http://msuperl.org/wikis/pcubed/ 2026-05-03T18:57:47+00:00 http://msuperl.org/wikis/pcubed/ http://msuperl.org/wikis/pcubed/lib/tpl/bootstrap3/images/favicon.ico text/html 2021-02-04T23:23:18+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:acceleration&rev=1612480998&do=diff Acceleration & The Change in Momentum As you read, the motion of a system is governed by the Momentum Principle (aka “Newton's Second Law of Motion). In these notes, you will learn another way to write the momentum principle, and how that relates to the concept of acceleration. Newton's Second Law The Momentum Principle (or Newton's Second Law) is a quantitative description for how a system changes its momentum when the system experiences an external force. You might already know that another… text/html 2021-06-04T04:12:18+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:ang_momentum&rev=1622779938&do=diff Section 11.1, 11.2 and 11.3 in Matter and Interactions (4th edition) 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. $\vec{p}$$$\vec{L}_{trans} = \vec{r}_A \times \vec{p}$$$\vec{r}_A$$x-y$$z$$$\left|\vec{L}_{trans}\right| = \left|\vec{r}_A \times \vec{p}\right| = \left|\vec{r}_A\right|\left|\vec{p}\right|… text/html 2021-05-31T15:47:28+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:angular_motivation&rev=1622476048&do=diff Section 5.4 in Matter and Interactions (4th edition) Why Angular Momentum? It seems like conservation of momentum and conservation of energy can helps us describe any and all observations that you have. Indeed, both of these principles are quite powerful and can be used in many situations. However, there are some situations where a new idea must be brought to bear to be able to predict or explain the motion of the system. $$\Delta \vec{p}_{sys} = \vec{F}_{ext}\Delta t$$$$\vec{F}_{floor} = \df… text/html 2014-11-20T18:06:56+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:ap_derivation&rev=1416506816&do=diff Derivation of the Angular Momentum Principle Consider a single particle (mass, $m$) that is moving with a momentum $\vec{p}$. This particle experiences a net force $\vec{F}_{net}$, which will change the particle's momentum based on the momentum principle, $$\vec{F}_{net} = \dfrac{d\vec{p}}{dt}$$ Now, if we consider the cross product of the momentum principle with some defined lever arm (e.g., the origin of coordinates), $\vec{r}$$$\vec{r} \times \vec{F}_{net} = \vec{r} \times \dfrac{d\vec{p}}… text/html 2022-12-22T15:43:24+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:center_of_mass&rev=1671723804&do=diff Section 3.11 in Matter and Interactions (4th edition) The Motion of the Center of Mass In a system of multiple particles, each particle has its own motion. But often, its useful to think about the collective motion of these objects. By “collective motion$m_i$$\vec{r}_i$$$\vec{r}_{cm} = \dfrac{m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3}{m_1+m_2+m_3} = \dfrac{1}{M_{tot}} \left(m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3\right)$$$$\vec{r}_{cm} = \dfrac{\sum_i m_i \vec{r}_i}{\sum_i m_i} = \dfrac{1}{… text/html 2021-04-01T02:01:22+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:colliding_systems&rev=1617242482&do=diff Section 10.1, 10.2, 10.3 and 10.4 in Matter and Interactions (4th edition) Collisions Collisions occur everywhere around you. Right now, molecules of gas are constantly colliding with each other in the air that surrounds you as well as colliding with your clothes and skin. $$\Delta \vec{p}_{sys} = \vec{F}_{ext}\Delta t$$$$\Delta \vec{p}_{sys} = \underbrace{\vec{F}_{ext}\Delta t}_{\approx 0}$$$$\vec{p}_{sys,f} - \vec{p}_{sys,i} = 0$$$$\vec{p}_{sys,f} = \vec{p}_{sys,i}$$$$\Delta E_{sys} = W_{su… text/html 2021-04-01T01:59:10+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:collisions&rev=1617242350&do=diff Section 3.10 and 3.12 in Matter and Interactions (4th edition) Colliding Objects One situation where the concept of a multi-particle system is incredibly useful, is when two objects collide with each other. In this situation, you will find that the momentum of the system before the collision and the momentum of the system after the collision are very nearly the same $$\Delta \vec{p}_{sys} = \vec{p}_{sys,f} - \vec{p}_{sys,i} = \vec{F}_{surr} \Delta t$$$$\vec{p}_{sys,f} = \vec{p}_{sys,i} + \v… text/html 2023-01-13T19:59:08+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:constantf&rev=1673639948&do=diff Section 2.5 in Matter and Interactions (4th edition) Constant Force Motion You read previously how to (separately) predict the final momentum and final location of a system. In these notes, you will read how to put those two ideas together for a system where the net force is a constant vector (unchanging magnitude and direction) to be able to predict the motion of such a system.