Newton’s Second Law of Motion
Claimed by Andrew Huot Spring 2017 (ahuot3)
History
Newton's Three Laws of Motion have been fundamental in the field of physics for hundreds of years. The three laws were first published in 1687, in Latin, in Newton's book titled "Principia Mathematica". In his original work, Newton stated the second law as the "change in momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed." In fact, Newton even created a whole new type of math, differential calculus, in order to further study and prove his laws. Eventually, the law was further simplified into Force equals mass times acceleration (F=ma). Newton's immense contributions lead to the unit of Force is named the "Newton".
Main Idea
A Mathematical Model
At the most basic level, Newton's Second Law of Motion states that force is equal to mass multiplied by acceleration, or F=ma. This means the force applied on an object is dependent on only two factors, the mass of the object and the acceleration, or change of momentum of the object. However, Newton's Second Law of Motion provides us with more information than simply that. First, it shows that the force applied on an object must be in the same direction as the acceleration, as mass is simply a positive constant. This can be further investigated to show that the force increases as the magnitude of acceleration increases, meaning acceleration, momentum, and force all have a positive relationship. It is important to realize that if an object is at constant velocity, even if nonzero, there is a zero net force acting on the object.
Additionally, this law can be re-written to show that [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where dp/dt represents change of momentum. Therefore, the greater the change in momentum, the greater the force being applied on the object.
A Computational Model
This is the code from this semester's "Fancart" Lab and is the closest related to the second law. "https://trinket.io/embed/glowscript/6aa33c7973"
Units
As with any physics problems we want to be sure that we use the proper units to ensure the accuracy of our answers. Furthermore, by knowing the desired units you may be able to better understand where to start a Second Law problem when you're confused. In terms of SI units, net force uses newtons, acceleration uses m/s^2, and mass uses kilograms. Generally, second law problems are very straightforward so one way that professors can complicate these problems is to provide inconsistent units, so consider yourself warned!
The Impulse Connection
Impulse is derivated from the second law. Impulse is defined as I= F(delta T), or force times change in time. This law allows one to calculate the force and the simply multiplying this force by a given change in time yeilds impulse. Impulse is the time integral of force, and since force is the time derivative of momentum it follows that impulse=change in momentum=mass times change in velocity. Impulse is important in predicting and studying the motion of collisions.
Example Problems
Simple
Given a object has a mass of 3.5 kg and an acceleration of 2.3 m/s^2. What is the force applied on the object?
Answer: 8.05 N
Explanation: Simply use the formula stated in Newton's Second Law of Motion. Force= 3.5(2.3)= 8.05 N.
Middling
A car has a mass of 200 kg. The car starts at rest. Ten seconds later, the car is moving at a speed of 20 m/s. What is the force applied on the object?
Answer: 400 N
Explanation: First, solve for the acceleration by finding the change in velocity, over the change in time. Therefore (20-0)/(10-0)=20/10=2 m/^2. Then use this acceleration value and the given mass to implement Newton's Second Law of Motion. Therefore, Force= 200(2)=400 N.
Difficult
A human named Julio has a mass of 40 kg and is running. Initially, Julio has a momentum of 240 kgm/s. Ten seconds later, Julio has a velocity of 8 m/s. What is the force applied on Julio?
Answer: 8 N
Explanation: This question is difficult because it has multiple parts to it. First, one must solve for the acceleration by dividing the momentum by the mass(240/40=6 m/s) and then finding the difference between the two velocities(8-6=2), and then divide the difference by the change in time(2/10=0.2 m/s^2) Next, Newton's Second Law must be applied in order to find the force(Force=0.2(40)= 8 N). Therefore the answer is 8 N
Connections to Real World
As far as connections to specific "real world" examples, Newton's Second Law of Motion can be used in almost any instance in which an object moves. Personally, a someone very interested in sports, I found it interesting how all three of Newton's Laws, specifically the second explained so much in how athletes interact with one another. One of these examples is given in the list of external links.
The Second Law Moving Forward
It is important to keep the second law in perspective as you move forward on your journey from physics Padawan to Jedi Master. Generally, the second law is superseded by the momentum principle and the conservation of momentum because force is the time derivative of momentum and thus becomes redundant. Additionally, it is important to note that at velocities near the speed of light (3e8 m/s) the second law breaks down and relativistic equations are far more accurate.
External Links
Use these links as further help in order to fully understand Newton's Second Law of Motion. Link number 3 offers a fun and unique real world application of this law.
[3]https://www.youtube.com/watch?v=qu_P4lbmV_I
References
[1]http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion3.htm
[2]http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law