183_notes:ang_momentum

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183_notes:ang_momentum [2021/06/04 04:09] – [Translational Angular Momentum] stumptyl183_notes:ang_momentum [2021/06/04 04:12] (current) – [Rotational Angular Momentum] stumptyl
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 ===== Angular Momentum ===== ===== Angular Momentum =====
  
-Angular momentum is a way to measure the rotation of a system. As we did with kinetic energy (which is a way to measure the motion of a system), we can separate the angular momentum into translational and rotational bits. In these notes, you will read about each of these two bits, how they are defined, and how to deal with systems that have both bits. You will also be introduced to the "into/out of the page" language that we often use in physics to describe the direction of vectors that do not point in the plane.+Angular momentum is a way to measure the rotation of a system. As we did with kinetic energy (which is a way to measure the motion of a system), we can separate the angular momentum into translational and rotational bits. **In these notes, you will read about each of these two bits, how they are defined, and how to deal with systems that have both bits.** You will also be introduced to the "into/out of the page" language that we often use in physics to describe the direction of vectors that do not point in the plane.
  
 ==== Catching a Ball ==== ==== Catching a Ball ====
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 {{youtube>itsnuo1DnRw?large}} {{youtube>itsnuo1DnRw?large}}
-==== Translational Angular Momentum ====+===== Translational Angular Momentum =====
  
-As with [[183_notes:torque#torque|torque]], angular momentum requires that you consider a particular rotation axis. That is, around what point will you determine the angular momentum of the system? In general any point can be chosen, but the point that is chosen will determine the value of the angular momentum.+As with [[183_notes:torque#torque|torque]], angular momentum requires that you consider a particular rotation axis. That is, around what point will you determine the angular momentum of the system? __//In general any point can be chosen, but the point that is chosen will determine the value of the angular momentum.//__
  
 Given that angular momentum is a measure of rotation, you probably have a sense that an object that rotates about itself can have angular momentum, which is true, and will be discussed in a moment. But, an object that is moving, but not rotating about its center can still have angular momentum about a point. In fact, this is how we define angular momentum, in general. To determine the value of this angular momentum requires that we choose a "rotation" axis. Given that angular momentum is a measure of rotation, you probably have a sense that an object that rotates about itself can have angular momentum, which is true, and will be discussed in a moment. But, an object that is moving, but not rotating about its center can still have angular momentum about a point. In fact, this is how we define angular momentum, in general. To determine the value of this angular momentum requires that we choose a "rotation" axis.
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 $$\vec{L}_{trans} = \vec{r}_A \times \vec{p}$$ $$\vec{L}_{trans} = \vec{r}_A \times \vec{p}$$
  
-where the vector $\vec{r}_A$ is the vector that points from the rotation axis to the object in question. The units of angular momentum are kilograms-meters squared per second ($\mathrm{kg\,m^2/s}$). This is how angular momentum is defined, but it is convenient to think a bit differently about angular momentum associated with a object that rotates about its own center.+where the vector $\vec{r}_A$ is the vector that points from the rotation axis to the object in question. The units of angular momentum are **kilograms-meters squared per second ($\mathrm{kg\,m^2/s}$)**. This is how angular momentum is defined, but it is convenient to think a bit differently about angular momentum associated with an object that rotates about its own center.
  
-=== Magnitude of the translational angular momentum ===+==== Magnitude of the translational angular momentum ====
  
 {{ 183_notes:week12_angm2.png?400}} {{ 183_notes:week12_angm2.png?400}}
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 $$\left|\vec{L}_{trans}\right| = \left|\vec{r}_A\right|\left|\vec{p}\right|\sin \theta = \left|\vec{r}_{A,\perp}\right|\left|\vec{p}\right|$$ $$\left|\vec{L}_{trans}\right| = \left|\vec{r}_A\right|\left|\vec{p}\right|\sin \theta = \left|\vec{r}_{A,\perp}\right|\left|\vec{p}\right|$$
  
-=== Direction of the translation angular momentum ===+==== Direction of the translation angular momentum ====
  
 {{ 183_notes:mi3e_11-010.png?300}} {{ 183_notes:mi3e_11-010.png?300}}
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 ==== Rotational Angular Momentum ==== ==== Rotational Angular Momentum ====
  
-As you [[183_notes:rot_ke|read with rotational kinetic energy]], it is often useful to think about the motion of a system about its center. In those notes, you [[183_notes:rot_ke#atoms_in_rotating_objects_can_move_with_different_speeds|read about how the translational kinetic energy of atoms in a solid as the move around some central rotation axis]] can be described with rotational kinetic energy. Rotational angular momentum is a similar construct. It is not that a translating and rotating object has a separate kinds of angular momentum, but that you can mathematically separate the angular momentum due to translation and due to rotation to think about the two parts more easily.+As you [[183_notes:rot_ke|read with rotational kinetic energy]], it is often useful to think about the motion of a system about its center. In those notes, you [[183_notes:rot_ke#atoms_in_rotating_objects_can_move_with_different_speeds|read about how the translational kinetic energy of atoms in a solid as the move around some central rotation axis]] can be described with rotational kinetic energy. Rotational angular momentum is a similar construct. //It is not that a translating and rotating object has a separate kinds of angular momentum, but that you can mathematically separate the angular momentum due to translation and due to rotation to think about the two parts more easily.//
  
 Consider the spinning ball, person, stool system from the demonstration. In this case, the whole system rotates with the same angular velocity ($\omega$) after the ball was caught. An atom in the ball at a distance of $r_{\perp}$ from the rotation axis is therefore moving with a linear speed $v = r_{\perp}\omega$. Here, $r_{\perp}$ is the perpendicular distance from the rotation axis to the atom in the ball. Consider the spinning ball, person, stool system from the demonstration. In this case, the whole system rotates with the same angular velocity ($\omega$) after the ball was caught. An atom in the ball at a distance of $r_{\perp}$ from the rotation axis is therefore moving with a linear speed $v = r_{\perp}\omega$. Here, $r_{\perp}$ is the perpendicular distance from the rotation axis to the atom in the ball.
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