Title: Relationship among net force, mass, and acceleration Introduction ● Background Information ○ The laws of motions, proposed by Isaac Newton, the most brilliant and influential person of the millenium and the father of classical physics, state that one, an object will not change its motion unless a force acts on it; two, the force on an object is equal to its mass times its acceleration; and three, when two objects interact, they apply forces to each other of equal magnitude and opposite direction. The objective of this experiment is to prove Newton’s laws of motion and the relationship among the net force on a macroscopic object, its mass, and its acceleration. In particular, Newton’s laws of motion state that force equals mass times acceleration. Therefore, we need to conduct two experiments on this. One is about the relationship between the mass and the acceleration, and the other one is about the relationship between the net force and the acceleration ○ Newton’s law of motion: F=ma ○ Equation for Atwood machine: a=g(m/(m+M)) ○ Equation for modified atwood machine: - Equation for the modified atwood machine is quite similar to the ordinary one. However, because there is also an inclination in the modified version, the cart will be pulled down by gravity. Therefore, we will have to minus Mgsin(theta) from mg as shown below. First experiment: relationship between net force and acceleration ● Variables ○ Independent variable: The mass added on the string. This is because in this experiment, we will increase the mass added on the string to see whether the acceleration of the cart increases or decreases. ○ Dependent Variable: the acceleration of the cart. This is because in this experiment, we will increase the mass added on the string to see whether the acceleration of the cart increases or decreases. ○ Other controlled variables that must be kept constant throughout the experiment - The inclination of the ramp (because according to the equation mentioned above, this can affect the gravitational force exerted on the cart by the earth) - Type of pulley (because this can affect the friction, and therefore, affect the acceleration of the cart) - Type of string (because this can affect the friction, and therefore, affect the acceleration of the cart) - Mass of cart (because according to the equation mentioned above, this can affect the force the cart being pulled down) - Type of ramp (because this can affect the friction, and therefore, affect the acceleration of the cart) ● Objective ○ The objective of this experiment is to prove Newton’s second law about net force is directly proportional to the acceleration. ● Hypothesis - The net force is directly proportional to the acceleration of the cart. This is because when there is more force pulling the cart, the faster it is going to run. Methodology ● List of materials Name of the materials Quantity 2m string 1 2m ramp 1 Cart (ideally 500g) 1 Weight(50g) 5 Camera 1 Tube grip stand holder 1 ● Procedure 1. Set up camera to record the experiments 2. Put sticky notes on the ramp so that the distance between the farther ends of them is 1 meter. 3. Put the cart on the ramp 4. Hang the string on the cart and take it to the other end of the pulley. 5. Slowly hang two weights on the string and leave the weights to fall down 6. Reset the experiment and do two more trials 7. After finishing three trials, add one more weight to the string and repeat from step 5 to step 6 8. Repeat step 7 for the last experiment. 9. Use data logger pro to analyze the videos and calculate the acceleration (https://www.youtube.com/watch?v=THnWsef745c) This is an example of the experiment: https://drive.google.com/file/d/1VFNPwERYHLM48-n5DBxC8GmYg_iHCVum/view?usp=sharin g Data Collection ● Qualitative observations ○ When there are only two weights to pull the cart down, the cart moves relatively slowly at first. After approximately 1 second, the cart accelerates and moves faster. When we add one more weight to the string, the cart also moves relatively slowly for the first 0.5 second. After that, the cart accelerates and starts moving faster. Finally, when there are 4 weights on the string, the cart moves very fast even at the start of the experiment. In conclusion, through qualitative observation only, the cart moves faster when there are more weights hung on the string, which means that the net force is directly proportional to the acceleration of the cart. ● Data Table: Relationship between the net force and the acceleration The acceleration of the cart (m/s/s) Mass added onto the string (g) Trial 1 Trial 2 Triall 3 100 0.36 0.41 0.34 150 0.55 0.53 0.46 200 0.67 0.72 0.63 Analysis of Results ● Sample calculations: To calculate the average acceleration of the cart, we divide the sum of the three trials to 3. E.g. (0.36+0.41+0.34)/3 = 0.37 m/s/s Table: Relationship between the net force and the acceleration Mass added onto the string (g) The acceleration of the cart (m/s/s) 100 0.37 150 0.51 200 0.67 Because this is the relationship between the net force and the acceleration of the cart, we change the label “mass added onto the string” to “force exerted on the cart”. To do this, we need to convert from gram to kilogram and multiply it to 9.81 which is the gravitational acceleration. E.g. 100 * 0.001 * 9.81 = 0.981 N Table: Relationship