Team: ACCELERATION Part I. Galileo s Experiment Galileo s Numbers Consider an object that starts from rest and moves in a straight line with constant acceleration. If the object moves a distance x during the first second of time, then how far will it move during the next one-second interval? How far will it move during the third second, the fourth second, etc.? Will the sequence of displacements be simple and regular, such as x, 2 x, 3 x, 4 x,... or complex and irregular such as x, 2.2 x, 4.5 x, 8.3 x,...? Note that the sequence of velocity increments is simple: v, v, v, v, etc. For uniformly-accelerated motion, the velocity changes by the same amount ( v) during equal time intervals ( t=1). Let the sequence of displacements be denoted by { x, A x, B x, C x,...}. Here is a picture that defines these displacements: t o t o +1 t o +2 t o +3 t o +4 1 x A x B x C x Note that t o is the initial time when the object is at rest and begins to accelerate. The object moves the distance x during the 1st second, A x during the 2nd second, B x during the 3rd second, etc. The initial displacement x serves as the reference length. The pure numbers 1, A, B, C,... are the displacements relative to x. The sequence {1, A, B, C,... } represents a dimensionless measure of the displacements. Let d denote the absolute displacements and let d r denote the relative displacements: d = 1 x, A x, B x, C x,... d r = 1, A, B, C,... The numbers A, B, C, D, etc. are the famous Galileo Numbers. This universal pattern of numbers describes the amount of space traversed in equal intervals of time for all phenomena of uniformlyaccelerated motion. Your quest is to find the values of these special numbers of nature. Galileo rolled a bronze ball down a wooden ramp. You will study the motion of a glider on a tilted air track. Experimental Procedure 1. Level the Track. If the track is level, then a glider placed at rest on the track should remain at rest. To adjust the level, turn (ever-so-slightly) the screws under the leg of the track. 2. Tilt the Track. Place one block under the leg of the track where the motion sensor is located. 3. Measure the Motion. Release the glider from rest at the top of the track and record its position x as a function of time t using the motion sensor [Open the Logger Pro file Changing Velocity 1]. 1
Make sure you start collecting data before you release the glider. Analyzing the Motion Data By looking at the velocity-time data (graph and table), carefully note the time t o at which the glider was released. Record the value of this initial time: t o =. Fill in the following table that displays the position x of the glider at time t = t o, t o +1, t o + 2, t o +3, t o +4, t o +5. Compute the distance traversed during each of the 1-second time intervals and record these displacements (absolute d and relative d r ) in the last two rows of the table. t (s) x (m) d (m) d r 1 Instead of t = 1 second, choose the time interval to be t = 1/2 second. Analyze the motion data to find the displacements. Note that in this case, t = t o, t o +0.5, t o +1.5, t o +2.0, t o + 2.5, etc. t(s) x(m) d (m) d r 1 Class Data: A data table displaying d r from all teams in the class appears on the chalkboard in front of the classroom. Please enter your team s values into this table. Conclusion Do you see a pattern in the values of d r? If all experimental errors could be eliminated, what seems to be the exact sequence of Galileo s numbers { 1, A, B, C, D,... }? Pure Theory 2
Consider an object moving in the x direction with constant acceleration a = 2 m/s 2. Derive the exact theoretical values of the displacement numbers A, B, C, D,... two different ways: 1. Algebraic Derivation. Use the kinematic relation x = ½ at 2. (Choose t o =0 and t=1) 2. Graphical Derivation. Graph velocity versus time and find areas. v t Theory versus Experiment Compare your measured experimental values of A, B, C, D,... with the calculated theoretical values. Carefully craft one sentence that summarizes the physics of uniformly-accelerated motion in terms of the space traversed and the time elapsed. Part II. The Physics of Free Fall 3
Consider an object of mass m that is released from rest near the surface of the earth. After a time t, the object has fallen a distance d and is moving with velocity v. The free-fall equations relating d, t, and v are where g = 9.8 m/s 2 is independent of m. d = ½ gt 2, v = gt, v 2 = 2gd, In this experiment, you will test these important properties of free-fall motion by studying the motion of a glider on a tilted air track. Strictly speaking, free fall refers to the vertical motion of a body that is free of all forces except the force of gravity. A body moving on a friction-free inclined track is falling freely along the direction of the track. It is non-vertical free fall motion. The track simply changes the direction of the fall from vertical to diagonal. This diagonal free fall is a slowed-down and thus easier-to-measure version of the vertical free fall. The acceleration along the track is the diagonal component of the vertical g. This acceleration depends on the angle of incline. It ranges from 0 m/s 2 at 0 o (horizontal track) to 9.8 m/s 2 at 90 o (vertical track). In other words, the track merely dilutes gravity. A frictionless inclined plane is a gravity diluter. 