PHY 105M Lab 5

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University of Texas *

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105M

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Mechanical Engineering

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Dec 6, 2023

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pdf

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PHY 105M Lab 5 Part 1: Hanging Wires Method: Our goal for this lab is to calculate the spring constant, k, for the restoring force given from stretching two different strings by attaching extra masses to them. We chose to experiment with the plastic string and the guitar string on the right (there were two hanging). We hypothesize that guitar string will be easier to stretch and will thus produce greater measured distances as mass and force increase, resulting in overall smaller k values compared to that of the plastic string. The displacement we are measuring is the distance (in meters) in the vertical direction that the string extends from the equilibrium point as a result of adding a certain mass (in kg). The force (in N) that restores this displacement is given by Hooke’s Law as . |𝐹| = 𝑘|∆𝑦| We conducted 3 trials for each string in which we varied the masses of the weights attached to the string (0.100kg, 0.0500kg, 1.000kg). We first recorded the initial measurement of each string without any mass attached by straightening it out (we were careful to not exert extra force on the string) and measuring where the meterstick we are holding horizontally intersects with the meterstick installed on the right side of the device – this value will be the equilibrium point. The position of the meter stick on the right side of the device hanging parallel to the wires will remain constant throughout all trials. Next, we attached our mass to the string and calculated displacement by subtracting the measurement we recorded after attaching the mass to the string from the initial measurement when no mass was attached. We repeated the absolute value of this calculation for each mass we are testing on each string. We then repeated these steps for the second wire, the plastic wire. Gravity (g) in the context of F = mg is equal to 9.8 m/s 2 due to the fact that we are measuring the displacement in the vertical direction (at an angle of 90º from the horizontal beam). The displacement was originally measured in centimeters and later converted to meters in order to calculate the unweighted and weighted spring constants and their associated uncertainties. We calculated the spring constant for each mass tested on each beam by using the equation Next, to find the unweighted uncertainty of the spring constant, k, we used 𝑘 = 𝐹 | | ∆𝑦 | | . the following equation and the associated uncertainties: δ𝑘 = − 𝐹 𝑦 2 • δ∆𝑦 ( ) 2 + 1 𝑦 • δ𝐹 ( ) 2 Uncertainty in Distance ( ∆y) = 0.0005 meters δ *This is the systematic uncertainty associated with the meterstick which is the primary source of uncertainty in this experiment. Uncertainty in Force = 0.005 N δ𝐹 = 𝑔 • δ𝑚 ( ) 2 *Uncertainty in Mass ( ) = 0.0005 kg δ𝑚 *g = 9.81 m/s Data: Plastic String Mass (kg) = 0.0005 kg δ𝑚 Displacement (∆y) ∆y = 0.0005 m δ Force (N) = mg (g=9.8m/s²) 𝐹 = 0.005 N δ𝐹 Spring Constant (k) Uncertainty of k δ𝑘 0.100 kg 0.001 0.98 N 980 490

0.500 kg 0.003 4.90 N 1633 272 1.000 kg 0.005 9.80 N 1960 196 Guitar String Mass (kg) = 0.0005kg δ𝑚 Displacement (∆y) ∆y = 0.0005 m δ Force (N) =mg (g=9.8m/s²) 𝐹 = 0.005 N δ𝐹 Spring Constant (k) Uncertainty of k δ𝑘 0.100 kg 0.002 0.98 N 490 123 0.500 kg 0.005 4.90 N 980 98 1.000 kg 0.008 9.80 N 1225 77 Conclusion: Plastic String k Weighted Average and Uncertainty = 1766.6 +/- 29.7 k weighted = 980 1 490 2 ( ) + 1633 1 272 2 ( ) + 1960 1 196 2 ( ) 1 490 2 + 1 272 2 + 1 196 2 k weighted = δ 1 2 980 1 490 2 ( ) + 1633 1 272 2 ( ) + 1960 1 196 2 ( ) 1 490 2 + 1 272 2 + 1 196 2 Guitar String k Weighted Average and Uncertainty = 1006.2 +/- 22.4 k weighted = 490 1 123 2 ( ) + 980 1 98 2 ( ) + 1225 1 77 2 ( ) 1 123 2 + 1 98 2 + 1 77 2 k weighted = δ 1 2 490 1 123 2 ( ) + 980 1 98 2 ( ) + 1225 1 77 2 ( ) 1 123 2 + 1 98 2 + 1 77 2 Conclusion: Overall, the experiment was successful in that we were able to obtain the weighted spring constant values

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