Physics 40 Lab 1: Tests of Newton s Second Law January 28 th, 2008, Section 2 Lynda Williams Lab Partners: Madonna, Hilary Clinton & Angie Jolie Abstract Our primary objective was to test the validity of Newton's Second Law of Motion which predicts that the acceleration due to gravity of an object is independent of object s mass. The acceleration of a glider on an inclined air track by the earth's gravity was measured as a function of glider mass. The motion was measured with an ultrasonic ranging device. We observed that the acceleration was 9.8 ± 0.05 m/s 2, independent of object mass, and consistent with the currently accepted value for the gravitational acceleration at sea level. We also studied the relationship between the angle of inclination of an inclined air track and the acceleration of gravity of an object traveling down it. We found that g = 9.82 ± 0.2 m/s 2, independent of the angle of inclination and consistent with the accepted value of g. We suggest methods by which even more accurate values of gravitational acceleration may be determined.
Theory/Purpose Galileo first observed that the acceleration of an object due to the earth s gravity is independent of the weight of the object, in the absence of air resistance. Newton s Second Law and Universal Law of gravity quantify Galileo s observations. For an object on a frictionless incline plane (μ=0), the acceleration of the object along the direction of motion is related to the angle of inclination and the acceleration due to gravity, g, by (see free body diagram): F = ma = mg sinθ a = gsinθ The angle of inclination, θ, for height, h, and 1 meter length of the air track is tan θ = h/1m Since the angle is less than 5 degrees, the small angle approximation can be applied here. Thus the angle (in radian measure) can be approximated as: tan θ θ h/1m Our experimental set up is very similar to the diagram above. We used an ultrasonic motion detector to determine position (and velocity) of the glider as a function of time; the acceleration is the slope of the velocity vs time graphs. The motion detector was placed at the top of the air track, level with the track. The purpose of our lab is to test the independence of mass and inclination angle on the acceleration due to earth s gravity. We will study the motion of a glider on an air track in two ways: 1. Constant Mass at Varying Inclination Angles 2. Varying Mass at Constant Inclination Angle
Procedure, Data and Graphs: Constant Mass at Varying Inclinations We measured the acceleration due to gravity of a glider on an inclined air track. The mass of the glider (including the reflecting flag) was 0.218 ±\ 0.001 kg. The separation of the supports under the glider is 1.0 ± 0.001 meters. The inclination angle was fixed by placing one or more spacers (each of height 0.008 meters) under one of the air track supports. We set the air track to two different heights, corresponding to two different angles. Four measurements were made for each angle of inclination. The uncertainty in the measurement of track elevation and track length was 0.001 meters (as measured with a mm scale meter stick.) Trial Height (meters) Acceleration (m/s 2 ) 1 2 3 4 5 6 7 8 0.008+/-0.1% 0.016+/-0.1% 0.077 0.080 0.076 0.080 0.158 0.152 0.160 0.161 Sample graphs (trial 5) of distance and velocity as a function of time are shown below.
Procedure, Data and Graphs: Varying Mass at Constant Inclination We next added mass to the glider in increments of 100 grams and again made several measurements of acceleration for air track set up with the angle of inclination, theta = 0.016 rads. Inclination angle: θ = 0.016 rads. Trial Mass of glider (kg) Acceleration (m/s 2 ) 1 2 3 4 0.218 0.158 0.152 0.160 0.161 5 6 7 8 9 10 11 12 0.318 0.418 0.157 0.158 0.156 0.157 0.156 0.154 0.159 0.157 Sample graphs (trial 3) of distance and velocity as a function of time are shown below.