1. Experimental Test of d t 2 In theory, the distance traversed by the glider along the track is proportional to the square of the time elapsed (after starting from rest). This means that if you double the time, t 2t, then the distance will quadruple, d 4d. More specifically, if it takes time t 1 to move distance d 1 and time t 2 to move distance d 2, then the proportionality d t 2 implies the equality d 2 /d 1 = (t 2 /t 1 ) 2. Hence if t 2 = 2t 1, then d 2 = 4d 1. Tilt the track by placing two blocks under the end of the track. Use a stop watch - not the motion sensor - to measure the time it takes the glider, starting from rest, to move a distance of 25 cm down the track. Repeat to find an average time. Next measure the time it takes to move a distance of 100 cm. t (25 cm) t (100 cm) Are your experimental results consistent with the theoretical relation d t 2? Explain. Average Calculate the value of the acceleration a of the glider along the track direction from your measured values of d and t. This acceleration is the diluted gravity: a = g (diluted). Show your calculation. g (diluted) = m/s 2. 2. Experimental Test of v 2 H 4
Physics Fact: The speed v of an object, starting from rest and falling down the frictionless surface of an inclined plane, depends only on the vertical height H of the fall and not the length of the incline. H v Since we are testing the proportionality, v 2 H, lets simply find how v depends on the number of blocks N that we stack vertically to elevate the track. Note that H is proportional to N (assuming uniform thickness of each block). Place one block under the motion-sensor end of the track. Position the glider at the point that is 20 cm away from the sensor. Release the glider from rest and measure its velocity (using the sensor) when it is 100 cm away from the sensor. Next place four blocks under the end. Once again, release the glider at 20 cm and measure its velocity at 100 cm. For N = 1, v =. For N = 4, v =. Do your experimental results support the theoretical relation v 2 H? Explain. 3. Experimental Test of the Universality of g One of the deepest facts of Nature is that the acceleration of an object due to gravity does not depend on the size, shape, composition, or mass of the object. Use two block to incline the track. Use the motion sensor to record the motion of the glider as it falls freely down the track. Find the acceleration of the glider from the slope of the best-fit line through the v versus t data. Confirm this value by averaging the a versus t data. Remember to carefully select the region of good data on the graph before you perform a curve fit or a statistical analysis of the data. To find the average of the a versus t data, click on the statistics icon [STAT]. To find the best-fit line through the v versus t data, click on the curve-fit icon [f(x)]. Add more mass (metal donuts) symmetrically to the glider and measure the acceleration. Include the uncertainty in a (see the special page on Uncertainty). m (kg) a ± uncertainty (m/s 2 ) Compare your values of a with the value in Experiment 1? What is the percent difference? Do your experimental results support the deep principle that g or g(diluted) = a is independent of mass? To answer this question, the role of uncertainty is vital. Part III. Designing a Diluted-Gravity System 5
In vertical free fall, an object released from rest moves about 45 m in 3.0 s. You need to slow this motion dilute gravity so that the object only moves 1.5 m in 3.0 s. Your goal is to find how much the track needs to be tilted to achieve this slowed-down motion. First work out the theory, then perform the experiment. Theory Architecture Diagram. H = height of blocks. L = distance between legs. θ = angle of incline. track leg glider H blocks L θ table Acceleration Diagram. g = full strength gravity. a = diluted gravity. a g θ θ The height H of the blocks determines the acceleration a of the glider. Derive the algebraic relationship between H and a. The constants L and g will also appear in this relation. Let t denote the time it takes the glider to move the distance d along the track. Derive the theoretical equation that expresses H as a function of t, d, L, and g. H = 6
The Design Specs say d = 1.5 m and t = 3.0 s. Measure the value of L. You know the value of g. Plug these numbers into your theoretical equation to compute the value of H. Show your calculation. H = cm. Experiment Raise the end of the track by the height H predicted above (use wooden blocks and thin metal plates). Release the glider from rest. Use a stop watch to measure the time t it takes the glider to move 1.5 m along the track. Repeat this measurement five times. List your five values of t below and compute the average time and the uncertainty in the time. Estimate the uncertainty simply from the spread in the values of time: Uncertainty = (max t + min t)/2. t = ± _ s. Compare this measured value of time with the design goal of t = 3.0 s. 7
