Newton's Second Law: the Momentum Principle - Revision history

Asaxena344: /* 4. (Difficult) */

2024-12-04T01:21:04Z

4. (Difficult)

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	The change in momentum <math>d\vec{p}</math> points towards the center of the circle, as shown in the diagram below:		The change in momentum <math>d\vec{p}</math> points towards the center of the circle, as shown in the diagram below:
	[[File:derivationofcentripetalforce.jpg]]		[[File:derivationofcentripetalforce.jpg]]



	Direction Explanation:		Direction Explanation:
	At any point on the circular path, the force responsible for changing <math>\vec{p}</math> acts perpendicular to the velocity vector and points radially inwards.		At any point on the circular path, the force responsible for changing <math>\vec{p}</math> acts perpendicular to the velocity vector and points radially inwards.

Asaxena344: /* 4. (Difficult) */

2024-12-04T01:20:27Z

4. (Difficult)

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	===4. (Difficult)===		===4. (Difficult)===

	A particle of mass <math>m</math> and speed <math>v</math> moves in a circular path~~. Its~~ angular frequency is <math>\Omega</math>. ~~(Angular frequency is the rate at which the angle of the particle's position is changing in radians per unit time.)~~ Using the momentum principle, ~~show that~~		A particle of mass <math>m</math> and speed <math>v</math> moves in a circular path with an angular frequency <math>\Omega</math>. Using the momentum principle, demonstrate the following:

	a) a nonzero net force is acting on the particle,		a) A nonzero net force is acting on the particle.

	b) ~~the~~ magnitude of ~~the~~ force is ~~given by~~ <math>f = mv\Omega</math>~~, and~~		b) The magnitude of this force is <math>F = mv\Omega</math>.

	c) ~~the~~ direction of the force is inwards towards the center of the circle.		c) The direction of the force is inwards, towards the center of the circle.

			a) Nonzero Net Force Acting on the Particle

	a) The particle~~'s momentum~~ vector is changing ~~over time;~~ its ~~magnitude is~~ constant ~~(mv), but its direction changes as its position along the circular path changes~~. ~~It is always tangential to the circular path. Since~~ <math>\vec{p}</math> changes ~~over time, a nonzero net force must be acting on~~ the ~~particle. In fact, the only way a particle~~ of ~~constant mass can have a net force of 0 acting on it is if it is travelling along a straight line at a constant speed~~, ~~or if it is at rest.~~		The particle moves in a circular path, which means its velocity vector is constantly changing direction even though its speed remains constant. The momentum vector <math>\vec{p} = m\vec{v}</math> changes as the direction of <math>\vec{v}</math> changes, which implies that:

	b)		<math>\frac{d\vec{p}}{dt} \neq 0</math>

			From Newton's Second Law, we know that:

			<math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>

			Since <math>\vec{p}</math> changes due to the particle's changing velocity direction, a nonzero net force must act on the particle. This force is responsible for continuously changing the direction of the particle's velocity, keeping it in circular motion.

			For a particle of constant mass, the only way <math>\vec{F}_{net} = 0</math> is if the particle is at rest or traveling in a straight line at constant speed, neither of which applies to circular motion.

			b) Magnitude of the Force

			To determine the magnitude of the net force, consider the centripetal force required to keep the particle moving in a circle. The centripetal force ensures that the particle maintains its circular path by continuously pulling it towards the center.

			Angular Frequency Relationship: The angular frequency <math>\Omega</math> is related to the particle's speed and the radius of the circular path: <math>v = \Omega R</math>
			Substituting for <math>R</math> gives: <math>R = \frac{v}{\Omega}</math>
			Centripetal Acceleration: The centripetal acceleration is: <math>a_c = \frac{v^2}{R}</math>
			Substituting <math>R = \frac{v}{\Omega}</math>: <math>a_c = v \Omega</math>
			Force Magnitude: Using Newton's Second Law: <math>F = ma_c</math>
			Substituting for <math>a_c</math>: <math>F = m v \Omega</math>
			Thus, the magnitude of the force is:

			<math>F = mv\Omega</math>

			c) Direction of the Force

			The direction of the net force acting on the particle is always towards the center of the circular path. This force is often referred to as the centripetal force.

