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183_notes:pp_vs_real [2015/10/14 01:36] – [An application] caballero183_notes:pp_vs_real [2021/06/02 22:36] (current) – [Energy in the Point Particle System] stumptyl
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 +Section 9.3 in Matter and Interactions (4th edition) 
 +
 ===== Point Particle and Real Systems ===== ===== Point Particle and Real Systems =====
  
-Until now, you read about the motion and energy of systems that are rigid, that is, they do not deform or change their shape. There are many applications where systems change their shape. Being able to analyze the motion and the transformation of energy in such systems is important to being able to predict and explain how these systems behave. Examples of these kinds of deformable systems are everywhere including children's toys (e.g., [[http://en.wikipedia.org/wiki/Yo-yo|yo-yo]]), engineering applications (e.g., engines such as [[http://en.wikipedia.org/wiki/Rhombic_drive|rhombic drives]]), and car accidents, which will be [[183_notes:colliding_systems|discussed later]]. In these notes, you will read about how to analyze simpler forms of these deformable systems using the //point particle// or center of mass system.+Until now, you read about the motion and energy of systems that are rigid, that is, they do not deform or change their shape. There are many applications where systems change their shape. Being able to analyze the motion and the transformation of energy in such systems is important to be able to predict and explain how these systems behave. Examples of these kinds of deformable systems are everywhere including children's toys (e.g., [[http://en.wikipedia.org/wiki/Yo-yo|yo-yo]]), engineering applications (e.g., engines such as [[http://en.wikipedia.org/wiki/Rhombic_drive|rhombic drives]]), and car accidents, which will be [[183_notes:colliding_systems|discussed later]]. **In these notes, you will read about how to analyze simpler forms of these deformable systems using the //point particle// or center of mass system.**
 ==== Lecture Video ==== ==== Lecture Video ====
 {{youtube>vL5yTCyRMGk?large}} {{youtube>vL5yTCyRMGk?large}}
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 ==== Energy in the Point Particle System ==== ==== Energy in the Point Particle System ====
  
-These deformable systems can be quite complicated because these systems have some structure and size, they are not [[183_notes:point_particle|point objects]]. External forces can be applied to different locations and the resulting motion can produce vibrations, rotations, and temperature changes, as well as translations. However, you can simplify the system significantly by modeling it as a point particle located at its [[183_notes:center_of_mass|center of mass]]. Doing so, allows you to determine the translation kinetic energy change of the system, but nothing else. A [[183_notes:point_particle|point particle has no internal structure]], so it cannot vibrate, rotate, or experience any temperature change. This is because the forces acting on the different parts of the system add to produce the net force, which (in this model) acts at the center of mass. The work done by this force over the displacement of the center of mass precisely determines the translation kinetic energy of the system.+These deformable systems can be quite complicated because these systems have some structure and size, they are not [[183_notes:point_particle|point objects]]. **External forces can be applied to different locations and the resulting motion can produce vibrations, rotations, and temperature changes, as well as translations.** However, you can simplify the system significantly by modeling it as a point particle located at its [[183_notes:center_of_mass|center of mass]]. Doing so, allows you to determine the translation kinetic energy change of the system, but nothing else. A [[183_notes:point_particle|point particle has no internal structure]], so it cannot vibrate, rotate, or experience any temperature change. This is because the forces acting on the different parts of the system add to produce the net force, which (in this model) acts at the center of mass. **The work done by this force over the displacement of the center of mass precisely determines the translation kinetic energy of the system.**
  
-[{{ 183_notes:pp_vs_real.007.png?450|Modeling a deformable, extended system as a point particle means tracking the motion of the center of mass.}}]+[{{ 183_notes:week11_realsystem.png?450|Modeling a deformable, extended system as a point particle means tracking the motion of the center of mass.}}]
  
  
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-The left hand side of this equation describes how the kinetic energy of the center of mass changes. A proof of that calculation is [[:183_notes:proof_of_pp|provided here]], but simply knowing that you can describe the kinetic energy of center of mass in this way is more important. The right hand side might be easier to see as the work done by the net external force in moving the center of mass from some initial location to some final location.+The left-hand side of this equation describes how the kinetic energy of the center of mass changes. A proof of that calculation is [[:183_notes:proof_of_pp|provided here]], but simply knowing that you can describe the kinetic energy of the center of mass in this way is more important. The right-hand side might be easier to see as the work done by the net external force in moving the center of mass from some initial location to some final location.
  
