Newton’s Second Law of Motion

Claimed by Rahul Singi Fall 2016

History

As you well know, Newton's Three Laws of Motion have been fundamental in field of physics for hundreds of years. The three laws were first published in 1687, in Latin, in a 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 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. At face value, 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

Connection to Newton's Other Laws

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