Abstract:  

The  objective of this experiment is to perform an experiment, which consists of validating the value of the  acceleration due to gravity g, which is  9.81 m/s2.  The experiment was conducted in two different ways to carry out the procedure.  The results displayed strong positive correlation between the experimental and the theoretical values as the errors were listed in the range from 0.2 – 0.71 %, with few exceptions.  Our conclusion suggested that the experiments could have had much accurate and precise results based on provision of better quality of equipment including carts, ramp and the sensor, with the sensor being the most essential one.   

 

Introduction:

The presence of gravity in this physical world changes the way science is viewed.  The purpose of the this experiment is to calculate the acceleration due to gravity on Earth.  There are many ways to calculate the little ‘g’. These ways to observe the nature of gravity can be traced back to the 16th Century when scientists such as Galileo Galilei and Isaac Newton first discovered this phenomenon (Lamb R. 2018). Their contribution in this field of study provided people with the essence of key concepts such as the benefits of gravity, its importance etc.  

The prime role of gravity is to hold the universe together.  The value of the acceleration due to gravity on the earth is constant and is 9.81 m/s2 (Henderson 2018).  Although this value slightly changes as the altitude increases, it is also worth noting that the little g is different on other planets within the solar system.  For example, the value of ‘g’ on the Moon is around 1.633 m/s2, while on Jupiter, it is around 25.95 m/s2.    

This marvelous concept in the world of Physics is what makes people walk on the ground, else people would be flying in the air like a flying saucer.  Over the years, there have been modifications in experiments to achieve the current value of 9.81m/s2 (which further depends on altitude).  

The goal of this lab is to conduct experiments where we drop various objects of different masses and try to calculate the acceleration due to gravity of those objects. (Henderson 2018) The lab carries out two essential ways of calculating the value of the acceleration due to gravity (g) and compare those values with the established value.     

 

Apparatus:

Ping Pong balls (1), Tennis ball (1), Practice Cricket Ball (1), Wooden block  (1) , A weight scale, Measuring tape, Meter stick, Stopwatch, Ramp (inclined plane track), Motion sensor, Logger pro software, Cart, and a ruler.

 

Procedure:

1. First, the group collected all the essential elements required to perform this experiment.
2. Second, we take all three objects and measure their weight one by one using the weight scale and register it.
3. Next, we measure the distance from which we are going to drop the objects from with the measuring tape. We are going to drop all three objects from the same height. We register the height, which is 1 meter
4. Now, we take the ping pong ball and drop it from 1 meter. Then, we register the time it took to hit the ground. To get the accurate time, we repeat the process thrice and the different times we get, we register the average of that time. We do the same thing for the tennis ball and the wooden block. Now, we have recorded all the timings of all the three objects to reach the ground.   

 

                                Figure 1                                                               Figure 2

                    

 

Figure 1 & 2 showing the materials used for the experiment (Mohammed, Shames, 2018)

 

1. Now since we have the time and the distance, we use the formula , the acceleration due to gravity will be calculated and then will be compared with the theoretical value of g i.e. 9.81 m/s2.     
2. For the next part of the experiment, we took a ramp and changed it into an inclined plane. For this, our team changed the angle of the ramp and then made a plastic cart to run on it.  The cart’s motion was obtained using a motion sensor device which enabled the logger pro program to produce a position – time graph of the cart’s total run.(LangaraDemoSquad 2013)
3. Thereafter, our group used the derivative and the second derivative functions to obtain the velocity – time graphs from the original graph. We repeated this process for 5 more trials.      
4. Next up, our team found the slopes of the velocity – time graphs and calculated the value of ‘g’ using the formula , where ‘a’ was the slope, and was the angle of the ramp.  
5. Lastly, we compared these experimental values with that of the theoretical value to verify if our data was reliable or not.  After this, the workstation was cleaned and all the materials were kept back in their original positions.

 

Results:                    

Below is the recorded data (raw data) for all the three objects along with their graphs, necessary for evaluation.  

Table-1

Name of the object	Mass (in Kg)
Practice Cricket Ball	0.102
Tennis ball	0.057
Ping – Pong ball	0.003
Small wooden block	0.018

 

Height from which the objects are dropped = 1 m.  

Following is the data collected for all the objects and the time (in seconds) they took when dropped from a height of 1 meter.  Each object was dropped five times, hence, five trials per object.

