Newton’s Second Law of Motion
Claimed by Andrew Huot Spring 2017
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.
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
How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript
Manipulate the code to see the different motions of the cart. See what changing the direction of the force, the net force, or the mass of the ball does to the momentum and final position of the cart.
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
This law helps find an objects impulse. Impulse is defined as I= F(deltat), 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 gives impulse. Calculating impulse is very important because impulse is key when predicting motion. Therefore, Newton's Second Law of Motion is important in order to predict motion of objects.
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.
(Edit by Lakshmi Krishnan (lkrishnan7))
For a real world example of Newton’s 2nd Law, take the scenario of a car crash.
Scenario: A lady has hit a lamppost. She appears fine, but the car has been totalled. Witness Statement: As a lady cruised by in her Honda Civic, she momentarily took her eyes off the road, and found her car in front of a lamppost and crashed into it.
Report: This case is a Newton’s 2nd law case. According to the law, F = MA, which means that F = M x (ΔV / ΔT), or F = M((Vf -Vi) / T) F ΔT = M(Vf - Vi) The extent of her injury is determined by the force that hit her. So, the size of the force she’s subjected to is determined by the interaction time. If the T becomes smaller, the force is bigger; if T is bigger, the force gets smaller. The sudden change of velocity also played a part in her injury. The car should’ve had safety features that made the interaction time as long as possible, because if you can spread the reaction time, you can do the deceleration with a smaller force - We can reduce the size of the force if we reduce the time.
One must also look at the impact force this car has faced. A crash like this one, that stops the car completely, must take away all the kinetic energy of the car, and as highlighted by the work-energy principle, it would take a longer stopping distance to decrease the impact force.
Net Work = (1/2)mv(final)^2 - (1/2)mv(initial)^2 The change in the kinetic energy of an object is equal to the net work done on the object.
The car is noted to be 1600 kgs, or 16,000 N, and travelling at a speed of 10 m/s. The work required by the lamppost to stop the car would be F(avg)D = -(1/2)mv^2 What would be the force on the car?
Mass = Weight/ Acceleration of Gravity Mass = 16,000 N / 10 (M/S^2) Mass = 1600 kgs
KE = (1/2)mv^2 KE = (1/2)x1600x(10m/s)^2 = 80000 Joules = kg x m^2 / s^2
d=1 metre, or 3 feet after impact
F(avg) = KE / d
80000 / 1 = 80,000 newtons, or 8.96 tons!
Conclusion: Thus, the “crumple zones” in the front of the car, and the airbags came in handy, because rather than allowing sudden stop, the car was allowed to halt over a larger period of time, making the force less than it would’ve been without them.
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