			Momentum Vector Analysis:
			The momentum vector <math>\vec{p} = m\vec{v}</math> is tangential to the circular path and changes direction as the particle moves.
			The change in momentum <math>d\vec{p}</math> points towards the center of the circle, as shown in the diagram below:
	[[File:derivationofcentripetalforce.jpg]]		[[File:derivationofcentripetalforce.jpg]]
			Direction Explanation:
			At any point on the circular path, the force responsible for changing <math>\vec{p}</math> acts perpendicular to the velocity vector and points radially inwards.
			For example, if the particle is at the top of the circle moving to the right, the force vector points downward towards the center. This is consistent throughout the motion.
			Thus, the force direction is always radially inward, ensuring the particle remains in circular motion.

			Final Summary
			a) A nonzero net force acts on the particle because its momentum changes direction over time.

			b) The magnitude of this force is:

	~~c) As one can see from the diagram above,~~ <math>d\~~vec{p}~~</math> is pointing towards the center of the circle. If the particle is exactly at the top of the circle, its momentum is to the right (or left if the particle is travelling counterclockwise), and as <math>d\theta</math> approaches 0, <math>d\vec{p}</math> approaches a downward direction, which is towards the center of the circle.		<math>F = mv\Omega</math>

	~~For more information~~, ~~see [[Centripetal Force and Curving Motion]]. Our answer agrees with~~ the ~~formulas given on that page~~.		c) The force always points inwards towards the center of the circle, perpendicular to the particle's instantaneous velocity.

	==Connectedness==		==Connectedness==

Asaxena344: /* 3. (Difficult) */

2024-12-04T01:19:14Z

3. (Difficult)

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 ===3. (Difficult)===
-A cylindrical rocket ship of mass <math>M</math> and speed <math>v</math> is cruising through outer space when it encounters a stationary dust cloud. The dust cloud has a number density of <math>N</math> particles per unit volume, and each dust particle has a mass of <math>m</math>. The rocket ship has a sticky surface on its front face that allows it to accumulate all of the dust that it hits, absorbing it into its mass. The radius of the rocket ship's circular face is <math>R</math>. What force does the rocket's thruster need to exert on it in order to preserve its speed as it moves through the dust cloud? (Assume that the consumption of fuel has a negligible impact on the mass of the rocket.)
+===Step 1: Rate of Mass Accumulation===
-[[File:Rocketdustcloudpart1.jpg]]
+As the rocket moves through the dust cloud, it accumulates mass at a rate proportional to its velocity and the density of the dust. The rate of mass increase (<math>\frac{dm}{dt}</math>) is determined as follows:
-[[File:Rocketdustcloudpart2.jpg]]
+The rocket sweeps out a cylindrical volume of dust per unit time: <math>\text{Volume swept per second} = \pi R^2 v</math>
-Advanced Note: in order to use <math>\frac{dm}{dt}</math> for the accumulation of mass, the accumulated mass (i.e. the dust) must initially be at rest, like in the above problem. Otherwise, one must treat the accumulation of dust particles as a series of collisions, or change one's frame of reference so that the accumulated mass is initially at rest.
+The momentum principle states that the net force acting on the system is equal to the rate of change of momentum:
 ===4. (Difficult)===

Asaxena344: /* 2. (Middling) */

2024-12-04T01:17:58Z

2. (Middling)