Uncertainty (Experimental Numerology) Due to experimental errors, there always exist some uncertainty in the measured value of any quantity. Consider measuring the height of the block used to elevate the air track. Lets use a ruler. If you repeat the measurement twenty times, or have each of the twenty people in the lab perform the measurement one time on the same block, then you will get twenty different readings. The heights will vary from 1.4 cm to 1.6 cm. The most likely value is 1.5 cm. The scientific way to report the measured value is 1.5 ± 0.1 cm. The typical value (average) is 1.5 cm and the uncertainty (deviation) is ± 0.1 cm. The distribution of measured heights has the following bell-shaped (Gaussian) shape: Probability 1.4 1.5 1.6 Height (cm) Relative Uncertainty is one of the most important statistical quantities in error analysis. An uncertainty of 0.1 cm out of 1.5 cm gives a relative uncertainty of 0.1/1.5, which is 1 part in 15, or 6.7 %. Thus you would say that the relative uncertainty in the height of a block is about 7%. If you measure a thick block to be 15.0 ± 0.1 cm, then the relative uncertainty is only 0.7%. Knowing that the height is between 14.9 cm and 15.1 cm is fairly accurate. If you measure a very thin block to be 0.15 ± 0.1 cm, then the relative uncertainty is 70%! In this case, there is not much confidence if all you know is that the height lies somewhere between 0.05 cm and 0.25 cm. Thus the relative uncertainty, not the absolute uncertainty, is the meaningful measure of accuracy Physics 150 Lab is a 10% Lab. A relative uncertainty of 10% or less is typical of all the measurements that you will make in this lab. Results and conclusions that are off by more than 10% need to be reexamined If the theory predicts the value of a velocity to be 1.0 m/s and your experimental measurement is 1.2 m/s, then this 20% error is a signal for you to ask your instructor if such a discrepancy is acceptable. The uncertainty ± 0.1 cm in the height of the block is due in part to the error in reading the ruler. The reader must estimate fractions (tenths of millimeters) that lie between the 1mm-spaced lines of the ruler. The uncertain spread in heights is also due to the uneven surfaces of the block. The thickness of the block depends on where the ruler is placed against the block. In general, there are two kinds of errors Random Errors and Systematic Errors. The classic example of a random error is the error in judgment in trying to estimate tenths of the smallest scale division of the instrument (meter stick, thermometer, ammeter, etc.). These errors are random because of their unpredictable nature and up-and-down variations sometimes too big (+) and sometimes too small ( ). In the acceleration lab, random errors are inherent in reading the ruler. They also result from the chaotic variations in the air currents in the room and the mechanical vibrations of the equipment. These random variations (miniature hurricanes and earthquakes) cause the velocity of the glider on the track to fluctuate sometimes too big, sometimes too small. 8
In contrast to the chance + and variations of a random error, a systematic error causes the measured value to always be too high (+) or always too low ( ). For example, when the block is placed under the leg of the air track, the block compresses. Thus the actual elevation of the track is always less than the measured height of the block. If the track was not level before tilting, then the measured value of the acceleration will always be greater than (or less than) the actual value. Air friction introduces a systematic error that causes the measured acceleration to be less than the friction-free value. Systematic errors can be corrected (compensated for) by adding or subtracting a corresponding correction factor to the measured value. Mistakes are not errors. Mistakes in reading numbers, recording data, and performing calculations are due to carelessness. Mistakes are avoidable. Systematic errors are correctable. Random errors are unavoidable and uncorrectable. There is an art to estimating uncertainties due to random errors. The straightforward but tedious method is to repeat the experiment many times and compute the average value and the standard deviation (uncertainty). If you want to perform the experiment only one time, then you must estimate the uncertainty from knowledge of the design specifications and limitations of the measuring apparatus and procedure. Suppose the apparatus is a ruler. Examine a ruler. Since the smallest interval between lines is one millimeter, the uncertainty in reading the ruler is about ± 0.3 mm or perhaps ± 0.5 mm. In theory, this uncertainty estimate based on ruler specs should be consistent with the uncertainty estimate (standard deviation) based on repeated experiments. The uncertainty in the measured value of the acceleration of a glider on the air track can be estimated as the standard deviation in the a versus t data. You could confirm this value by finding the error in the slope of the best-fit line through the v versus t data. The uncertainty in the acceleration is due to many factors, including air friction, track friction, unleveled track, dirt on track, dirt on glider, warped track, bent glider, and malfunctioning motion sensor. In the laboratory, it is not always obvious if one measured number is equal to another measured number. Suppose you measure the length of two metal rods to be 100 cm and 97 cm. Do these rods have the same length? This question cannot be answered until you specify the uncertainty. How was the length measured? Did you use a ruler (low tech) or a laser (high tech)? Using a certain ruler, suppose the uncertainty was 1%: 100 ± 1cm and 97 ± 1 cm. Since the range of values ( 99 101cm and 96 98 cm) do not overlap, you can conclude that these two lengths are not equal. Using a laser and an interferometer, the uncertainty in length can be made as small as ± 1 nanometer = 10 9 m. Suppose you used a bent, jagged ruler with an uncertainty of 5%: 100 ± 5 cm and 97 ± 5 cm. Since the range of values ( 95 105 cm and 92 102 cm) now overlap, you can conclude that these two lengths are equal within the experimental errors. 9