"Acceleration Due to Gravity"
Acceleration Due to Gravity Introduction When forces on an object become unbalanced, an acceleration will ensue. This dictum has been in our vernacular for several centuries now. It accurately describes behavior that we have all experienced since childhood. Release your grip on a ball or a block, and it will accelerate to the ground. Push hard on a wheeled wagon, and it will speed up in the direction of pushing. Of course, our study of this behavior has always been confounded by forces that we cannot readily see or measure. Drop a balloon, and it does not accelerate downward in any noticeable fashion. Stop pushing the wagon and it accelerates to a stop even though there are no apparent forces on it. This type of behavior has lead humankind on a convoluted journey in our search for an answer to the questions of how and why things move. They have caused quite knowledgeable people to develop incorrect theories of the motion of objects. One of the more famous explanations as to why and how things fell was given to the world by the Greek scholar Aristotle (384- 322 B.C.). He believed in the concept of teleology, which is the philosophy that the explanation of an occurrence was found in its final causes. Applied to the motion of objects, Aristotle believed that objects always wanted to return to their beginnings where they could be with like objects, i.e. a rock that was from the ground wanted to return to the ground to be amongst other rocks. At the time, this was as good of an explanation as any other that had been given, and since it emanated from the mind of Aristotle, who was held in high regard, it was accepted as the Fig. 1: Aristotle (NIH) correct explanation for over a thousand years in the Western world. Aristotle did not stop at just giving a reason for why things fall down. He also went on to describe the motion. In particular, he stated that heavier objects fall faster than lighter objects. For instance, he posited that an object that was twice as heavy as another object would fall twice as fast. If you dropped the two at exactly the same time, then the heavier object would hit the ground at the same time that the lighter object had only fallen half of the distance. This, too, was accepted by the public, based somewhat on the authority of Aristotle and somewhat on observations. For instance, if you drop a light piece of paper and a heavy rock at the same time, the rock hits the ground well before the paper. Galileo It was not until the early 1600’s that Galileo Galilei showed that Aristotle’s theory of heavier objects falling faster was wrong. He did this by running experiments wherein the weight of the object did not affect the rate at which the object moved. Legend has it that he dropped two different sizes of cannonball from the Leaning Tower of Pisa (Galileo was a professor at the University of Pisa), which both hit the ground at the same time. Unfortunately, while this story has a certain amount of romance to it, there is no evidence that Galileo ever did any such thing, although he does discuss theoretically dropping cannonballs to disprove Aristotle. Fig. 2: Galileo (Sustermans) Acceleration Due to Gravity 1 The experiments that Galileo performed to study gravity were done with a ball rolling down an inclined plane, rather than a ball falling through the air. The reason was quite simple: Galileo wanted to time how long it took the ball to move a given distance, and the clocks of his time did not allow for accurate measurements on such short time scales as a second or so. By having the balls roll down an inclined plane, Galileo lengthened the time over which the balls moved. Using the most accurate pendulum and water clocks of his time, he was able to make fairly accurate measurements of the elapsed time. These measurements allowed Galileo to do more than to show that Aristotle was wrong. He was able to test and discern mathematical relationships between various factors, which led to the first accurate equations about the motions of objects. What Galileo found was that, under the influence of gravity, the distance that a ball travels from rest is proportional to the square of the amount of time that the ball is allowed to travel. Mathematically, we write this as ∆x = k t2 where ∆x is the distance traveled, t is the elapsed time from rest, and k is a proportionality constant. Knowing that the average velocity of an object is related to the displacement by vavg = ∆x/t and that the final velocity of an object is related to the acceleration (for an object starting at rest) by vfinal = at, it was simple enough for Galileo to show that the proportionality constant was related to the acceleration, thus giving ∆x = ½ a t2 Galileo’s experimental work on gravity also led him to the conclusion that the natural state of an object was to be in a state of constant motion, which was later credited to Newton as his First Law of Motion. This, too, was contrary to Aristotlian view mentioned above that the natural state was to return to the state that they originated in, i.e. at rest. Biological Systems Gravity affects all organisms, but not necessarily at the same magnitude. Life on this planet runs the gamut from the smallest (a bacterium) at 3x10-11 g to the largest (a coastal Red Wood Tree) at 3,600 metric tons. Since the force due to gravity (weight) is mass dependent (weight = mass * gravity), larger organisms must exert a considerably higher proportion of their energy to resist gravity than smaller organisms do. The force due to gravity on an elephant (7,000 Kg) is 350,000 times greater than the same force on a mouse (20 g). Thus, gravity’s influence can be found throughout the biological world as a key element of natural selection, driving the evolution of countless morphological traits. The feet of elephants are cushioned by a heavy layer of tissue because the impact from simple walking would be enough to damage bone. Blue whales are the largest animal ever to inhabit the planet (bigger than any dinosaur known) and they only evolved in a marine environment where the density of the fluid (buoyancy) was able to help balance the dramatic force of gravity (~2x106 N) on these mammoth animals; there is still debate about whether animals with the mass of a Blue whale could ever have evolved in a terrestrial environment where gravity plays a stronger role due to the lower density of air. Acceleration Due to Gravity 2 Experiment The Galileo’s equation above