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 ===2. (Middling)===
-If this collision is inelastic, then we know that kinetic energy is not conserved. However, momentum is conserved for all types of collisions (elastic, inelastic, partially elastic) due to the momentum principle.
+A 1.2 kg cart moving with an initial velocity of <math>v_i = 5 , \text{m/s}</math> collides with a stationary 0.8 kg cart on a frictionless track. After the collision, the two carts stick together. Determine their final velocity after the collision, assuming it is totally inelastic. Additionally, calculate the energy lost during the collision.
-[[File:momentumconserved.jpg|200px|thumb|left|alt text]]
+In a perfectly inelastic collision, the objects stick together after impact. While kinetic energy is not conserved, momentum is always conserved.
-[[File:momentumconserved.jpg]]
+The momentum principle states:
-[[File:MCP2.jpg|200px|thumb|left|alt text]]
+<math>\vec{p}{initial} = \vec{p}{final}</math>
 ===3. (Difficult)===

Asaxena344: /* 1. (Simple) */

2024-12-04T01:16:50Z

1. (Simple)

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	===1. (Simple)===		===1. (Simple)===

	At t=0, a ~~4kg~~ particle is released from rest and a constant ~~12N~~ force begins acting on it. How far has the particle moved after 5 seconds?		At <math>t = 0</math>, a 6 kg particle is released from rest, and a constant 18 N force begins acting on it. How far has the particle moved after 4 seconds?

	~~For~~ this problem, ~~it is easier to~~ use <math>\vec{F}_{net} = m\vec{a}</math> ~~than <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math> because it allows us to solve for <math>\vec{a}</math>~~ which ~~can then be used in~~ a ~~kinematic equation. We know this~~ form of Newton's ~~second law is accurate for this problem because the particle travels at~~ non-relativistic speeds and ~~is of constant mass~~.		To solve this problem, we use <math>\vec{F}_{net} = m\vec{a}</math>, which is a simpler form of Newton's Second Law in scenarios with constant mass and non-relativistic speeds. This allows us to calculate the acceleration first and then use kinematic equations to determine the displacement.

	~~<math>\vec{F}_{net} = m\vec{a}</math> can be rearranged into <math>\vec{a} = \frac{\vec{F}_{net}}{m}</math>~~		Step 1: Calculate Acceleration

	<math> ~~a =~~ \~~frac~~{12}{4} = 3 </math> ~~(direction doesn't matter for this problem.)~~		Rearranging <math>\vec{F}_{net} = m\vec{a}</math>, we find:

	<math> x_f = \frac{1}{2}at^2 + v_it + x_i = \frac{1}{2}(3)(5^2) = \frac{75}{2}</math> ~~= 37.5~~ m.		<math>\vec{a} = \frac{\vec{F}_{net}}{m}</math>

			Substitute the given values:

			<math>a = \frac{18}{6} = 3 , \text{m/s}^2</math>

			Step 2: Apply Kinematic Equation

			The displacement can be calculated using the kinematic equation:

			<math>x_f = \frac{1}{2}at^2 + v_it + x_i</math>

			Since the particle starts from rest (<math>v_i = 0</math>) and at the origin (<math>x_i = 0</math>), the equation simplifies to:

			<math>x_f = \frac{1}{2}a t^2</math>

			Substitute the values for <math>a</math> and <math>t</math>:

			<math>x_f = \frac{1}{2}(3)(4^2)</math>

			<math>x_f = \frac{1}{2}(3)(16) = 24 , \text{m}</math>

			Final Answer

			The particle has moved a total distance of <math>24 , \text{m}</math> after 4 seconds.

	===2. (Middling)===		===2. (Middling)===

Asaxena344: /* History */

2024-12-04T01:14:54Z

History

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	==History==		==History==

	In his 1687 work ~~<i>~~Principia Mathematica~~</i>~~, Isaac Newton (~~1643-1727~~) ~~published~~ his three laws of motion. ~~Although his~~ first and third laws were inspired by ~~other~~ scientists' ~~work, his second law~~ was entirely original. ~~Below was~~ the ~~law as it appeared in his book:~~		In his groundbreaking 1687 work Philosophiæ Naturalis Principia Mathematica (commonly referred to as Principia Mathematica), Isaac Newton (1643–1727) introduced his three laws of motion. While the first and third laws were inspired by the work of earlier scientists, such as Galileo Galilei and René Descartes, Newton's Second Law was an entirely original contribution to the understanding of dynamics. This law is considered one of the cornerstones of classical mechanics.