$$p_{fx} = p_{ix} + F_{net,x} \Delta t$$$$x_{f} = x_{i} + v_{avg,x} \Delta t$$$\Delta t$$$p_{fx} = p_{ix} + F_{net,x}… text/html 2014-11-18T15:33:02+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:cross_product&rev=1416324782&do=diff Taking the cross product in general To take the cross product of two vectors ($\vec{B} \times \vec{C}$) in Cartesian coordinates in general, we set up a special 3-by-3 matrix that has as its rows the Cartesian unit vectors ($\hat{x}$, $\hat{y}$, and $\hat{z}$), the components of the first vector ($B_x$, $B_y$$B_z$$C_x$$C_y$$C_z$$$\vec{B} \times \vec{C} = det\begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ B_x & B_y & B_z \\ C_x & C_y & C_z \\ \end{vmatrix}$$$$\vec{B} \times \vec{C} = det\begin{vm… text/html 2021-03-04T12:56:48+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:curving_motion&rev=1614862608&do=diff Section 5.5, 5.6 and 5.7 in Matter and Interactions (4th edition) Modeling Curved Motion The motion of objects is not limited to straight line motion. As you read earlier, forces can change the momentum of objects (including the direction of that momentum). These interactions can produce projectile motion, circular motion, oscillations, or more generalized trajectories. $\Delta \vec{p} = \vec{F}_{net} \Delta t$$\Delta t$$$\dfrac{d\vec{p}}{dt} = \vec{F}_{net}$$$$\vec{p} = |\vec{p}|\hat{p}$$$$\… text/html 2021-03-17T20:43:44+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:define_energy&rev=1616013824&do=diff Section 6.1 and 6.2 in Matter and Interactions (4th edition) What is Energy? “It is important to realize that in physics today, we have no knowledge what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way.$\sum m_i v_i^2$$\sum m_i \vec{v}_i$$\rightarrow$$\rightarrow$$\rightarrow$$\rightarrow$$\rightarrow$$E_{sys}$$W_{surr}$$Q$$$\Delta E_{sys} = W_{surr} + Q$$ text/html 2021-05-06T20:22:47+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:discovery_of_the_nucleus&rev=1620332567&do=diff Section 10.7 and 10.8 in Matter and Interactions (4th edition) Discovery of the Nucleus There is a long history about what makes up matter starting with Democritus in Ancient Greece and continuing through modern times. A critical experiment that spurred many future experiments was Rutherford's discovery of the nucleus in 1911. This experiment used a simple collision between Helium nuclei and gold foil and demonstrated the existence of the nucleus. text/html 2021-02-18T21:17:10+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:displacement_and_velocity&rev=1613683030&do=diff Section 1.4 and 1.6 in Matter and Interactions (4th edition) Constant Velocity Motion Our job in mechanics is to predict or explain motion. So, all the models and tools that we develop are aimed at achieving this goal. The simplest model of motion is for an object that moves in a straight line at constant speed. You can use this simple model to build your understanding about the basic ideas of motion, and the different ways in which you will represent that motion. $\Delta \vec{r}$$$\Delta \ve… text/html 2021-02-04T23:39:15+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:drag&rev=1612481955&do=diff Drag In most real world situations, there is some kind of resistive force. Some of these are due to contact between solid objects (e.g., friction) and you will learn about those later. For now, we will consider resistive forces due to some kind of fluid, which might be air, water, oil, or $$\vec{F}_{drag} = -b\vec{v}$$$b$$r$$$\vec{F}_{drag} = -6\pi\eta r \vec{v}$$$$\vec{F}_{drag} = -cv^2\hat{v}$$$c$$c$$$\vec{F}_{drag} = -\dfrac{1}{2} \rho C_d A v^2 \hat{v}$$$\rho$$A$$C_d$$$\vec{F}_{drag} = -b\v… text/html 2021-05-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:energy_cons&rev=1621958002&do=diff Section 6.1 6.6 and 6.7 in Matter and Interactions (4th