Point Particle Systems - Revision history

YorickAndeweg: /* A Mathematical Model */

2019-08-03T19:14:44Z

A Mathematical Model

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	===A Mathematical Model===		===A Mathematical Model===

	The mathematical concepts used to analyze point particle systems depend on the system. Often, the ~~work-energy theorem (see~~ [[Work/Energy]]) is used. This section explains how to use the work-energy theorem for a point particle system because this is the concept that varies the most significantly from its application to real systems.		The mathematical concepts used to analyze point particle systems depend on the system. Often, the [[Work/Energy\|work-energy theorem]] is used. This section explains how to use the work-energy theorem for a point particle system because this is the concept that varies the most significantly from its application to real systems.

	The work done on a point particle system is defined as		The work done on a point particle system is defined as

YorickAndeweg: /* 5. (Difficult) */

2019-07-12T20:18:31Z

5. (Difficult)

← Older revision		Revision as of 16:18, 12 July 2019
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	[[File:Pointparticlesystemtwomass.png]]		[[File:Pointparticlesystemtwomass.png]]

	[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/~~twomasssystem~~ Here is a simulation of the system to help you visualize it.]		[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/TwoMassSystem Here is a simulation of the system to help you visualize it.]

	Note: this problem requires the system to be modeled both as a real system and as a point particle system.		Note: this problem requires the system to be modeled both as a real system and as a point particle system.

YorickAndeweg: /* The Main Idea */

2019-07-09T16:50:43Z

The Main Idea

← Older revision		Revision as of 12:50, 9 July 2019
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	When work is done on a system, the energy imparted on it may take on multiple forms. These include [[Translational, Rotational and Vibrational Energy]], [[Potential Energy]], and [[Thermal Energy]]. Translational kinetic energy is kinetic energy due to the movement of the system's center of mass. All other types of energy the system has are considered "internal" types of energy because they do not affect the system's motion through its environment, but rather indicate its local properties. Sometimes, when work is done on a system, it imparts both translational kinetic energy and internal types of energy. For example, consider a ball rolling down a ramp. Gravity does work on the ball and increases its energy. It gains some translational kinetic energy (its center of mass gains a velocity down the ramp) and some rotational kinetic energy (its rolling motion causes it to rotate about its center of mass).		When work is done on a system, the energy imparted on it may take on multiple forms. These include [[Translational, Rotational and Vibrational Energy]], [[Potential Energy]], and [[Thermal Energy]]. Translational kinetic energy is kinetic energy due to the movement of the system's center of mass. All other types of energy the system has are considered "internal" types of energy because they do not affect the system's motion through its environment, but rather indicate its local properties. Sometimes, when work is done on a system, it imparts both translational kinetic energy and internal types of energy. For example, consider a ball rolling down a ramp. Gravity does work on the ball and increases its energy. It gains some translational kinetic energy (its center of mass gains a velocity down the ramp) and some rotational kinetic energy (its rolling motion causes it to rotate about its center of mass).

	The purpose of modeling a system as a point particle system is to easier to calculate how forces acting on it affect its translational motion through its environment.		Forces acting on a real system can affect both the system's translational acceleration and, depending what part of the system they act on, the internal state of the system. The purpose of modeling a system as a point particle system is to easier to calculate how forces acting on it affect its <i>translational</i> motion through its environment. By modeling the system as a point particle, all forces are assumed to act directly on the system’s center of mass, allowing us to examine only the translational effects of the forces. The net force acting on the point particle causes it to accelerate according to [[Newton's Second Law: the Momentum Principle]].

	Forces acting on a real system can affect both the system's translational acceleration and, depending what part of the system they act on, the internal state of the system. By modeling the system as a point particle, all forces are assumed to act directly on the system’s center of mass, allowing us to examine only the translational effects of the forces. The net force acting on the point particle causes it to accelerate according to [[Newton's Second Law: the Momentum Principle]].		From an energy perspective, a real system might have multiple types of work done on it at once; it might experience an increase in translational kinetic energy as well as an increase in some types of internal energy. By modeling the system as a point particle, we find only some of the work that is actually done on the system because we look at the displacement of the system's center of mass rather than the part of the system on which the force acts. Specifically, we find the work that affects only its <i>translational</i> kinetic energy because translational kinetic energy is the only energy a system can have when it is reduced to a point particle.