is only valid under the assumption that there is no wind resistance. This presents some problems, as we happen to live on a planet that has air (this is not a problem when it comes to other processes, like breathing). What we want to test in this week’s activity is how valid this equation is for work carried out in the atmosphere. Galileo thought the equation was accurate, but he was using pendula and water clocks to measure his experiments. The effects of wind resistance on his experiment might have been so small so as to not be noticeable with the level of accuracy achieved in his equipment. Today, scientists have measuring equipment that is precise to the nearest picosecond. While our equipment in the lab is not this sophisticated, we are still much more accurate than Galileo and should be able to accurately measure errors to within microseconds. To aid us in our endeavors, we will be using commercially available photogates from Pasco, Inc. These photogates work by passing an infrared beam across an opening to a receiver unit. If something crosses the opening, it stops the beam from reaching the receiver, which causes the photogate to send a signal to the computer to signify that an object has entered the area. When the object leaves the opening, the beam is once again picked up by the receiver, which causes the photogate to send another signal to the computer to let it know that the object has left the area. In this manner, the photogate is able to communicate the timing of the movements of an object, which the computer is able to time and track. To measure gravity, we will be dropping a transparent ruler that has Fig. 3: Ruler (Pasco) opaque stripes placed on it at regular intervals (5.0 cm). As each stripe passes through the gate, the computer will note the time of the event. At the conclusion of the experiment, it will display these arrival times, along with the position of each stripe on the ruler. We will then process this data as below to calculate the acceleration due to gravity. Procedure 1. Plug the photogate into the digital input #1 on the Pasco interface. Make sure that the interface is turned on by checking that the green light is on. 2. Position the photogate close to the edge of the table so that the ruler can fall through the photogate. Place a piece of foam rubber below the photogate so that the ruler does not directly hit the floor. 3. Within the DataStudio software, open the activity entitled “P02-gravity”. Delete any data runs that are pre-existing in the table. 4. Test that the photogate is operational by clicking MON on the software and moving your hand through the photogate. Numbers should appear in the table as you block and unblock the gate. Click stop when you are finished. 5. Place the lower edge of the ruler so that it is just above the gate. Click Start in the software and drop the ruler. After the ruler has hit the foam, click Stop. 6. Record the data on the activity sheet. Acceleration Due to Gravity 3 7. Repeat the experiment 4 more times with the ruler starting at different heights above the photogate. Data Analysis The data measured by the computer is the time at which each opaque picket after the first picket enters the photogate. If we imagine the edge of the first picket to be at x = 0 cm, then the second picket is at 5.0 cm, and each subsequent picket is 5.0 cm further away than the previous. The table of data that we have for each run, then, consists of position-time pairs for the ruler. The differences in each run are due primarily to two factors: the amount of time that elapses between when we press start and when we release the ruler, and the initial velocity with which the ruler enters the photogate because we dropped it from a different height. In the analysis that follows below, these differences should not matter, if our theory has been properly accounted for in the experiment. Using our position time pairs, we can reconstruct the average velocity of the ruler between each time interval using the average velocity formula, vavg,n = ∆x/∆t = (xn – xn-1)/(tn – tn-1) where vavg,n is the average velocity over the nth interval. If the acceleration due to gravity is a constant, then we know that this average velocity over the time interval is equal to the instantaneous velocity at the midway point of the time interval. Thus, we can use the formula above as the instantaneous velocity if we merely post the velocity at the average time of the time interval. Let us show an example to explain this. Suppose we have the following data point in our table Interval Time Position n-1 1.50 s .35 m n 1.60 s .40 m n+1 1.68 s .45 m Using the formula above, this gives us the average velocities over the two intervals of (.40 m - .35 m)/(1.60 s - 1.50 s) = .05 m/.10 s = .5 m/s (.45 m - .40 m)/(1.68 s – 1.60 s) = .05 m/.08 s = .6 m/s This velocity is also the instantaneous velocity at the time tn = (1.50 s + 1.60 s)/2 = 1.55 s tn+1 = (1.60 s + 1.68 s)/2 = 1.64 s Thus, we can use the above data to calculate a new set of instantaneous velocity data of Interval Time Velocity n 1.55 s .5 m/s n+1 1.64 s .6 m/s Acceleration Due to Gravity 4 From the five different data sets on the activity sheet, we need to calculate five different sets of velocity data using the above technique. Place these data sets in the five tables on the activity sheet. If we were to plot any one of these sets of instantaneous velocity versus time, we should find that our data falls along a straight line with a slope equal to the acceleration due to gravity. The reason for this is because v = vo + a t for a system undergoing constant acceleration. Calculating a best fit line through this plot will give us this acceleration for the experiment, which we can use to compare to the accepted value of 9.80 m/s2. Acceleration Due to Gravity 5 Name: Instructor: Position Data Run 1 Run 2 Run 3 Run 4 Run 5 Time Position Time Position Time Position Time Position Time Position Velocity Data Run 1 Run 2 Run 3 Run 4 Run 5 Time Velocity Time Velocity Time Velocity Time Velocity Time Velocity After plotting each of these sets of data, calculate the acceleration from the best fit line of each graph Run 1 Run 2 Run 3 Run 4 Run 5 Acceleration 1. What are the possible sources of random errors in this experiment? How have you attempted to account for them? 2. What are the possible sources of systematic errors in this experiment? Are their effects noticeable? If so, is the error large? Acceleration Due to Gravity 6