Lab Report #4

Report for Experiment #4 Work and Energy on a Frictionless Track Serena Le Lab Partner: Isabella Cameron TA: Jinzheng Li February 27, 2024 Abstract Throughout Investigations 1 and 2, we observed the motion of a glider along a pressurized air track using a motion sensor to prove the work energy theorem and determine an experimental value for the acceleration due to gravity, g. Using an inclined air track in Investigation 1, we proved the work energy theorem through a linear graph of v 2 vs x and determined an experimental value of ¿ 9.86 m / s 2 ± 0.595 . In Investigation 2, we observed the motion of a glider attached to a hanging mass along a horizontal air track. We again proved the work energy theorem through a linear graph of v 2 vs x and determined two experimental values of g using the slope and y-intercept of the best fit lines, to get g = 10.15 m / s 2 ± 0.730 and g = 8.81 ± 1.333 m / s 2 , respectively.

Introduction In this experiment, we observed the motion of a glider on an air track to verify the work-energy theorem and find the acceleration due to gravity using energy concepts. The first investigation consisted of a glider on an inclined track and the second investigation used a horizontal, level track. The work energy theorem states that the net work done by the forced on an object is equal to the change in kinetic energy, or: kinetic energy, or: W = ∆ KE , (I1) where the equation for kinetic energy is given by: KE = 1 2 m v 2 . (I2) We also know that the equation for work can be given as W = F x ∆x (I3) We used a pressurized air track that helped to reduce work on the system due to friction by pushing pressurized air out of the track to “lift” the glider off the track and reduce the normal force of the track onto the glider. On an inclined plane, this leaves only the component of gravity along the track to do work on the system. This allows us to derive the following: W = F x ∆x = ∆ KE = ¿ mgΔx sin θ = ¿ 1 2 m v 2 − 1 2 m v i 2 ¿ ¿ > v 2 = 2 g sin θ ( x − x i ) + v i 2 (I4) To capture the position of the glider throughout time, we used a sonar motion sensor on one end of the track, with the direction away from the sensor being the positive direction. Since the sensor tracks position and not velocity, we used the equation for average velocity: V avg = ΔX Δt = X 2 − X 1 t 2 − t 1 (I5) for each point in time, t avg , as given by: t avg = Δt 2 − Δ t 1 2 (I6) Throughout the experiment, we also encountered various sources of error and limits of precision due to the instruments used, error propagated through calculations, and random error. To quantify the error of velocity squared, we used the error propagation formula: δ v 2 = √ 8 v 2 ( δx Δt ) . (I7) When it came to considering the error of the angle, θ , of the incline in investigation 1 and the associated trigonometric calculations, we used the error propagation formula: δsinθ = √ ( δh h ) 2 + ( δd d ) 2 (I8)

6 6.5 7 7.5 8 8.5 9 0 2 4 6 8 10 12 Position vs Time (initial drop) Time (s) Position (m) Figure 2: Position vs. time plot of the glider on the inclined track between the first two collisions. The blue portion represents the motion of the glider moving towards the motion sensor and the orange is of it moving away. We then calculated the velocity, v avg and velocity squared, v 2 , of the glider using Eq (I5) as well as the uncertainty for velocity squared using Eq (I7 ) for each position data point. We also calculated the average position between adjacent points for each time using the equation: x avg = x 2 − x 1 2 (1.1) as well as its error, given by: δ x avg = √ 1 2 δx . (1.2) The error in the average position remains constant throughout the motion, given its determined using only constants. A sample of our data table for all quantities is given below in Table 1.2. Table 1.2: Sample data for recorded data of position and time as well as calculated quantities velocity, velocity squared, average position, and the respective calculated errors. x (m) t (s) v (m/s) v 2 (m/s) 2 δ( v 2 ) (m/s) 2 x avg (m) δx avg (m) 1.2777 4.2 -0.62 0.3844 0.0350724 96 1.2622 0.0007071 1.2467 4.25 -0.688 0.47334 4 0.0389191 57 1.2295 1.2123 4.3 -0.658 0.43296 4 0.0372221 01 1.19585 1.1794 4.35 -0.642 0.41216 4 0.0363170 04 1.16335 1.1473 4.4 -0.626 0.39187 0.0354119 1.13165