	From an energy perspective, a real system might have multiple types of work done on it at once; it might experience an increase in translational kinetic energy as well as an increase in some types of internal energy. By modeling the system as a point particle, we find only some of the work that is actually done on the system because we look at the displacement of the system's center of mass rather than the part of the system on which the force acts. ~~Apecifically~~, we find the work that affects only its translational kinetic energy because translational kinetic energy is the only energy a system can have when it is reduced to a point particle.

	Modeling a system as a point particle offers no insights about the internal state of the system because only forces and work affecting its translational motion can be found.		Modeling a system as a point particle offers no insights about the internal state of the system because only forces and work affecting its translational motion can be found.

YorickAndeweg: /* 4. (Middling) */

2019-07-07T18:32:08Z

4. (Middling)

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	[[File:Pointparticlesystemsramp.png]]		[[File:Pointparticlesystemsramp.png]]

	Solution: Let us analyze this system as a point particle system because we are only interested in its translational motion.		Solution:

			Let us analyze this system as a point particle system because we are only interested in its translational motion.

	The component of the disk's weight perpendicular to the ramp is balanced by the normal force, so only the force acting to accelerate the disk down the ramp is the component of the disk's weight parallel to the ramp. This can be found to be		The component of the disk's weight perpendicular to the ramp is balanced by the normal force, so only the force acting to accelerate the disk down the ramp is the component of the disk's weight parallel to the ramp. This can be found to be

YorickAndeweg: /* 4. (Middling) */

2019-07-04T19:08:13Z

4. (Middling)

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	===4. (Middling)===		===4. (Middling)===

	A uniform disk of mass <math>M</math> starts at rest and rolls without slipping down a ramp. The plane of the ramp makes an angle of <Math>\Theta</Math> with the horizontal. The force of [[Static Friction]] acting on the bottom of ~~the~~ disk an causing it to ~~roll~~ always has a magnitude of 1/3 the force acting to move the disk down the ramp. What is the translational speed of the disk after it has traveled a distance of <math>d</math> diagonally down the ramp?		A uniform disk of mass <math>M</math> starts at rest and rolls without slipping down a ramp. The plane of the ramp makes an angle of <Math>\Theta</Math> with the horizontal. The force of [[Static Friction]] acting on the bottom of such a disk, causing it to rotate, always has a magnitude of 1/3 the force acting to move the disk down the ramp. What is the translational speed of the disk after it has traveled a distance of <math>d</math> diagonally down the ramp?

	[[File:Pointparticlesystemsramp.png]]		[[File:Pointparticlesystemsramp.png]]

YorickAndeweg: /* The Main Idea */

2019-06-14T17:15:55Z

The Main Idea

← Older revision		Revision as of 13:15, 14 June 2019
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	Forces acting on a real system can affect both the system's translational acceleration and, depending what part of the system they act on, the internal state of the system. By modeling the system as a point particle, all forces are assumed to act directly on the system’s center of mass, allowing us to examine only the translational effects of the forces. The net force acting on the point particle causes it to accelerate according to [[Newton's Second Law: the Momentum Principle]].		Forces acting on a real system can affect both the system's translational acceleration and, depending what part of the system they act on, the internal state of the system. By modeling the system as a point particle, all forces are assumed to act directly on the system’s center of mass, allowing us to examine only the translational effects of the forces. The net force acting on the point particle causes it to accelerate according to [[Newton's Second Law: the Momentum Principle]].