edition) Conservation of Energy The observational fact that the energy of a system and its surroundings does not change has become a principle that underlies all of physics. When we look at a system, we can count up all the energy at different times and determine how energy is moving between the system and its surroundings. This help us to be able to predict and explain the motion of objects. $$\Delta E_{sys} + \Delta E_{surr} = 0$$$$\De… text/html 2021-06-02T23:15:35+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:energy_dissipation&rev=1622675735&do=diff Section 7.10 in Matter and Interactions (4th edition) Dissipation of Energy You have read that energy is always conserved. This is a true observable fact of the universe. Energy cannot be created or destroyed, it simply changes forms. However, sometimes those forms are less useful to us. For example, the increased thermal energy of a box due to the frictional interaction with the surface it is pushed across is unlikely to be useful. This type of transformation of energy is often referred to a… text/html 2021-06-02T23:25:31+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:energy_sep&rev=1622676331&do=diff Section 9.1 in Matter and Interactions (4th edition) Separating Energy in Multi-Particle Systems You have read about the motion of the center of mass of a system from the perspective of the momentum principle. In these notes, you will read about how this motion can be connected to the energy of a multi-particle system, and how different kinetic energy terms can separated out from the total kinetic energy to be discussed and thought about separately.$$K_{tot} = \sum_i K_i = \sum_i \dfrac{1}{2}… text/html 2021-04-01T12:37:12+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:escape_speed&rev=1617280632&do=diff Section 6.10 in Matter and Interactions (4th edition) Escape Speed Gravitational systems are particularly interesting because there are so many examples of such systems. The formation of our universe from immense galactic structures to solar systems with planets and moons, and even the orbits of asteroids and comets are all examples of gravitational systems. $v$$v<v_{esc}$$v>v_{esc}$$r \rightarrow \infty$$v=v_{esc}$$r \rightarrow \infty$$v \rightarrow 0$$v_{esc}$$M$$R$$m$$r=R$$v=v_{esc}$$r=\i… text/html 2015-01-31T21:01:42+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:fbds&rev=1422738102&do=diff Free-body Diagrams Free-body diagrams are one of the most useful tools in mechanics. They catalog all the forces acting on an object, and provide you with the necessary representation to make use of the Momentum Principle (i.e., to find the direction of the net force acting on the object). In these notes, you will read how to construct a free-body diagram and how it can help you reason about the direction of the net force on an object.$\vec{F}_{table}$$\vec{F}_{Earth}$$\Delta \vec{p}_{book} = 0… text/html 2023-11-30T20:35:36+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:force_and_pe&rev=1701376536&do=diff Force and Potential Energy The work done by a force is the integral of the force along the path that the force acts. This definition of the work gives rise to a relationship between the potential energy due to the interaction between the objects and the force responsible for that interaction. In these notes, you will read about the relationship between the force and the potential energy and how a graphical representation of the potential energy can also illustrate this force.$$\Delta U = -W_{in… text/html 2021-03-13T18:53:02+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:freebodydiagrams&rev=1615661582&do=diff Section 2.2, 5.1 and 5.2 in Matter and Interactions (4th edition) Free-body Diagrams Free-body diagrams are one of the most useful tools in mechanics. They catalog all the forces acting on an object, and provide you with the necessary representation to make use of the $\vec{F}_{table}$$\vec{F}_{Earth}$$\Delta \vec{p}_{book} = 0$$\vec{F}_{Earth}$$\vec{F}_{wire}$$\Delta \vec{p}_{ball} = 0$$\vec{F}_{Earth}$$\vec{F}_{wire}$$\Delta \vec{p}_{ball} \neq 0$ text/html 2021-02-18T21:23:48+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:friction&rev=1613683428&do=diff Section 4.7 and 4.8 in Matter and Interactions (4th edition) Contact Interactions: The Normal Force & Friction A microscopic perspective