	Original Latin:		==The Original Statement==
			The Second Law was originally written in Latin as:

	“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”		“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”
			This was closely translated in Andrew Motte's 1729 English edition of Principia Mathematica as:

	This was ~~translated closely~~ in ~~Motte~~'s ~~1729~~:		“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”
			In modern terms, this can be interpreted as:

			The rate of change of momentum of a body is proportional to the net force acting on it.
			The direction of the change in momentum is the same as the direction of the applied force.
			Defining Force
			This law was the first rigorous definition of force, linking it quantitatively to changes in momentum. Newton’s formulation unified and clarified many previous ideas about motion and force, enabling scientists to make precise predictions about physical systems.

			==Legacy==
			Newton's contributions to the study of forces have had a lasting impact on physics and engineering. The unit of force in the International System of Units (SI), the Newton (N), is named in his honor. One Newton is defined as the force required to accelerate a one-kilogram mass by one meter per second squared:

	~~“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”~~		<math>1 , \text{N} = 1 , \text{kg} \cdot \text{m/s}^2</math>

	In other ~~words, the rate~~ of ~~change of momentum of a body is proportional to the force impressed on the body, and happens in~~ the ~~direction~~ of ~~that force.~~		==Broader Implications==
			Newton's Second Law not only revolutionized mechanics but also laid the foundation for other fields of physics. It enabled the development of:

	~~This law was~~ the ~~first time force had ever been defined~~, and ~~it is~~ a ~~definition that is still used today~~. ~~As a result of~~ Newton's ~~contribution to~~ the science ~~of forces~~, the ~~unit of force most commonly still used today is named after him~~.		Orbital Mechanics: Used to predict the motion of planets, moons, and artificial satellites.
			Engineering Design: Governs the behavior of structures and machines under forces.
			Modern Dynamics: Serves as a precursor to advanced theories, such as Lagrangian and Hamiltonian mechanics.
			Newton's Principia Mathematica remains a seminal text in the history of science, and the Second Law continues to be a fundamental tool for understanding and modeling the physical world.

	== See also ==		== See also ==

Asaxena344: /* Connectedness */

2024-12-04T01:11:26Z

Connectedness

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	Newton's Second Law is a universal principle that connects forces, momentum, and motion. It applies to any scenario where forces are present, causing changes in a system's momentum. These applications span countless fields, from everyday situations to cutting-edge technologies. Below are examples that highlight the versatility and importance of this law.		Newton's Second Law is a universal principle that connects forces, momentum, and motion. It applies to any scenario where forces are present, causing changes in a system's momentum. These applications span countless fields, from everyday situations to cutting-edge technologies. Below are examples that highlight the versatility and importance of this law.

	Scenario: Bungee Jump		===Scenario: Bungee Jump===
	A bungee jump is a dynamic example of Newton's Second Law in action. Initially, the jumper accelerates downward due to gravity. As the cord stretches and becomes taut, it exerts an upward elastic restoring force, which increases with the cord's extension.		A bungee jump is a dynamic example of Newton's Second Law in action. Initially, the jumper accelerates downward due to gravity. As the cord stretches and becomes taut, it exerts an upward elastic restoring force, which increases with the cord's extension.

	Force Interaction:		==Force Interaction:==
	At maximum stretch, the cord's force becomes greater than the gravitational force, causing a net upward force.		At maximum stretch, the cord's force becomes greater than the gravitational force, causing a net upward force.
	This net upward force reduces the jumper's downward velocity and eventually reverses their motion.		This net upward force reduces the jumper's downward velocity and eventually reverses their motion.
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	This interplay of forces and momentum illustrates the principle of force-driven motion and how Newton's Second Law governs complex oscillatory systems like a bungee jump.		This interplay of forces and momentum illustrates the principle of force-driven motion and how Newton's Second Law governs complex oscillatory systems like a bungee jump.