The data for this part of the experiment was collected using a stopwatch (± 0.01 s).

Table-2

Object	Practice Cricket Ball	Tennis ball	Ping – Pong ball	Wooden block
Trial 1	0.44 s	0.43 s	0.69 s	0.53 s
Trial 2	0.43 s	0.48 s	0.46 s	0.49 s
Trial 3	0.45 s	0.52 s	0.56 s	0.46 s
Trial 4	0.42 s	0.39 s	0.65 s	0.53 s
Trial 5	0.46 s	0.43 s	0.46 s	0.46 s

Below is the % error data along with the ‘g’ values of the four objects during their free fall.

 

Table-3

Object	Acceleration due to gravity (in m/s2)	Percent error (%)
Practice Cricket ball	10.33	5.3%
Tennis Ball	9.88	0.71%
Ping -Pong ball	6.29	– 35.88%
Wooden Block	8.19	-16.5%

 

Sample Calculations:   

1. Value of ‘a’ (acceleration):  

1. Formula:
2. a = 2(0.45)2 (Tennis ball, avg. time)
3. a = 9.88 m/s-2

 

1. Percent Error:

1. Formula:
2. % error: (9.88 ms-2– 9.81 ms-2  9.81 ms-2 )*100%
3. % error = 0.71%

 

3. Average of time taken for the fall by an object:

1. Formula: Sum of all valuesTotal no. of values
2. Average: (0.44 s + 0.43 s + 0.45 s + 0.42 s + 0.46 s)5
3. Average = 0.44 s (Tennis ball)

Part 2

For this part of the experiment, the group collected data for determining the acceleration due to gravity (g) while making an object (a cart) roll on a ramp (inclined at an angle) and obtaining the slope of its velocity during its run.   

1. Mass of the cart: 0.501 Kg
2. Height of the ramp: 7.5 cm (0.075 m)
3. Hypotenuse of the ramp: 106.8 cm (1.068 m)
4.                           

Hence, the value of will be sin-1(Opposite Hypotenuse), which in this case was 1.663°.  (Bourne n.d)

 

Given these measurements and based on the graphs depicted below, the slope of the velocity – time graph i.e. acceleration of the rolling cart was found and later, the value of ‘g’ was determined using the formula .   These graphs were obtained through a motion sensor device connected to the ramp.  Furthermore, the cart was made to move from the point where the motion sensor was located to achieve better quality results.   

Following is sample graph which was obtained by the motion sensor during the total run of the cart on the ramp.  

Figure 3

     

 

Graphical Analysis:  

The graph illustrated above shows two graphs, the top one is the position – time graph and the bottom one is the velocity – time graph.  The motion sensor depicts the total run of the moving cart on an inclined ramp and highlights the distance (velocity in the other graph) when the velocity of the cart increases.  We then took the slope of this highlighted area from the velocity – time graph, which is shown in the graph as 0.2843 m/s2 (acceleration of the cart during that time interval).

 

The table below gives a summary of the data collected for the various values of ‘g’ given the different slopes of velocity – time graphs for the moving cart on the inclined ramp.  This procedure was also repeated five times for better precision of data.

 

Table-4

No. of trials	Acceleration due to gravity (g) (m/s2)
Trial 1	8.96 m/s2
Trial 2	8.39 m/s2
Trial 3	9.65 m/s2
Trial 4	8.96 m/s2
Trial 5	9.79 m/s2

 

Sample Calculations:  

1. Value of g:

1. Formula:
2. g = 0.2843sin(1.663°)

g =  9.79 m/s2  

1. Percent Error:

1. Formula:  
2. % error = (9.79 ms-2 – 9.81 ms-2  9.81 ms-2 )*100%

% error = 0.204 % (ignoring the ‘-’ sign)  

 

Discussion:  

