02_Physics_205_Lab_2_1D_Motion_

Physics 205L Calabrese 1D Motion Introduction The quantitative investigation of motion for real-life situations is always complicated by external influences such as friction and the necessity of making measurements (interacting with the system). Ideal theoretical models, which are so useful for discussion purposes, generally cannot be duplicated in the laboratory. In this laboratory exercise, you will look at an example of one-dimensional accelerated motion down an inclined plane. You will soon realize that a careful consideration of external influences along with good experimental technique is essential to the outcome of an experiment. It is highly recommended that you fully understand the procedure before attempting it, and that you always pay close attention to detail. Objectives Experimental 1. To graphically describe the motion of an accelerated object in one dimension. 2. To calculate the acceleration due to gravity from measured quantities. 3. To understand error analysis. 4. To understand the difference between systematic, random, and personal errors. Learning 1. To practice making measurements and working with significant figures. 2. To review and practice the techniques of graphing 3. To reinforce your understanding of one-dimensional kinematics Theory Before beginning, be sure that you understand the basic concepts of position, displacement, velocity and acceleration in one dimension, and how they relate to each other. You should also review freefall, and how the displacement mathematically depends on the acceleration due to gravity (refer to derivation in your textbook). 1. Define the terms displacement, position, velocity, and acceleration using complete sentences. 2. Prove that the average velocity in a time interval from t 1 to t 2 =t 1 +  t is equal to the instantaneous velocity in the middle of the time interval between t 1 and t 2 { e.g. (t 1 + t 2 )/2} for an object moving at constant acceleration. t 2 – t 1 = V i + at t 2 – t 1 = ∫ t 1 t 2 V i + at dt t 2 – t 1 =[ V i t + 1 2 at 2 ] t2 t1 t 2 – t 1 =[ V i ( t 2 − t 1 ) ¿ +[ 1 2 a ( t 2 − t 1 )( t 2 + t 1 ) ] ] 1 Experiment 2

Physics 205L Calabrese 0 = [ V i ( t 2 − t 1 ) ] + [ 1 2 a ( t 2 − t 1 ) ( t 2 + t 1 ) ] t 2 − t 1 V i = 1 2 a ( t 2 − t 1 ) 3. Starting with the expressions for average acceleration (refer to your textbook), average velocity at constant acceleration; algebraically (NO CALCULUS) derive the equation for one- dimensional motion that relates displacement to the constant acceleration and time. a x = V xf − V xi t V f = V i t + at V avg = V 1 + V 2 2 = Δ x Δ t x 2 − x 1 = V xavg t = 1 2 ( V 2 + V 1 ) t x f = x 1 + 1 2 ( V 2 + V 1 ) t  x = _____________________________________ Procedure Fig 1: Photo of experimental set-up. 1. Locate the experimental setup and check to see that it is set up correctly. 2. Measure and record the angle the incline makes with the horizontal. Record your values in Table 1, 3. Cut a piece of paper tape as long as the distance between the timer and the end of the incline. 4. Attach the paper tape to the cart. 5. Thread the other end of the paper tape through the tape timer with the cart at the top of the incline. 6. Set the Timer to 40Hz and turn it on. 7. Release the cart being careful to allow the tape to thread smoothly through the timer. 8. Tape the paper tape to a table and measure the position of the dots. Don’t use the first dot because the mass may not have begun to move when the dot was made. Record the data in Table 2. Note: Only one strip of tape is needed for each group. The tape is costly and inconvenient to purchase, so please do not waste it. 2

Physics 205L Calabrese Table 1: Measured Incline Angle For Setup B Trial  1 31.8 2 32.0 3 31.9 4 32.1 5 31.7  = __32.08 _ ± _0.5 _ Table 2: Position & Time Data t (s) x (cm)  x (cm) V avg (cm/s) x 10 2 0.025 1.76 1.07 0.428 0.050 2.83 1.46 0.584 0.075 4.29 1.62 0.648 0.100 5.91 1.99 0.796 0.125 7.90 2.23 0.892 0.150 10.13 2.56 1.02 0.175 12.69 2.85 1.14 0.200 15.54 3.08 1.23 0.225 18.62 3.57 1.43 0.250 22.19 3.62 1.45 0.275 25.81 3.97 1.58 0.300 29.88 4.34 1.74 0.325 34.22 4.65 1.86 0.350 38.87 4.96 1.99 0.375 43.83 5.25 2.10 0.400 49.08 0.425 54.63 5.55 2.22 4

