Momentum Principle
This page discusses the Momentum Principle and examples of how it is used.
Claimed by hyk96610
The Main Idea
The Momentum Principle is the first fundamental principle of mechanics where it describes the relationship between the change in momentum of a system and the total amount of interaction (or total amount of force) with the surroundings. In terms of the system and surroundings, both can be set in any way necessary, where the system may just include a person or the entire Earth. The Momentum Principle can be used in nearly all situations, and it is always advised to start a problem by first writing out the Momentum Principle and then branching out (by rearranging or substituting values) in order to solve a problem.
A Mathematical Model
The Momentum Principle is defined as [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net} }[/math] (or [math]\displaystyle{ ∆\vec{p} = \vec{F}_{net} * {∆t} }[/math]).
p is the momentum of the system. In the equation, momentum (measured in kg*m/s) is expressed as the "change in momentum" ([math]\displaystyle{ ∆\vec{p} = \vec{p}_{final} - \vec{p}_{initial} }[/math]), which includes both the magnitude and direction of the momentum.
F is the net force from the surroundings. Force (measured in Newtons, or N) includes the interactions between system and the surroundings, like the gravitational force exerted by the Earth on us or the force that a compressed spring exerts on a mass. In the Momentum Principle, the force includes both the magnitude and direction. Also, it is important to note that the Momentum Principle calls for the net force, which is the sum of all the different forces from the surroundings, like adding both the force of gravity and the force of the spring together to calculate the net force. Because of this, it is even more crucial to pay attention to the direction of the forces as a positive or negative sign error could cause an error in the calculated net force.
t is the time (measured in seconds, or s). Specifically, the Momentum Principle calls for the "change in time" ([math]\displaystyle{ ∆\vec{t} = \vec{t}_{final} - \vec{t}_{initial} }[/math]), or in other words, the duration of the interaction is needed.
A Computational Model
Click on the link to see the Momentum Principle through VPython!
Make sure to press "Run" to see the principle in action!
Examples
Simple
Two external forces <40,-70,0>N and <20,10,0>N, act on a system. What is the net force acting on the system?
Answer: <60,-60,0>N
Explanation:
[math]\displaystyle{ \vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} }[/math]
[math]\displaystyle{ \vec{F}_{net} }[/math] = <40,-70,0)N + <20,10,0>N = <60,-60,0>N
Middling
A hockey puck is sliding along the ice with nearly constant momentum <10,0,5>kg*m/s when it is suddenly struck by a hockey stick with a force <0,0,2000>N that lasts for only 3 milliseconds (3e-3s). What is the new (vector) momentum of the puck?
Answer: <10,0,11>kg*m/s
Explanation:
[math]\displaystyle{ ∆\vec{p} = \vec{F}_{net} * {∆t} }[/math]
[math]\displaystyle{ \vec{p}_{final} - \vec{p}_{initial} }[/math] = \vec{F}_{net} * {∆t}</math>
[math]\displaystyle{ \vec{p}_{final} }[/math] - <10,0,5>kg*m/s = <0,0,2000>N * (3e-3)s
[math]\displaystyle{ \vec{p}_{final} }[/math] = <10,0,11>kg*m/s
Difficult
In outer space a rock of mass 5kg is acted on by a constant net force <29,-15,40>N during a 4s time interval. At the end of this time interval the rock has a velocity of <114,94,112>m/s. What is the rock's velocity at the beginning of the time interval?
Answer: <90.8,106,80>m/s
Explanation:
[math]\displaystyle{ ∆\vec{p} = \vec{F}_{net} * {∆t} }[/math]
[math]\displaystyle{ \vec{p}_{final} - \vec{p}_{initial} = \vec{F}_{net} * {∆t} }[/math]
[math]\displaystyle{ \vec{p} = m * \vec{v} }[/math]
[math]\displaystyle{ m\vec{v}_{final} - m\vec{v}_{initial} = \vec{F}_{net} * {∆t} }[/math]
(5kg * <114,94,112>m/s) - (5kg * [math]\displaystyle{ \vec{v}_{initial} }[/math]) = <29,-15,40>N * 4s
[math]\displaystyle{ \vec{v}_{initial} }[/math] = <90.8,106,80>m/s
Connectedness
All over the world and at every point in time, interactions are continuously occurring, and I thought it was interesting to see how the Momentum Principle was the most fundamental principle that would used in starting to the analyze the different interactions. Although there is not a direct relationship between the concept of the Momentum Principle and my major in Biochemistry (which would have more connections with the Energy Principle), there are many industrial applications of the Momentum Principle. Again, the Momentum Principle is not directly connected to the applications, but it is used in the process (especially in the beginning) of industrial application. For example, when creating life saving airbags and seat belts for cars, the Momentum Principle is used. The final momentum of a car during an accident would be zero, or would stop, and the initial momentum would be based on the mass and velocity of the car. With the change in momentum fixed, the airbag and seat belt would focus on increasing the time taken for the body's momentum to reach zero (final momentum), which would consequently reduce the force of the collision and protect the body from getting as injured. With the Momentum Principle being applicable in so many areas of my life, I found the concept even more interesting.
History
Although the Momentum Principle is credited as Newton’s second law of motion, it is difficult to just credit Isaac Newton (1643AD – 1727AD) for the development of the principle. As the Momentum Principle is the quantitative and more in-depth representation of Newton’s first law of motion (“An object tends to be at rest or moves in a straight line and a constant speed except to the extent that it interact with other objects”), the development of the first law also serves an important role in the history of the Momentum Principle. Aristotle (384BC – 322BC) initially proposed that objects had the natural tendency to be at rest and that a push (or a force) was absolutely needed to keep the object moving. His proposal was challenged by Galileo (1564AD – 1642AD), who introduced the idea that objects had the natural tendency to travel in a straight line at constant speed unless something (or a force) was interacting with something. Likewise, Descartes (1596AD – 1650AD) also contributed as he proposed three laws of nature in his “Principle of Philosophy,” which actually outlined the later published Newton’s first law of motion. After studying Descartes, Newton adopted Descartes’ principles as his first law of motion, and alongside the famous story of Newton sitting under an apple, Newton was able to create the Momentum Principle, or his second law of motion.
See also
As the Momentum Principle is the first of three fundamental principles of mechanics, the next possible topics to examine would be the other fundamental principles, the Energy Principle and the Angular Momentum Principle. Also, although the momentum principle is an extremely important concept that usually signals the start of a momentum related problem, the principle branches out into other momentum topics like Impulse and Iterative Prediction, which are used to solve other types of problems.
External links
[2] https://www.youtube.com/watch?v=ZvPrn3aBQG8
References
[1] http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion2.htm
[2] Matters & Interactions: Modern Mechanics 4th Ed. Vol. 1 (Chabay & Sherwood)
[3] Dr. Flavio Fenton's Lecture Notes on the Momentum Principle (8/26/15)