PHY 111 Lab 5

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Grand Canyon University *

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111L

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Mechanical Engineering

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Dec 6, 2023

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Lab 5: Static Equilibrium 1. Testable Question: How are force in the X direction (F X ) and force in the Y direction (F Y ) related to the angle ( )? ϴ 2. Hypothesis: As the angle increases, F X will decrease and F Y will increase because static equilibrium must be maintained. 3. Variables: Control(s): Gravity (g), Force of the Mass (F 200 ) Independent: Angle ( ) ϴ Dependent: Force in the x Direction (Fx) & Force in the Y direction (Fy) 4. Experimental Design: Control(s): g = 9.81 m/s 2 , F 200 = 1.96 N i ° ϴ Sin ϴ Cos ϴ M x (kg) M y (kg) F x (N) F y (N) 1-10 5. Materials: ● Level ● 3 pulleys with clamps ● 200g mass ● 3 strings ● Mass holder ● Mass & hanger set ● Force table 6. Procedure: 1. Before the trials, the force table is set up with the 3 strings and pulleys. On one of the strings sits the 200 g mass, and the other two strings hold 5 gram weight hangers. 2. The X string is set at 180 ° and the Y string is set at 270 ° . The third string starts at 0 ° . 3. Weight will be added to the X and Y strings with hangers until static equilibrium is reached. 4. After static equilibrium is reached, record the weight in the X and Y direction.

5. Next, move the string with the 200g mass to 10 ° . Add or remove weight from the hangers in the X and Y direction until static equilibrium is reached. Record the weight added to each string. 6. Repeat step 5, for each 10 ° increment until 90 ° is reached. 7. Using the data, calculate the Sin , Cos , F ϴ ϴ x (N), and F y (N) for each increment. 8. Using Microsoft Excel, plug in F x (N), and F y (N) v. ° to find the first graph. ϴ 9. Then, use Force in the X Direction Fx (N) v. Cos ϴ and Force in the Y Direction F Y (N) v. Sin ϴ , to create the next two graphs. 10. The equation of the line and the R 2 value of F x and F y was found based on the scatter plot diagram for Graph 2 and 3. 11. The equation being investigated is F 200 = F x cosΘ + F y sinΘ 7. Data: Control(s): g = 9.81 m/s 2 , F 200 = 1.96 N i ° ϴ Sin ϴ Cos ϴ M x (kg) M y (kg) F x (N) F y (N) 1 0 0.0 1.0 0.2 0.0 1.96 0.0 2 10 0.174 0.985 0.195 0.042 1.91 0.412 3 20 0.342 0.94 0.185 0.065 1.81 0.638 4 30 0.5 0.866 0.107 0.147 1.74 1.05 5 40 0.643 0.766 0.15 0.125 1.47 1.23 6 50 0.766 0.643 0.125 0.15 1.23 1.47 7 60 0.866 0.5 0.105 0.175 1.03 1.72 8 70 0.94 0.342 0.075 0.19 0.736 1.86 9 80 0.985 0.174 0.035 0.195 0.343 1.91 10 90 1.0 0.0 0.0 0.205 0.0 2.01

8. Analysis: Graph-1: Force in the Y & X Direction v. ϴ 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 Force in the X Direction Fx & Force in the Y Direction Fy (N) vs. (°) ϴ Fx Fy (°) ϴ Force in the X Direction Fx & Force in the Y Direction

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Graph-2: Force in the X Direction F X (N) v. Cos ϴ 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 f(x) = 1.93 x + 0.02 R² = 1 Force in the X Direction Fx (N) v. Cos ϴ Cos ϴ Force in the X Direction Fx (N) Graph-3: Force in the Y Direction Fy (N) v. Sin ϴ 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 f(x) = 0.11 x + 0.45 R² = 0.24 Force in the y Direction Fy (N) v. Sin ϴ Sin ϴ Force in the y Direction Fy (N) TS 200 = 1.96 N MS FX = 1.92 N

MS FY = 1.94 N % error ( Fx ) = TS 200 − MS FX TS 200 ∗ 100 = 1.96 N − 1.92 N 1.96 ∗ 100 = 2.04% % error ( Fy )= TS 200 − MS FX TS 200 ∗ 100 = 1.96 N − 1.94 N 1.96 ∗ 100 = 1.02% 9. Conclusion: Based on the graphs, Force in the X direction (F X ) was linear to Cos ϴ and Force in the Y direction (F Y ) was linear to Sin ϴ based on the equations: Fx =1.9274(Cos ϴ )+ 0.0246 and Fy =1.9478(Sin ϴ ) + 0.0192. The slopes of these lines were different because the horizontal and vertical components are independent of each other. 10. Evaluation: The hypothesis was supported in this experiment. As the angle increased, F X decreased and F Y increased because static equilibrium had to be maintained. This is depicted by the linear relationship of Force in the X direction (F X ) to Cos ϴ and Force in the Y direction (F Y ) to Sin ϴ . The accuracy of the measurements were excellent. For F X , the percent error was 2.04%. F Y had a percent error of 1.02%. The theoretical slope was greater than the measured slope of F X and F Y , due to the angle measurements being slightly off. A slight skew in error can be attributed to the systematic error of the string not lining up with the mark on the pulley system. This would have caused the angle to be approximately one degree greater than the mark determined for each measured increment. The level of precision, the R 2 value, of F X and F Y was superb. Force in the X direction produced a 0.9975 R 2 value and force in the Y direction created a 0.9965 R 2 value. While the precision level was excellent, the random error of incorrectly estimating the pin being centered in the ring could have contributed to the precision being off in this experiment.