 The resulting energy principle for a point particle system is given by, The resulting energy principle for a point particle system is given by,
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 $$\Delta K_{trans} = \dfrac{1}{2}M_{tot}v_{cm,f}^2 - \dfrac{1}{2}M_{tot}v_{cm,i}^2 = \int_i^f \vec{F}_{ext} \cdot d\vec{r}_{cm} = W_{cm}$$ $$\Delta K_{trans} = \dfrac{1}{2}M_{tot}v_{cm,f}^2 - \dfrac{1}{2}M_{tot}v_{cm,i}^2 = \int_i^f \vec{F}_{ext} \cdot d\vec{r}_{cm} = W_{cm}$$
  
-To analyze the translation kinetic energy using the point particle system, imagine taking your deformable system and compressing it down to a point, which is located at the center of mass of the system. Allow the net external force to act at the center of mass (this is the //point of application of the net force// in the point particle system). Follow the displacement of the center of mass. The work done in the point particle system is the scalar product of the net force and displacement of the center of mass. By the point particle energy principle, this work is mathematically equivalent to the translation kinetic energy of the system. +To analyze the translation kinetic energy using the point particle system, imagine taking your deformable system and compressing it down to a point, which is located at the center of the mass of the system. Allow the net external force to act at the center of mass (this is the //point of application of the net force// in the point particle system). Follow the displacement of the center of mass. The work done in the point particle system is the scalar product of the net force and displacement of the center of mass. By the point particle energy principle, this work is mathematically equivalent to the translation kinetic energy of the system. 
 ==== Energy in the Real System ==== ==== Energy in the Real System ====
  
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 What is particularly important here is that you are able to determine the total work that is done by different forces acting on the system. Each force might act through a different displacement, so you will have to keep track of that when calculating the total force. The total energy of the system may take many forms: translational, rotational, and vibrational kinetic energy, as well as potential and thermal forms. What is particularly important here is that you are able to determine the total work that is done by different forces acting on the system. Each force might act through a different displacement, so you will have to keep track of that when calculating the total force. The total energy of the system may take many forms: translational, rotational, and vibrational kinetic energy, as well as potential and thermal forms.
  
-In the real system, you need information about each force, the point of application of each force, and the distance through which each forces acts. From the real system, you gain information about the total energy of the system.+In the real system, you need information about each force, the point of application of each force, and the distance through which each force acts. From the real system, you gain information about the total energy of the system.
  
 In the point particle system, you need information about the net force and the center of mass; the point of application of the net force is the center of mass and it acts through the displacement of the center of mass. From the point particle system, you gain information about the translational kinetic energy of the system only. In the point particle system, you need information about the net force and the center of mass; the point of application of the net force is the center of mass and it acts through the displacement of the center of mass. From the point particle system, you gain information about the translational kinetic energy of the system only.
  
-==== An application ====+===== An Application =====
  
-This might all seem abstract right now, but consider two pucks with same mass ($m$) that are pulled across an icy surface. Puck 1 has the rope attached to its center and Puck 2 has the rope wound about its edge (see below).+This might all seem abstract right now, but consider two pucks with the same mass ($m$) that are pulled across an icy surface. Puck 1 has the rope attached to its center and Puck 2 has the rope wound about its edge (see below).
  
 {{ 183_notes:hand_puck.png }} {{ 183_notes:hand_puck.png }}
  
-=== Both pucks travel the same distance ===+==== Both Pucks Travel The Same Distance ====
  
 Both pucks start from rest and are pulled with the same force ($F_T$) over the same amount of time. But, Puck 2 will rotate as the rope is unwound from the edge of the puck. Because both pucks experience the same force over the same time, both will have the same final momentum, Both pucks start from rest and are pulled with the same force ($F_T$) over the same amount of time. But, Puck 2 will rotate as the rope is unwound from the edge of the puck. Because both pucks experience the same force over the same time, both will have the same final momentum,
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 $$\Delta K_{trans} = \dfrac{1}{2}m\,v_{cm}^2 = W_{cm} = F_T\,d$$ $$\Delta K_{trans} = \dfrac{1}{2}m\,v_{cm}^2 = W_{cm} = F_T\,d$$
  
-=== Where does the rotational energy come from? ===+==== Where does the rotational energy come from? ====
  
-But why does Puck 2 rotate? You know there's [[183_notes:rot_ke|energy associated with rotation]]. If both puck experience the same work in the point particle system, why does Puck 2 rotate?+But why does Puck 2 rotate? You know there's [[183_notes:rot_ke|energy associated with rotation]]. If both pucks experience the same work in the point particle system, why does Puck 2 rotate?
  
-The point particle system can only describe the translation kinetic energy of a system. For Puck 1, the point particle and real system are exactly the same. Puck 1 only has translational kinetic energy.+The point particle system can only describe the translation kinetic energy of a system. For Puck 1, the point particle and real system are exactly the same. **Puck 1 only has translational kinetic energy.**
  
-But Puck 2 has rotational kinetic energy also. So, you must analyze the energy in the real system. For now, let's assume there's only changes in kinetic energy (translation and rotation) and no thermal energy changes (and no Q).+**But Puck 2 has rotational kinetic energy also.** So, you must analyze the energy in the real system. For now, let's assume there's only changes in kinetic energy (translation and rotation) and no thermal energy changes (and no Q).
  
 $$\Delta K_{tot} = \Delta K_{trans} + \Delta K_{rot} = W_{surr}$$ $$\Delta K_{tot} = \Delta K_{trans} + \Delta K_{rot} = W_{surr}$$
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