between the net force and the acceleration ● Force exerted on the cart (N) The acceleration of the cart (m/s/s) 0.98 0.37 1.47 0.51 1.96 0.67 Graphs Title: Relationship between net force and acceleration of the cart A positive correlation is clearly illustrated in this graph, which means that the net force and acceleration of the cart are directly proportional. This is because as the force exerted on the cart increases, the acceleration of the cart also increases. Discussion ● ● Conclusion Statement: ○ Claim ■ Net force is directly proportional to acceleration ○ Evidence ■ This is shown clearly by the positive correlation between the force exerted on the cart and the acceleration of the cart in the graph. ○ Reasoning ■ As we can see, as the net force applied on the cart increases, its acceleration decreases. This is because when more force is applied, the object would be pushed harder, and therefore, accelerate faster. Evaluate Your Experiment: - Validity ○ Throughout this experiment, we have controlled the angle of inclination and the mass of the cart. Firstly, when we increase the angle of inclination, the force exerted on the cart by gravity is also going to increase. Therefore, the cart would be pulled backwards more, and this can significantly impact the acceleration of the cart as well as the validity of the experiment. Similarly, when we increase the mass of the cart, the force exerted on the cart by gravity also increases. ○ ○ ○ Therefore, the cart would be pulled backwards more, and this can significantly impact the acceleration of the cart as well as the validity of the experiment. Thirdly, another factor that can also considerably impact the acceleration of the cart is friction. This is because friction will make the cart move slower. Therefore, we will also need to use the same pulley, string, and ramp throughout the whole experiment. Reliability: ■ The range between the trials of the first variable is 0.07 ■ The range between the trials of the second variable is 0.09 ■ The range between the trials of the third variable is 0.09 ■ The ranges between the trials of the experiment are very small and insignificant, which indicates that this experiment is reliable Sources of Error: ■ There may be one source of error that we did not have time to check. It is the height of the ramp or the angle of inclination. Because we did not have enough time to conduct the experiments all in one period, when another student comes to the class, they may have changed the height of the ramp. Additionally this can also be because over time, the height of the ramp has fallen down a little bit by gravity and would impact the acceleration of the cart. Improvements: - If we have an opportunity to conduct this experiment again, we will try to conduct all the experiments in one day because as I have shown, the height of the ramp may change over time and can affect the acceleration of the cart. Also, if the experiments are splitted into different days, we can also check the height of the ramp and make sure that it is constant before conducting the experiment. Second experiment: relationship between mass and acceleration ● Variables ● (Second experiment) Relationship between the mass and the acceleration ○ Independent variable is the mass added on the cart. This is because in this experiment, we will increase the mass added on the cart to see whether the acceleration of the cart increases or decreases. ○ Dependent variable is the mass added on the cart. This is because in this experiment, we will increase the mass added on the cart to see whether the acceleration of the cart increases or decreases. ○ Other controlled variables that must be kept constant throughout the experiment - The inclination of the ramp (because according to the equation mentioned above, this can affect the gravitational force exerted on the cart by the earth) - Type of pulley (because this can affect the friction, and therefore, affect the acceleration of the cart) - Type of string (because this can affect the friction, and therefore, affect the acceleration of the cart) - Mass of the weight (because according to the equation mentioned above, this can affect the force the cart being pulled up) - Type of ramp (because this can affect the friction, and therefore, affect the acceleration of the cart) ● Objective ○ The objective of this experiment is to prove Newton’s second law about mass is inversely proportional to the acceleration. ● Hypothesis - The mass is inversely proportional to the acceleration of the cart. This is because the force must be stronger in order to pull an object with the same acceleration but heavier mass. Methodology ● List of materials Name of the materials Quantity 2m string 1 2m ramp 1 Cart (ideally 500g) 1 Weight(50g) 5 Camera 1 Tube grip stand holder 1 ● Procedure 1. Set up camera to record the experiments 2. Put sticky notes on the ramp so that the distance between the farther ends of them is 1 meter. 3. Put the cart on the ramp 4. Hang the string on the cart and take it to the other end of the pulley. 5. Place 1 weight on the cart 6. Slowly hang two weights on the string and leave the weights to fall down 7. Reset the experiment and do two more trials 8. After finishing three trials, add one more weight to the cart and repeat from step 6 to step 7 9. Repeat step 8 for the last experiment. 