Experiment 2: Motion in one dimension
Introduction:
Motion of a body in one dimension is basic study in physic and to have knowledge about motion in one dimension it is necessary to be acquainted with two fundamental laws of physics. One of them is Newton’s laws of motion and another is Newton’s law of gravity. There are three laws in Newton’s laws of motion, but in this paper only Newton’s second law of motion has been discussed. If force is applied on a body, the same will move as a result of applied force on the body. The body will remain in same position if no force is applied on it. Applied force also causes acceleration of the body. Newton’s law relates the body, the load applied upon the body and motion of the body caused by the applied force. Newton’s second law is used to carry out quantitative measurement of the applied force on a body. Motion in one dimension has been discussed in this experiment with the help of Newton’s laws.
Objective:
Purposes of this experiment were to compute an object’s velocity from its position, to measure the acceleration of a mass as it falls under the influence of gravity and to compare experimental value with theoretical value quantitatively. Procedure of obtaining the value of acceleration due to gravity by plotting graphs was also an objective of this experiment.
Theory:
Basis of this experiment was Newton’s second law of motion and Newton’s law of gravity.
Newton’s second law of motion basically deals with momentum of a body. If a body is in motion then multiplication of its mass and its velocity at a particular time is termed as momentum of that body at that particular time. At a particular time velocity of an object has a direction, so momentum of a moving object also has a direction and momentum of a body is vector quantity. According to Newton’s Second law of motion, the rate of change of momentum of a body is proportional to the applied force on that body and the direction of the force as well as direction of change of momentum is same. In other word, vector sum of forces applied on a body is equal to multiplication of mass of the body and its acceleration caused by that force. If “P” be the applied force on an object, “m” be the mass of that object, “f” be the acceleration of that body caused by applied force, then following expression indicates the relation between these quantities
P = mf
Another theory used in this experiment was Newton’s law of gravity. Gravitational attraction between two objects can be described by Newton’s law of gravity. It can be expressed by following expression -
F = G (m1.m2/r^2)
Where, F is the force of attraction, G is the universal gravitational constant and equal to 6.67 x 10^(-11) N-m^2/kg^2.The m1 and m2 are masses of two bodies and r denotes the distance between the centers of the two bodies.
Acceleration of gravity can be evaluated from above mentioned theories.
When an object of mass m1 falls near the surface of the earth, it will accelerate due to force F= m1.a , according to Newton’s Second law of motion . In this expression “a “is acceleration of that object caused by that gravitational attraction force. But according to Newton’s law of gravity, F = G ( m1 . m2 / r^2).
Now, equating the value of force rather equating above mentioned two equations,
F = G ( m1 . m2 / r^2) = m1 .a ,
where m2 is taken as mass of earth , mearth.and rearth is radius of earth. Distance between centers of the mass m1 and earth is almost equal to radius of the earth.
Or, a = ( G . mearth )/ r^2
Or, a = [(6.67 x 10^(-11) ) x ( 5.98 x 10^24) ] / (6.37 x 10^6)^2 = 9.8 m/sec^2
Methodology:
The methodology used to carry out this experiment is called “The Spark Tape Method of Determining g”. As the name suggest, a spark tape was used in this experiment. Apart from the tape, a wood block, a rod clamped with table, 50g mass, a spark generator were required.
A weighted paper tape was dropped during this experiment through a slot cut into a block of wood. Two electrodes were screwed into the block. Electrodes were connected to a high voltage power supply which caused an electrical spark to jump from one electrode to the other at regular time interval. Sketch of the apparatus and the arrangement has been shown below.
In this experiment a long meterstick of 2m long was used to measure distance from floor to ceiling. The height of the block containing the electrodes was adjusted in such a way that it was half of the distance from floor to ceiling. This adjustment was done to maximize the amount of data on the spark tape. A spark tape of 1.4 m long was torn off and used in this experiment. After arranging the spark tape a foam pad was placed on the floor to catch the weight. Frequency of sparker was adjusted to its highest setting so that all sparkers had a maximum setting of 60 sparks per second. Care was taken so that not to touch the wires or electrodes while sparks were generated.
At first an “experiment run “was performed by dropping the weighted spark tape through the slot. It was made sure that the tape was held vertically. Sparks left a small burn mark on the paper because of falling under influence of Earth’s gravity. Spark tape was secured to the table by using masking tape. 2m.-long meterstick edgewise was placed along the line of sparks. First large black burn mark was neglected and positions of burn marks were recorded into a blank spreadsheet.
After successful drop, the positions of burn marks were measured and then the continuously increasing velocity of paper tape was calculated. After this a graph of velocity vs. instantaneous time was plotted with a best-fit straight line. Then Regression routine was run in the Data Analysis command to evaluate the error in the slope and y –intercept. A text box was added to the graph and the slope, y –intercept and their error were entered in the box. This slope was equivalent to the local acceleration due to gravity.
There was another part in this experiment where a data acquisition system was used. This system was consisted of a computer, a data acquisition circuit card which is called the Labpro, a sensor and a software package entitled LoggerPro. The sensor used was Ultrasonic Motion Sensor. To evaluate the velocity vs. time graph, the slope of a tangent was considered at any point along a position vs. time graph. LoggerPro permitted to visualize that. While utilizing the software a short line segment, which was always tangent to one point on the curve, was displayed. During moving the cursor through the graph window, the tangent live was seen race along the data. By considering the pattern of changing of the slope of the line, shape of velocity vs. time graph was determined. Given worksheet was guide to perform Part -2 of the experiment. On the first page of the In-Lab, each graph was labeled based on whether it demonstrated constant velocity or constant acceleration. For each graph, its corresponding velocity vs time graph was drawn on the back page of the In-Lab exercise. By using LoggerPro comparison between this graph and predicted graph was carried out on the In-Lab.
Experimental data and analysis of data:
Following table contains data obtained from experiment.
Following graph has been obtained
Discussion:
Following result has been obtained. It has also been found the data obtained are more or less along a straight line which was deserved.
Equation of the line obtained is y = 914.4 x + 93.9. Now, comparing this equation with basic equation of straight line y = mx + c
slope of this straight line is 914.4 and Y-intercept is 93.9
It can be obtained that,
So, slope of the graph has been obtained as 914.4 which denotes acceleration due to gravity is 914.4 cm/sec ^2 or 9.144 m / sec^2. It differs with the calculated value of “a” i.e. 9.8 m/sec^2.
So, percent difference between obtained value and established value of acceleration due to gravity is = 100 x [(9.8 -9.144)/9.8] = 6.7 % approximately.
Difference was found between theoretical value of acceleration due to gravity and experimental value of acceleration due to gravity. This may be caused by some random error occurred during the experiment or error in measuring tool. There may be other sources of errors like approximating experimental record data, not drawing best-fit straight line through the experimental record data in the graph etc.
Conclusion:
The experiment was performed satisfactorily and the experimental value of acceleration due to gravity has been found slightly deviated from theoretical value of acceleration due to gravity. In this experiment velocity of the body was also determined. Knowledge of using Labpro and Loggerpro was gained through this experiment. So objectives were fulfilled by carrying out this experiment. Though some errors might have been occurred and caused a little deviation of the value of acceleration. However, the experiment was performed safely and without any accident. If methodology of experiment would be adhered more perfectly the percent of difference between obtained value and established value of acceleration due to gravity could be less.