6 08 1.116 4.45 -0.602 0.36240 4 0.0340542 63 1.10095 After calculating these quantities for each time throughout the glider’s motion, we moved onto data analysis. We began by plotting a velocity squared vs average position graph. The data was plotted at two separate series denoted by blue and orange. Like in the position vs time graph in Figure 1.2, the blue points represent the motion towards the sensor and the orange points represent the motion away from the sensor. 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 f(x) = 0.57 x − 0.33 f(x) = NaN x + NaN Velocity Squared vs Average Position Average Position (m) Velcotiy Squared (m/s)^2 Figure 1.2: Plot of velocity squared vs average position , broken into two series with respective error bars. Each series was fit with a best fit line, and the slope of the line is displayed on the graph. The straight line fit of v 2 vs x verifies the work energy theorem, showing a linear relationship between the two values as suggested in the equation for kinetic energy, as given in Eq (I2).The slope of the blue line is steeper than the orange line, meaning the glider had more energy going away from the sensor than moving towards it. This is because some energy is lost in the collision (sound, heat, some friction on the track), and not all energy was conserved and converted to kinetic energy. The data was also plotted in the IPL Straight Line Fit Calculator. The IPL plot is more accurate and precise because it accounts for the error of each measurement as well as the uncertainty in slopes. After plotting in IPL, we got v 2 = 0.674 ± 0.016 and v 2 = 0.568 ± 0.014 . To find g , began by expanding Eq (I4) to get: v 2 = 2 g sin θ x − 2 g sin θ x i + v i 2 (1.3)

Table 1.3: Slope and g values of each segment of motion, experimental (average) value for g, standard error in the mean, and percent different and any necessary quantities to calculate these quantities. slope (B) g using B g avg std dev of avg error of avg known g % diff 0.674 10.69841 27 9.857142 86 0.8412698 4 0.594867 61 9.81 0.48055 92 0.568 9.015873 02 The biggest source of error in this investigation was the systematic and random error from the collision. Making the collision more elastic so that less energy was lost would help make the system truly conservative and would yield better values. Investigation 2 This investigation was similar except instead of observing the motion of a solitary glider on an inclined plane, we observed the motion of a glider with an attached mass on a horizontal plane. We began by removing the wooden block used in the previous setup and making sure the track was level again. Then, we measured the mass of the glider, m , as well as the mass of a hanging lead weight, m' . These masses as well as their error, denoted as half the smallest increment on the scale, are shown in Table 2.1 below. Table 2.1: Mass of the glider and hanging lead weight with respective errors. m (kg) δ m (kg) m' (kg) δ m' (kg) 0.3797 0.00005 0.029 0.00005

We then cut a strip of paper that was attached to a clip on the glider on one end, extended over an air pulled, and attached to a clip on the lead mass on the other end. The data collection process was similar to that of Investigation 1, except instead of releasing the glider from rest 20 cm from the motion sensor, it was released from 40 cm. After recording the motion, the data was moved into Excel, and the same computations for velocity, velocity squared, average position, and all respective errors were calculated. The plot for position vs. time for all data, position vs. time for between collision, and velocity squared vs. average position are shown in Figures 2.1- 2.3, respectively, below. 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Position vs Time (initial drop) Time (s) Position (m) Figure 2.1: Position vs time graph for entire motion

