03 Physics 205L Atwood Machine S24

Physics 205L Spring 2024 Apparatus 1. PASCO Interface 850. 2. Smart Pulley. 3. Ring stands, table clamps and right-angle utility clamps. 4. 1.5 m of string and two mass hangers. 5. A set of numbered masses 6. PASCO Capstone. Procedure 1. Locate the Atwood Machine file on the desktop and open it. 2. Using the top-loading electronic balance, measure and record the total mass of M 1 and M 2 . 3. Put M 1 on one side of the pulley and M 2 on other side connected with a string. 4. Move a numbered 2-, 5-, 10-, 20- grams mass (as assigned by the instructor) from M 2 to M 1 . Record the mass of M 1 and M 2 in the table below. 5. Raise M 1 up, dampen any movement, click RECORD, release, and monitor the velocity, and then click STOP. You should see a linear graph with a positive slope. 6. Perform a linear curve fit to the velocity vs. time graph.The slope of the fitted data yields the acceleration of the cart. Record this value in Table I along with the uncertainty. Caution: To report the measured acceleration in a manner that is consistent with its uncertainty, you might need to modify the display of the fitting parameter for each run. To change the display of the fitting parameters, follow the instruction below:  Click the curve fit display box, right click within this box, and select “Curve Fitting Properties.”  Choose “Numerical Format.”  Click the upside-down triangle next to “Coefficients” and select “Fixed Decimal” from the drop-down menu next to “Number Style.”  Choose a number greater than 4 from the drop-down menu “Number of Decimal Places” and click the upside-down triangle next to “Coefficients.”  Click the upside-down triangle next to “Coefficient Uncertainties” and select “Fixed Decimal” from the drop- down menu next to “Number Style.”  Choose a number greater than 4 in the drop-down menu “Number of Decimal Places” and click the upside-down triangle next to “Coefficient Uncertainties.”  Click “OK” located at the bottom right of the “Properties” window. 7. Repeat steps 4-6 until all numbered masses have been moved from M 2 to M 1 . Keep the numbered masses in order when you move it. 3

Physics 205L Spring 2024 Average of M Ratio = 0.2 + 0.122 + 0.117 + 0.1158 + 0.1155 + 0.115 + 0.1147 + 0.115 + 0.1151 11 = 0.124 kg Average Deviation : ∑ ( x i − x avg ) n ( 0.2 − 0.124 )+( 0.122 − 0.124 )+( 0.117 − 0.124 )+( 0.1158 − 0.124 )+( 0.1155 − 0.124 )+( 0.115 − 0.124 )+( 0.1147 11 = 0.001kg Standard Deviation: σ = √ ∑ ( x i − x ) 2 n − 1 = 0.02 kg √ ( 0.2 − 0.124 ) 2 + ( 0.122 − 0.124 ) 2 + ( 0.117 − 0.124 ) 2 + ( 0.1158 − 0.124 ) 2 + ( 0.1155 − 0.124 ) 2 + ( 0.115 − 0.124 ) 2 + ( 0.11 10 δ M ratio= 0.02 √ 11 = 0.006 kg 4. Compare the values of a meas and a theory . Does the percent error between the two values decrease as  m increases? Explain. No, the percent error of between the two values would actually increase when the change of m increases because it requires more force to accelerate the object therefore the difference would be much larger between both a measure and a theory. 5. Use the equation M Ratio =F NET /a MEAS to derive the theoretical expression for the uncertainty in the total mass of the system, d M Ratio , in terms of the measured quantities M 1 , M 2 and a MEAS along with estimated uncertainties d M 1 , d M 2 and d a MEAS . Then derive the expression for theoretical relative uncertainty d M Ratio /M Ratio . δM Ratio = √ ( ∂M Ratio ∂ M 1 ) 2 ( δ M 1 ) 2 + ( ∂M Ratio ∂ M 2 ) 2 ( δ M 2 ) 2 + ( ∂ M Ratio ∂a measure ) 2 ( δ a measure ) 2 + ( ∂ M Ratio ∂ g ) 2 ( δg ) 2 M Ratio = Fnet a measure M Ratio = ∆m 1000 x a measure g δM Ratio = √ ( g 1000 x a measure ) 2 ( δ M 1 ) 2 + ( − g 1000 x a measure ) 2 ( δ M 2 ) 2 + ( ∆m 1000 √ a measure g ) 2 ( δ a measure ) 2 6