: Experimental Period with Varying Mass of Stiff Spring Stiff Spring at incline Added Total Period Mass Mass 0g g g g g g
3 Dammer, 3 Analysis Graph 1: Experimental Determination of Spring Constants Force (N) y = x y = 17.47x Medium Spring Strong Spring Displacement (m) Considering uncertainty associated with the endpoints of each experimental segment, the slope might vary to extremes determined as the difference in force plus or minus two times associated uncertainty divided by the difference in displacement plus or minus two times the uncertainty associated with these measurements. Because of far greater precision in the mass measurements, only distance measurements figuring into angle determination associated with force calculation, and the displacement measurements themselves need be considered. Assuming uncertainty of 0.003m in any measurement, [arcsin(-.003) - arcsin(0.003)]/2 may be ignored, leaving uncertainty to be accounted only by displacement. Thus, in the slope determination of F/x, x can be at most inaccurate by twice the uncertainty, thus, the factor of uncertainty for either of the above slopes is ratio given by slope with denominator adjusted by ± This translates to uncertainty of ±(30.9-(30.9/0.994)), yielding a k strong value for the tight spring of 30.9 ± 0.19; k medium, then, is 17.5 ± Theoretical period (T), as 2π (m/k) is then found to be 0.799s for T strong and 1.06s for T medium. Using [(δτ/δk)²(0.19) + (δτ/δm)²(0.003)] for uncertainty with upward rounding gives ± 0.045s for T strong and ± 0.060s for T medium. Regarding data from table 2, it is noteworthy that the natural period of each spring did not change within the standard error while varying the angle of the incline. This fits with the idea that each spring has a natural frequency, and hence period. Hence, were the angle of incline vertical (90 ), the period would remain unchanged, assuming no plasticity in the spring at this magnitude of force. The 1.08 to 1.09s experimental period of the medium spring also fits squarely with the theoretical period of 1.06 ±.06s as calculated above, while the same is true for the strong spring period experimentally found at 0.82s, within the limits of the above
4 Dammer, 4 calculation of ± 0.045s. There is no significant discrepancy for which the theory does not account! In the graph to the left, the data from table 3 are graphed and the exponent of 0.58 is calculated by excel While the theoretical equation for period y = x 0.58 relates mass to period as a factor with exponent 0.5, it is important to remember that the mass in question was applied to the spring at the indicated incline. To make sense of the data, it is worthwhile to examine the quantities for correlation with the known and experimentally proven equation for period. Doing this in table 4 below shows that the theoretical k for data originally shown in table 3 would appear Mass (kg) to vary with mass, but this does not Graph 2: Mass vs. period of an inclined spring correspond to the known fact that k must at with exponential best fit. be constant. Apparently, the exponential of.08 in graph two is in fact nothing more than experimental error due to manual timing methods using the stopwatch that cannot approach the accuracy of measurements used in tables one and two. The one remaining anomaly is that the measurements appear to falsely indicate a nonlinear trend towards a weaker spring constant with greater weight. In fact, there may be some truth to this, though the trend should be at least smooth if not linear. The mass of the spring, which was ignored in all calculations to this point, must become less significant as additional weight is added to the load on the spring of course, ideal k and period cannot vary, but this phenomenon in the experimental data takes on theoretical significance. Stiff Spring at incline Period (s) Added Total Period Theoretical Mass Mass (s) k=4π 2 m/t 2 0g g g g g g Table 4: Theoretical k based on table 3 data. k-trend (kg/s 2 ) y = Ln(x) Mass (kg) Graph 3: Finding a true spring constant with varying mass. By varying mass load on a spring that cannot be weighed, an opportunity to calculate the weight of the spring may in fact become possible! Arbitrarily given that
5 Dammer, 5 about 1/3 of the mass of the spring participates in the effective harmonic motion which it is powering (Giancolli 364), and assuming a logarithmic trend towards a limit at (only estimated roughly with the less reliable timing data) k=27.5 (let s call this the true experimental spring constant), it would seem that the strong spring would have a mass m such that the original equations tested hold true. I.e., k original = 30.9 = 4π 2 (m/ )/t 2, so m = 3 { [30.9 (0.82)²/ 4 ( )² ] }, or ~ kg. Conclusions The theory regarding simple harmonic motion is indeed elegant and experimentally verifiable with few if any conditions affecting the validity of the relationships set forth in the relation T = 2π (m/k). Using theoretical knowledge that natural period and the spring constant are indeed constant, any significant experimental variation from the expected can be interpreted as another variable or constant which was not considered prior in the above case, the trick would appear within reason for estimating a mass of the spring itself. Were the stopwatch replaced by a more precise device such as a strobe or regular laser pulse at the cart exposed on time-elapse film, the determination may in fact be quite accurate, while the guess that a logarithmic relation in table 4 approaching a limit k value should then be verified.
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