Calculations/Error Analysis Constant Mass, Varying Inclination: From the data shown, we find that the average accelerations for the 0.008 meter and 0.016 meter heights are 0.0785 ± 0.002 m/s 2 and 0.158 ± 0.003 m/s 2, respectively. The uncertainty is estimated from average deviation of the data points. The angle of inclination, theta, for height, h, and 1 meter separation of the air track supports is theta ~ tan θ = h/1 = h. (The small angle approximation is valid here.) Our uncertainty on the measurement of theta is derived by summing the percentage errors in our length measurements. The percentage error for h and for the one meter length used for angle determinations were 10% and 0.1%, respectively. Hence, the percentage error in theta is about 10% (the greater of the two.) We compute g from the acceleration data using the relation a = g sin θ, the acceleration of an object on a frictionless incline at angle theta to the horizontal. Our values for g for each trial are: Trial Inclination angle (radians) value of g (m/s 2 ) Average value of g (m/s 2 ) 1 2 3 4 0.008 9.625 10.00 9.500 10.00 9.781 5 6 7 8 0.016 9.875 9.500 10.00 10.06 9.859 Instead of propagating errors mathematically for g, we averaged our results and estimated the error from the average deviation or about 2% fractional error. For the data shown in the table, we find g = 9.82 ± 0.2 m/s 2 Varying Mass and Constant Inclination: The analysis of these trials follows exactly as described above for the unweighted glider. In each case, the uncertainty on the glider mass is 0.001 kg and the uncertainty on the determination of g is 0.2 m/s 2, or about 2% fractional error. Our results for all measurements are summarized below: Trial Glider mass (kg) 1-4 5-8 9-12 0.218 0.318 0.418 Average value of g (m/s 2 ) 9.82 9.80 9.79 Again, instead of propagating errors mathematically for g, we averaged our results and estimated the error from the average deviation or about 2% fractional error. For the data shown in the table, we find g = 9.80 ± 0.01 m/s 2
Analysis and Conclusions The best estimate we have found for the value of g in Philadelphia is 9.802 ± 0.001 m/s (reference Physics by Eugene Hecht, pg. 73). According to Newtonian theory, this value should be the same for all objects regardless of mass. Our result for three different glider masses shows that g is independent of mass and are consistent within their 2% uncertainty. The average value of g from all three masses is 9.80 ± 0.01. Our results are consistent with the value given in Hecht. There are several sources of uncertainty in the determination of the gravitational acceleration: 1) the measurement of lengths with a meter stick; 2) the intrinsic measurement accuracy of acceleration with the MacMotion program; and 3) the influence of air resistance on the glider. The measurement of the length between the air track supports is appropriate for the meter stick since the length to be measured is comparable to stick length and the fractional error is small. However, using the same stick to measure the small height of the blocks used for inclination leads to a much larger fractional error. These lengths would be better measured using calipers or a micrometer. The uncertainty in the height accounts for essentially all of the uncertainty on the determination of the inclination angle. Since the angle stays fixed for several trial measurements of the acceleration, this error does not play a role in determining the spread of acceleration values. The uncertainty in the determination of acceleration accounts for essentially all of the spread in our values of g. This spread is presumably partially due to actual small differences between the trials and partially to limitations in the instruments and software. Fitting various parts of a section of good data and repeating this for several data plots indicate that we have 2% uncertainty in our acceleration measurements, which leads to the 2% spread in our determination of g for a fixed incline angle. Some method for directly determining velocity, e.g. with direct timing measurements over short intervals might reduce this uncertainty. Finally, air resistance needs to be reduced. Air resistance gives us lower values of acceleration than we would otherwise measure, hence it should tend to push our measured values of g lower than the true value. It is difficult to measure the extent of this effect. We see little evidence for it in our measurements, presumably because air resistance increases as the speed of the glider increases and all our velocities for small inclination angles are small. This effect might be minimized by switching to an ultrasonic reflection flag which is streamlined. Conclusions The determination of the independence of gravitational acceleration on mass or content of a falling body represents a major achievement in scientific thought. It allows the use of gravitationally accelerated objects to be used in testing concepts of inertia and the relationship between force, mass, and acceleration. In addition to verifying Galileo's original observation, our experimental technique can be used to measure g to about 1% in any local environment.