Physics 205L Calabrese Table 3: Velocity & Time Data t (s) V (cm/s)  V (cm/s) a(cm/s 2 ) 0.0375 0.428 0.156 6.24 0.0625 0.584 0.064 2.56 0.0875 0.648 0.148 5.92 0.1125 0.796 0.096 3.84 0.1375 0.892 0.130 5.20 0.1625 1.02 0.120 4.80 0.1875 1.14 0.090 3.60 0.2125 1.23 0.200 8.00 0.2375 1.43 0.020 0.800 0.2625 1.45 0.130 5.20 0.2875 1.58 0.160 6.40 0.3125 1.74 0.120 4.80 0.3375 1.86 0.130 5.20 0.3625 1.99 0.110 4.40 0.3875 2.10 0.120 4.80 0.4125 2.22 Graphical Presentation of Data: You are to construct the following two graphs that are to be attached to this report (Remember to follow the “Rules of Graphing”): A. Position vs. Time B. Instantaneous Velocity vs. Time 1. Perform a “fit” to your data for the position vs time graph and determine the object’s acceleration along with the uncertainty in its value, and the object’s initial velocity along with the uncertainty in its value. Record these values in Table 4. You should only keep one significant figure for the uncertainty. How many significant figures should you report for your value of the acceleration and the initial velocity? Paste your graph below. Graph A & Data: Distance (cm) vs. Time (s) - (10 2 is for conversions) - (Multiply a(2) by 2 x 10 2 for conversion to m/s/s) a(0) = (0.0103 +/- 0.0003) x 10 2 cm a(1) = (0.248 +/- 0.004) x 10 2 cm/s a(2) = (4.76 +/- 0.025) x 10 2 cm/s/s 5

Physics 205L Calabrese 3. From the acceleration data in Table 3, calculate the average acceleration, its standard deviation ( s ), and the δa = σ √ N precision of the mean. Record these values in Table 4. σ = √ ∑ ( a − a ) 2 ( N − 1 ) a = ∑ a N = 4.75 x 10 2 cm σ = 2.96 cm δa = σ √ N δa = 0.76 cm 4. Free body analysis of the set up predicts that the acceleration is related to gravitational acceleration by: a = gsinθ Where, g is the acceleration due to gravity and  , is the angle the incline makes with the horizontal. Use the theoretical expression that relates the measured accelerations to g to calculate g. Report your results in Table 4. Show a sample calculation below. g = a sin ( θ ) = 4.75 sin ( 0.5599 ) = 8.9 m / s 2 5. Use error propagation to determine the uncertainty in your calculation of g for each case. Report your results in Table 4. Show your work below. Quadratic: √ ( θ ) 2 x ( δa ) 2 + ( a ) 2 x ( δθ ) 2 = √ ( 0.5599 ) 2 x ( 0.01 ) 2 + ( 0.247 ) 2 x ( 0.000009 ) 0.005m/s 7

Physics 205L Calabrese Linear: √ ( θ ) 2 x ( δa ) 2 + ( a ) 2 x ( δθ ) 2 = √ ( 0.5599 ) 2 x ( 0.01 ) 2 + ( 0.25 ) 2 x ( 0.000009 ) 0.005m/s Average of Data Table: √ ( θ ) 2 x ( δa ) 2 + ( a ) 2 x ( δθ ) 2 = √ ( 0.5599 ) 2 x ( 0.2 ) 2 + ( 0.7 ) 2 x ( 0.000009 ) 0.01m/s Results Report the extracted values of g (one from each graph) in the table below and calculate the percentage of the error (g Rocklin = 9.8006 m/s 2 ). Show your work for the percent error calculation below. g rocklin = 9.8006 m/s 2 = 980.6 cm/s 2 Quadratic: %error = | gobserved − gtrue | gtrue x 100% = | 890 − 980.06 | 980.6 x 100% = | − 90.86 | 980.06 x 100% %error = 9% Linear: %error = | gobserved − gtrue | gtrue x 100% = | 890 − 980.06 | 980.6 x 100% = | − 110.06 | 980.06 x 100% %error = 9% Average of Data Table: %error = | gobserved − gtrue | gtrue x 100% = | 870 − 980.06 | 980.6 x 100% = | − 90.86 | 980.06 x 100% %error = 11% Table 4: Results for Initial Velocity and Acceleration Initial Velocity (m/s) Uncertainty (m/s) a (m/s 2 ) Uncertainty (m/s 2 ) Quadratic Fit (x vs. t plot) 0.247 0.005 4.73 0.01 Linear Fit (v vs. t plot) 0.249 0.05 4.7 0.01 Average from Table 3 4.75 σ √ N = ¿ 0.7 8