	Application: Automobile Industry		===Application: Automobile Industry===
	Newton's Second Law is critical in automotive design, as it governs vehicle motion under forces such as engine thrust, braking, and turning.		Newton's Second Law is critical in automotive design, as it governs vehicle motion under forces such as engine thrust, braking, and turning.

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	The application of Newton’s Second Law in automotive design ensures vehicles operate safely and efficiently under various forces.		The application of Newton’s Second Law in automotive design ensures vehicles operate safely and efficiently under various forces.

	Additional Examples of Connectedness		===Additional Examples of Connectedness===
	Rocket Propulsion:		==Rocket Propulsion:==
	Rockets eject fuel at high velocity, creating a thrust force due to the conservation of momentum. This force propels the rocket forward and is directly tied to Newton's Second Law.		Rockets eject fuel at high velocity, creating a thrust force due to the conservation of momentum. This force propels the rocket forward and is directly tied to Newton's Second Law.
	Sports Dynamics:		==Sports Dynamics:==
	In sports, athletes use force to change the momentum of objects like balls or their own bodies. For example, a soccer kick applies a net force that accelerates the ball, and the force direction determines its trajectory.		In sports, athletes use force to change the momentum of objects like balls or their own bodies. For example, a soccer kick applies a net force that accelerates the ball, and the force direction determines its trajectory.
	Space Exploration:		==Space Exploration:==
	Spacecraft rely on Newton’s Second Law for trajectory corrections, where even small thrusters provide precise changes in momentum to adjust the spacecraft's path.		Spacecraft rely on Newton’s Second Law for trajectory corrections, where even small thrusters provide precise changes in momentum to adjust the spacecraft's path.
	Biomechanics:		==Biomechanics:==
	Human motion, such as running or jumping, is governed by forces exerted by muscles that cause changes in the body's momentum. Understanding these forces helps in injury prevention and optimizing athletic performance.		Human motion, such as running or jumping, is governed by forces exerted by muscles that cause changes in the body's momentum. Understanding these forces helps in injury prevention and optimizing athletic performance.

Asaxena344: /* Connectedness */

2024-12-04T01:10:12Z

Connectedness

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 ==Connectedness==
-Newton's second law is applicable to any situation where a force is applied and a system's momentum changes. There are nearly limitless examples of such situations, as well as nearly limitless applications. A few can be found below.
+Newton's Second Law is a universal principle that connects forces, momentum, and motion. It applies to any scenario where forces are present, causing changes in a system's momentum. These applications span countless fields, from everyday situations to cutting-edge technologies. Below are examples that highlight the versatility and importance of this law.
-===Scenario: bungee jump===
+Scenario: Bungee Jump
-When a bungee jumper falls and their cord is pulled taut, the cord applies a force in the direction opposite to the jumper's direction of travel. When this force is greater in magnitude than the gravity force, the net force is upward, which cuses causes the jumper's momentum, which is initially downward, to change over time in the upward direction. Eventually, the jumper is pulled back to a high altitude, the cord becomes slack, and gravity is once again the dominant force. The net force is downward once more, and the jumper's momentum changes over time in the downward direction.
+Force Interaction:
-===Application: automobile industry===
+Application: Automobile Industry
-Cars frequently undergo all sorts of acceleration; they speed up, turn, and slow down. Forces are responsible for all of those accelerations, and these forces ultimately come from the tires' contact with the road. This force is, in fact, a form of [[Static Friction]]. The magnitude of this static friction force can only be up to a certain quantity known as the maximum static friction force. This maximum is determined by the shape and materials of the tires, as well as the weight of the car. Automobile manufacturers must always be aware of this maximum friction force when designing their car, as it places a limit on the ability of the car to accelerate, turn, and brake without slipping.
+Acceleration and Braking:
 ==History==

Asaxena344: /* A Computational Model */

2024-12-04T01:08:58Z

A Computational Model

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	Model Description		===Model Description===
	This simulation models the motion of a falling object under the force of gravity, using the momentum principle. The object's position and velocity are updated iteratively, and the output is visualized in real-time using graphs and arrows representing forces.		This simulation models the motion of a falling object under the force of gravity, using the momentum principle. The object's position and velocity are updated iteratively, and the output is visualized in real-time using graphs and arrows representing forces.