10. Use data logger pro to analyze the videos and calculate the acceleration (https://www.youtube.com/watch?v=THnWsef745c) This is an example of the experiment: https://drive.google.com/file/d/15tQHG3y0uUfhmiaG57DZbeq_vKWIsx6a/view?usp=sharing Data Collection ● Qualitative observations - When there is only one weight on the cart, the cart moves relatively fast and hits the blocker very hard. When we add one more weight to the cart, the cart moves relatively slowly for the first 0.5 second. After that, the cart accelerates and starts moving faster. However, this time, the cart did hit the blocker, but not as strong as with only one weight, which shows that the acceleration of the cart is slower than with only one weight. Finally, when there are 3 weights on the string, the cart moves very slowly throughout the experiment, and it could not even reach the blocker. In conclusion, through qualitative observation only, the cart moves slower when there are more weights placed on it, which means that the mass is inversely proportional to the acceleration of the cart. ● Data Acceleration of the cart(m/s/s). Mass added to the cart (kg). Trial 1 Trial 2 Trial 3 0.05 0.35 0.32 0.33 0.1 0.25 0.24 0.28 0.15 0.21 0.20 0.21 Analysis of Results ● Sample calculations: To calculate the average acceleration of the 3 trials, we take the sum of the three trials and divide it by 3 For example: (0.37+0.31+0.33)/3 = 0.34 (m/s/s) Mass added to the cart (kg). Acceleration of the cart with a margin of error of 0.01(m/s/s). 0.05 0.34 0.1 0.26 0.15 0.21 Title: Relationship between mass and acceleration of the cart A negative correlation is clearly illustrated in this graph, which means that the mass and acceleration of the cart are inversely proportional because as the mass added on the cart increases, the acceleration of the cart decreases. Discussion ● Conclusion Statement: ● Claim - Mass is inversely proportional to acceleration ● Evidence - This is clearly shown by the negative correlation between the mass and the acceleration of the cart. ● Reasoning - As we can see, as the mass of the cart increases, its acceleration decreases. This is because when the mass increases, it would require more force to push at the same acceleration, or on the contrary, for the same amount of force applied, the acceleration would be slower. ● Evaluate Your Experiment: ● Validity - Throughout this experiment, we have controlled the angle of inclination and the mass of the weight. Firstly, when we increase the angle of inclination, the force exerted on the cart by gravity is also going to increase. Therefore, the cart would be pulled backwards more, and this can significantly impact the acceleration of the cart as well as the validity of the experiment. Similarly, when we increase the mass of the weight, the force exerted on the cart by the weight increases. Therefore, the cart would be pulled towards the pulley more, and this can significantly impact the acceleration of the cart as well as the validity of the experiment. Thirdly, another factor that can also considerably impact the acceleration of the cart is friction. This is because friction will make the cart move slower. Therefore, we will also need to use the same pulley, string, and ramp throughout the whole experiment. ● Reliability - The range between the trials of the first variable is 0.03 - The range between the trials of the second variable is 0.02 - The range between the trials of the third variable is 0.01 - The ranges between the trials of the experiment are very small and insignificant, which indicates that this experiment is reliable ● Sources of Error - There may be one source of error that we did not have time to check. It is the height of the ramp or the angle of inclination. Because we did not have enough time to conduct the experiments all in one period, when another student comes to the class, they may have changed the height of the ramp. Additionally this can also be because over time, the height of the ramp has fallen down a little bit by gravity and would impact the acceleration of the cart. ● Improvements - If we have an opportunity to conduct this experiment again, we will try to conduct all the experiments in one day because as I have shown, the height of the ramp may change over time and can affect the acceleration of the cart. Also, if the experiments are splitted into different days, we can also check the height of the ramp and make sure that it is constant before conducting the experiment. Summary ● ● Summary Statement ○ Throughout the experiment, we have found that acceleration is directly proportional to the net force and inversely proportional to the mass. Real World Connection ○ This relationship or formula is fundamental and a foundation to classical physics. It is applied to everyday activities as well as physical phenomena. For example, when we push a car, in order for the car to accelerate, the force we exert on the car has to be bigger than the mass of the car because the net force is directly proportional to the acceleration and inversely proportional to the mass of the car. Another real-life example of this relationship is rocket launching. In order for the rocket to escape the earth’s atmosphere, the force it has to exert must outweigh its mass or the force of gravity.