	From an energy perspective, a real system might have multiple types of work done on it at once; it might experience an increase in translational kinetic energy as well as an increase in some types of internal energy. By modeling the system as a point particle, we find only some of the work that is actually done on the system~~; specifically~~, we find the work that affects only its translational kinetic energy because translational kinetic energy is the only energy a system can have when it is reduced to a point particle.		From an energy perspective, a real system might have multiple types of work done on it at once; it might experience an increase in translational kinetic energy as well as an increase in some types of internal energy. By modeling the system as a point particle, we find only some of the work that is actually done on the system because we look at the displacement of the system's center of mass rather than the part of the system on which the force acts. Apecifically, we find the work that affects only its translational kinetic energy because translational kinetic energy is the only energy a system can have when it is reduced to a point particle.

	Modeling a system as a point particle offers no insights about the internal state of the system because only forces and work affecting its translational motion can be found.		Modeling a system as a point particle offers no insights about the internal state of the system because only forces and work affecting its translational motion can be found.

YorickAndeweg: /* The Main Idea */

2019-06-13T16:33:16Z

The Main Idea

← Older revision		Revision as of 12:33, 13 June 2019
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	When work is done on a system, the energy imparted on it may take on multiple forms. These include [[Translational, Rotational and Vibrational Energy]], [[Potential Energy]], and [[Thermal Energy]]. Translational kinetic energy is kinetic energy due to the movement of the system's center of mass. All other types of energy the system has are considered "internal" types of energy because they do not affect the system's motion through its environment, but rather indicate its local properties. Sometimes, when work is done on a system, it imparts both translational kinetic energy and internal types of energy. For example, consider a ball rolling down a ramp. Gravity does work on the ball and increases its energy. It gains some translational kinetic energy (its center of mass gains a velocity down the ramp) and some rotational kinetic energy (its rolling motion causes it to rotate about its center of mass).		When work is done on a system, the energy imparted on it may take on multiple forms. These include [[Translational, Rotational and Vibrational Energy]], [[Potential Energy]], and [[Thermal Energy]]. Translational kinetic energy is kinetic energy due to the movement of the system's center of mass. All other types of energy the system has are considered "internal" types of energy because they do not affect the system's motion through its environment, but rather indicate its local properties. Sometimes, when work is done on a system, it imparts both translational kinetic energy and internal types of energy. For example, consider a ball rolling down a ramp. Gravity does work on the ball and increases its energy. It gains some translational kinetic energy (its center of mass gains a velocity down the ramp) and some rotational kinetic energy (its rolling motion causes it to rotate about its center of mass).

	The purpose of modeling a system as a point particle system is to easier to calculate its translational motion through its environment ~~by isolating this motion from its internal energy changes~~. Forces acting on a real system can affect both the system's translational acceleration and, depending what part of the system they act on, the internal state of the system. ~~Any forces acting on a~~ system ~~modeled~~ as a point particle are assumed to act directly on the system’s center of mass ~~and therefore affect~~ only ~~its~~ translational ~~acceleration, which is given by~~ [[Newton's Second Law: the Momentum Principle]]. ~~Forces that affect the real system's internal state are not found when treating it like a point particle because point particles cannot undergo internal state changes. Similarly~~, a real system might have multiple types of work done on it at once; it might experience an increase in translational kinetic energy as well as an increase in some types of internal energy. ~~When~~ modeling the system as a point particle, only some of the work ~~acting~~ on ~~it can be found~~; specifically, ~~any~~ work that affects only its translational kinetic energy ~~is found~~ because translational kinetic energy is the only energy a system can have when it is reduced to a point particle. Modeling a system as a point particle offers no insights about the internal state of the system because only forces and work affecting its translational motion can be found.		The purpose of modeling a system as a point particle system is to easier to calculate how forces acting on it affect its translational motion through its environment.

			Forces acting on a real system can affect both the system's translational acceleration and, depending what part of the system they act on, the internal state of the system. By modeling the system as a point particle, all forces are assumed to act directly on the system’s center of mass, allowing us to examine only the translational effects of the forces. The net force acting on the point particle causes it to accelerate according to [[Newton's Second Law: the Momentum Principle]].