	Key Features		===Key Features===
	Interactive Visualization: The simulation shows the position, velocity, and force vectors in real time.		Interactive Visualization: The simulation shows the position, velocity, and force vectors in real time.
	Graphing Tools: Position and velocity vs. time graphs are plotted, enabling analysis of motion dynamics.		Graphing Tools: Position and velocity vs. time graphs are plotted, enabling analysis of motion dynamics.

Asaxena344: /* A Computational Model */

2024-12-04T01:08:37Z

A Computational Model

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	===A Computational Model===		===A Computational Model===

	The relationship between net force and change in momentum can be computationally modeled using the momentum update formula: <math>p_2=p_1+F_{net}\Delta~~\text{~~t}</math>. This is a rewriting of ~~Fnet~~= dp/dt, where mass is assumed to be constant. ~~First, momentum~~ is ~~defined~~ as the object’s mass ~~times~~ its velocity in the initial conditions ~~of the simulations~~. ~~For [[~~Iterative Prediction~~]], a while loop for the duration of the movement first updates~~ <math>F_{net}</math> by summing all possible ~~net~~ forces acting on the object during the simulation. ~~Then~~ the momentum principle ~~is updated using~~ the ~~formula described previously~~. ~~Finally, using~~ the ~~new~~ momentum~~, the position is updated using the formula~~: <math>\~~overrightarrow~~{r}_2=\~~overrightarrow~~{~~r}_1+(~~\~~overrightarrow~~{p}/m~~)\Delta\text{t~~}</math>~~, where velocity can be substituted as~~ <math>(\~~overrightarrow~~{p}~~/m)~~</math>.		The relationship between net force and change in momentum can be computationally modeled using the momentum update formula:

			<math>p_2 = p_1 + F_{net} \cdot \Delta t</math>.

			This is a rewriting of <math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>, where mass is assumed to be constant.

			Steps in the Computational Model
			Define Momentum: Momentum is calculated as the object’s mass multiplied by its velocity in the initial conditions.
			Iterative Prediction: For each time step:
			Update <math>F_{net}</math> by summing all possible forces acting on the object during the simulation.
			Use the momentum principle to update the momentum: <math>p = p + F_{net} \cdot \Delta t</math>.
			Calculate the velocity from momentum: <math>\vec{v} = \frac{\vec{p}}{m}</math>.
			Update the position: <math>\vec{r}_2 = \vec{r}_1 + \vec{v} \cdot \Delta t</math>.

	[[File:UpdateLoopforMomentumPrinciple.png]]		[[File:UpdateLoopforMomentumPrinciple.png]]


	~~Such~~ a ~~simulation~~ is ~~shown below:~~		Model Description
			This simulation models the motion of a falling object under the force of gravity, using the momentum principle. The object's position and velocity are updated iteratively, and the output is visualized in real-time using graphs and arrows representing forces.

	The ~~vPython computational model calculates~~ the ~~changing~~ position, velocity, and ~~momentum~~ vs. time graphs of ~~a free-falling ball that is only subject to the force of gravity near~~ the ~~surface of the Earth.~~		Key Features
			Interactive Visualization: The simulation shows the position, velocity, and force vectors in real time.
			Graphing Tools: Position and velocity vs. time graphs are plotted, enabling analysis of motion dynamics.
			Customizability: Parameters like mass, initial position, initial velocity, and time step (<math>\Delta t</math>) can be adjusted for different scenarios.
			You can interact with the simulation on GlowScript here:


	https://www.glowscript.org/#/user/~~rgovind~~/folder/MyPrograms/program/~~Wiki~~		https://www.glowscript.org/#/user/Arayna%5fSaxena/folder/MyPrograms/program/Lab2onlygravity

	==Examples==		==Examples==