The data was also plotted in the IPL Straight Line Fit Calculator. The IPL plot is more accurate and precise because it accounts for the error of each measurement as well as the uncertainty in slopes. After plotting in IPL, we got v 2 = 1 . 585 ± 0.035 and v 2 = 1.300 ± 0.030 . Since the track is no longer inclined, we must adjust Eq (I4) to attribute the work done by gravity to the hanging lead mass. This gives: W = Fg = m ' g = 1 2 ( m + m' ) v 2 − 1 2 ( m + m' ) v i 2 ¿ > v 2 = 2 m ' g m + m ' ( x − x i ) . (2.1) Expanding this equation, as we did in the previous investigation, we get: v 2 = 2 m ' g m + m ' x − 2 m ' g m + m ' x i , (2.2) where the slope, B , is given by equating the coefficient of the first term to the slope: slope = B = 2 m ' g m + m ' . (2.3) Solving for g , we get: g = ( slope )( m + m ' ) 2 m' . (2.4) Using the values from motion of the glider going up the incline, we get g = 11 . 17 m / s 2 , and using the value of the glider going down the incline, we get g = 9.10 m / s 2 . Averaging these values, we get g avg = 10.16 . Like in Investigation 1, this is done to correct for the effects of friction, as friction acts in different directions going up and down the incline, as shown in the force diagram in Figure 1.3. Using standard deviation and standard error in the mean, given by Eq (1.6) and Eq (1.7), we get an error in g avg , δg = 0.730 m / s 2 . Given the known value of g = 9.81 m / s 2 , our experimental value g = 10.13 m / s 2 ± 0.730 does fall within the range of uncertainty that we calculated. Using the percent difference equation in Eq (1.8), we get a percentage difference of 3.33%, which is low. Further, we observed that the y-intercept, C, could also be used to determine the experimental value for g. By equating the y intercept of the best fit lines to the “C” term of Eq (2.2), we get: y − intercept = 2 m ' g m + m ' x i . (2.5) Solving for g, we get: g = ( y − intercept )( m + m ' ) 2 m' x i . (2.6) Using the values from motion of the glider going up the incline, where x i is the position of the glider at the collision, we get g = 7.48 m / s 2 , and using the value of the glider going down the incline, where x i is the position of the glider at the top of the incline when it is closest to the sensor, we get g = 10.15 m / s 2 . Averaging these values, we get g avg = 8.81 m / s 2 . Using standard deviation and standard error in the mean, given by Eq (1.6) and Eq (1.7), we get an error in g avg ,

δg = 1.333 . Given the known value of g = 9.81 m / s 2 , our experimental value of g = 8.81 m / s 2 ± 1.333 does fall just within the range of uncertainty that we calculated. Using the percent difference equation in Eq (1.8), we get a percentage difference of 10.15%, which is just about acceptable for such a calculation. All experimental values for both the B and C methods, errors, and percentage differences are shown in Table1.3 below. Table 1.4: Slope and g values of each segment of motion, experimental (average) value for g, standard error in the mean, and percent different and any necessary quantities to calculate these quantities slope (B) x int (C) g using B g using C 1.626 -1.1329 11.1687 84 7.48112 1 1.292 -0.8921 9.10414 48 10.1478 48 g calculation method g avg std dev δ g known g % diff B 10.1364 65 1.03231 98 0.72996 04 9.81 3.32787 62 C 8.81448 45 1.33336 35 0.94283 04 10.1479 67 Calculating g using the y-intercept is less accurate, as it introduces another quantity, x i that has error. The biggest source of error in this investigation was the systematic and random error from the collision. Making the collision more elastic so that less energy was lost would help make the system truly conservative and would yield better values. Conclusion Throughout this experiment, the objective was to verify the work energy theorem and to find an experimental value for the acceleration due to gravity, g, using energy concepts. For. Investigation 1, we used an inclined, pressurized air track and recorded the position data of a glider moving down the track using a motion sensor. By plotting velocity squared, v 2 , as a function of position, x , and yielding a linear plot, we proved the work energy theorem, showing the proportional relationship between v 2 and x . From the best fit lines, we were able to determine an experimental value for g using the slope, where g = 9.86 m / s 2 ± 0.595 , which falls within the range of uncertainty for the real value, g = 9.81 m / s 2 . We determined a 0.48% difference between the real and experimental value, which is extremely low. In Investigation 2, we used a horizontal pressurized air track and recorded position data of a glider attached to a hanging mass strung over a pulley along the track. By, again, plotting

4. For the configuration of Investigation 2, what is the acceleration of the glider if m' ⟶ ∞ ? The acceleration would still be 9.81 m/s 2 . 5. For the configuration of Investigation 2, what is the change in potential energy from the moment of release to the moment of collision with the bumper? (Hint: Look at the change in x in the data list. Remember that only the weight is changing potential energy.) Considering the kinetic energy of the system just before it crashes into the bumper, what is the change in total energy of the system? Is the change in energy positive or negative? Explain whether your result makes sense. The change in potential energy is equal to -m’gh, where h is the displacement of the objects from their original position. This gives a change of -0.365 J. The kinetic energy od the system right before the collision can be found using the equation KE=1/2(m+m’)v 2 . This gives a value of 0.18 J. This means there is a total change in energy of -0.19 J. This is a negative change and is likely due to friction along the track or energy lost to sound.