			From an energy perspective, a real system might have multiple types of work done on it at once; it might experience an increase in translational kinetic energy as well as an increase in some types of internal energy. By modeling the system as a point particle, we find only some of the work that is actually done on the system; specifically, we find the work that affects only its translational kinetic energy because translational kinetic energy is the only energy a system can have when it is reduced to a point particle.

			Modeling a system as a point particle offers no insights about the internal state of the system because only forces and work affecting its translational motion can be found.

	Point particle systems are in contrast to [[Real Systems]] (also known as extended systems), which analyze each part of a system individually instead of reducing it to a single point. Real systems can model a system's internal behavior in addition to its motion through its environment, although they can be very complicated and difficult to analyze quantitatively. Often, to obtain a complete model of a system's behavior, both point-particle and extended models are used.		Point particle systems are in contrast to [[Real Systems]] (also known as extended systems), which analyze each part of a system individually instead of reducing it to a single point. Real systems can model a system's internal behavior in addition to its motion through its environment, although they can be very complicated and difficult to analyze quantitatively. Often, to obtain a complete model of a system's behavior, both point-particle and extended models are used.

	You may not be aware of it, but you have been modeling systems as point particles for most problems through week 9 because you have mostly been analyzing the translational motion of rigid bodies. Now that we are starting to look at internal types of energy, it becomes more important to separate point particle systems and real systems, and to understand what each model is able to accurately represent.		You may not be aware of it, but you have been modeling systems as point particles for most problems through week 9 because you have mostly been analyzing the translational motion of rigid bodies without regard for their internal energies. Now that we are starting to look at internal types of energy, it becomes more important to separate point particle systems and real systems, and to understand what each model is able to accurately represent.

	===A Mathematical Model===		===A Mathematical Model===

YorickAndeweg: /* 5. (Difficult) */

2019-06-12T22:27:09Z

5. (Difficult)

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	===5. (Difficult)===		===5. (Difficult)===

	A pair of 10kg masses are connected by a spring of unknown spring constant. They begin at rest at the spring's equilibrium length. A 30N rightward force is applied to the mass on the right for 4s. At the end of the 4 second period, the force ceases to act and the mass on the right has traveled 20m. The system continues to travel to the right, and the distance between the two masses oscillates sinusoidally. How much vibrational kinetic energy does the system have? (Vibrational kinetic energy is the energy of the mass' oscillatory movement, as opposed to their translational kinetic energy, which is the energy of the rightward movement of their center of mass.)		A pair of 10kg masses are connected by a massless spring of unknown spring constant. They begin at rest at the spring's equilibrium length. A 30N rightward force is applied to the mass on the right for 4s. At the end of the 4 second period, the force ceases to act and the mass on the right has traveled 20m. The system continues to travel to the right, and the distance between the two masses oscillates sinusoidally. How much vibrational kinetic energy does the system have? (Vibrational kinetic energy is the energy of the mass' oscillatory movement, as opposed to their translational kinetic energy, which is the energy of the rightward movement of their center of mass.)

	[[File:Pointparticlesystemtwomass.png]]		[[File:Pointparticlesystemtwomass.png]]

YorickAndeweg at 21:35, 12 June 2019

2019-06-12T21:35:55Z

← Older revision		Revision as of 17:35, 12 June 2019
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	==The Main Idea==		==The Main Idea==

	~~[[File:RealPointParticleDifference.PNG\|thumb\|left]]~~

	A point particle system is a physical system, usually composed of multiple parts, modeled as though it were a single particle at its [[Center of Mass]].		A point particle system is a physical system, usually composed of multiple parts, modeled as though it were a single particle at its [[Center of Mass]].

YorickAndeweg: /* The Main Idea */

2019-06-12T21:32:12Z

The Main Idea

← Older revision		Revision as of 17:32, 12 June 2019
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	==The Main Idea==		==The Main Idea==

			[[File:RealPointParticleDifference.PNG\|thumb\|left]]

	A point particle system is a physical system, usually composed of multiple parts, modeled as though it were a single particle at its [[Center of Mass]].		A point particle system is a physical system, usually composed of multiple parts, modeled as though it were a single particle at its [[Center of Mass]].