Our team tried out two different methods to find the value of ‘g’. The first one is the free fall experiment, where we dropped objects of different masses from the same height and got as close as possible to 9.8ms-2. We used four different objects, for the practice cricket ball we found, g to be 10.33 ms-2, while the tennis ball, ping pong ball and the wooden block had the ‘g’ values as 9.88, 6.29 and 8.19 ms-2 respectively.  After evaluating these values and comparing them with the established value of 9.81 ms-2, our group found out the % error to be the following: cricket ball = 5.30 %, tennis = 0.71 %, ping pong ball = – 35.88 % and lastly, the wooden block = – 16.5 % errors.  This showed that the tennis ball was most precise and reliable source for our experiment. For other objects, such as ping pong balls, there were several factors that played a role in their fall.  One of them is the wind resistance. Being a lightweight ball, the wind might have been a huge factor in deteriorating the impact of its weight with respect to the ground and hence, had the largest error.  For the wooden block, it encountered the second biggest error due to its shape. Wind resistance works largely on flat surfaces. As a result, it is plausible that the block’s measurements might have been affected by it.  Furthermore, the height from which the objects were dropped was also another factor, which played a role in determining our values. Since our group did not have a marked point. Instead, we measured it with a meter stick and kept our finger pointed there.  Due to this human error, it might be a possibility that the objects dropped from the height of 1 m may have actually been dropped from a few centimeters above or below. Timing could have been a factor as we relied on the sound of the object dropping on the floor to stop the timer. There is a delay between hearing the sound and the actual sound being done. Even though this delay is very small it still exists.   

For the second part of the experiment, we were using logger pro, a device that plots graphs based on object’s motion was used to graph the position (m) and the velocity (m/s), the graph of the velocity was very off due to the problems in the ramp, or motion sensor, or in the cart itself. This goes on to show that, the motion in which the cart moves in the inclined ramp has to be very accurate in order to find the accurate value of g. We switched to another ramp and applied the same method, and because of the fact that the ramp, motion sensor, and the cart was working properly, we were able to pull off the slope from the velocity graph from the logger pro, using the formula , we were able to get as close as possible to 9.81 ms-2. To be more accurate next time, especially in case of light objects such as ping – pong ball and wooden block, we need to make sure the wind doesn’t affect the free fall of such objects. Also, in our inclined gravity experiment, due to the differences in kinetic friction of ramps, we had the errors in our graphs and as a result, a value of g with error. So in order for us to make the value of g as accurate as possible in the inclined gravity experiment, we have to keep in mind that the kinetic friction between the ramp and the cart’s wheel also plays an important role in the shape of position time graph and velocity time graph, which eventually helped us find the accurate value of g.  

 

Conclusion:  

The data collected by our group and its results proved that although there were few inconsistencies, the lab carried out by our team showed varied results, with most of them keeping the outcomes relatively similar to those in the scientific world. These inconsistencies, however, were caused by various reasons including human error when timing the fall of objects on freefall,  minimal friction, particularly kinetic friction, present between the ramp and the cart’s wheels, and the superposition of the motion sensor with the cart’s wheels. Although the ramps were supposed to be frictionless, in reality it’s physically impossible to have a frictionless ramp. We disregarded the force of friction in our calculations because it offered very little change to our results and unnecessarily complicated our calculations. While the second experiment was in progress, the graphs obtained during the cart’s run had numerous hill-shaped portions in the acceleration – time graph.  This was a rare case since the cart was going downhill and was not supposed to have any flaws in the graph obtained. The acceleration graph was supposed to be a graph with straight lines with one huge fall due to rebounding. Since this was not the case, we believe that there was some flaw within the motion sensor. If our group were to redo this experiment, we would ensure that the ramp and all other materials including the cart, the motion sensor are all without flaws. All in all, I believe that through the course of this lab, our team could achieve the real purpose behind the true nature of science through these although not accurate but highly precise results.

 

References:   

Bourne, M.(n.d) “2. Sine, Cosine, Tangent and the Reciprocal Ratios.” Intmathcom RSS, www.intmath.com/trigonometric-functions/2-sin-cos-tan-csc-sec-cot.php.

Henderson, T. (2018). Free Fall and Air Resistance. Retrieved from http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Free-Fall-and-Air-Resistance

Henderson, T. (2018). Acceleration of Gravity. Retrieved from https://www.physicsclassroom.com/class/1DKin/Lesson-5/Acceleration-of-Gravity

 

LangaraDemoSquad, (2013, August),  director. g On an Incline. YouTube, YouTube, www.youtube.com/watch?v=E4J92AsZMAg.

 

Lamb, R. (2018, June 28). What is gravity? Retrieved from https://science.howstuffworks.com/environmental/earth/geophysics/what-is-gravity.htm