of materials helps to explain how contact interactions occur in nature. Contact interactions (forces) are not themselves $$\vec{F}_{net} = \vec{F}_{normal} + \vec{F}_{grav} = 0$$$$\vec{F}_{normal} = - \vec{F}_{grav} = \langle 0, mg \rangle$$$$F_{normal} = mg$$$$F_{friction,sliding} \approx \mu_k F_{normal}$$$\mu_k$$$F_{friction,static} \leq \mu_s F_{normal}$… text/html 2021-05-11T18:50:27+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:fundamental_principles&rev=1620759027&do=diff The Three Fundamental Principles of Mechanics In this course, you have worked with the 3 central principles to mechanics, the momentum principle, the energy principle, and the angular momentum principle. These 3 principles can predict or explain all motion in the universe as we know it. They are incredibly powerful principles that have been tested in many experiments. In this last set of notes, you will be reminded of them. $$\Delta \vec{p}_{sys} = \vec{F}_{ext} \Delta t$$$$\Delta \vec{p}_{sys}… text/html 2026-01-04T20:13:45+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:graphing_motion&rev=1767557625&do=diff Section 1.6 and 1.7 in Matter and Interactions (4th edition) Graphing Motion Predicting or explaining motion often requires you to use some sort of representation (or visual aid). A common (and incredibly useful) one is the graph. In these notes, you read about graphs of motion and how to translate between different graphs. $$\vec{r}_f=\vec{r}_i + \vec{v}_{avg} \Delta t$$$\vec{v}_{avg}$$\vec{v}$$$\vec{v}_{avg} = \dfrac{\Delta \vec{r}}{\Delta t}$$$$\vec{v} = \lim_{\Delta t \rightarrow 0} \dfrac… text/html 2021-02-05T00:02:51+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:grav_accel&rev=1612483371&do=diff Section 3.2 and 3.3 in Matter and Interactions (4th edition) Gravitational Acceleration Earlier, you read about Newton's Universal Law of Gravitation or, rather, the model we use to describe the gravitational interaction between two objects with mass. In these notes, you will read about how the gravitational acceleration of system depends only on the system that attracts it and the relative position of the systems.$$\vec{F}_{grav} = -G\dfrac{m_1 m_2}{|\vec{r}|^2}\hat{r}$$$m_1$$m_2$$\vec{r}$$$\… text/html 2021-03-12T02:45:15+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:grav_and_spring_pe&rev=1615517115&do=diff Section 6.8 and 7.2 in Matter and Interactions (4th edition) Types of Potential Energy Potential energy is the energy associated with interactions between pairs of objects. In these notes, you will read about two particular types of potential energy: the energy associated with the gravitational interaction and the energy associated with a spring-mass system.$m$$h$$m$$h$$y_i=0$$y_f=h$$$W_{grav} = \vec{F}_{grav}\cdot\Delta\vec{r} = -mg(y_f-y_i) = -mgh$$$$W_{grav} = -mg(y_f-y_i)$$$y_i=0$$y_f=h$$… text/html 2024-01-31T14:45:14+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:grav_pe_graphs&rev=1706712314&do=diff Section 6.10 in Matter and Interactions (4th edition) Graphing Energy for Gravitationally Interacting Systems Knowing the equation for the Newtonian gravitational potential energy might help you solve certain problems, but graphing the energy can help you reason about the motion of different systems. In these notes, you will read about the graph of the gravitational potential energy, how it can tell you about the motion of systems, and how the $$U(r) = -G\dfrac{Mm}{r}$$$$\underbrace{E_{tot}}_… text/html 2014-10-10T12:46:30+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:grav_pe&rev=1412945190&do=diff Gravitational Potential Energy text/html 2021-02-05T00:01:17+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:gravitation&rev=1612483277&do=diff Section 3.1, 3.2, 3.3 and 3.4 in Matter and Interactions (4th edition) Non-constant Force: Newtonian Gravitation Earlier, you read about the gravitational force near the surface of the Earth. This force was constant and was always directed “downward” (or rather toward the center of the Earth). In these notes, you will read about Newton's formulation of the gravitational force that (in his day) helped explain the motion of the solar system including why the Sun was at the center of the solar sy… text/html 2021-06-02T23:05:33+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:heat&rev=1622675133&do=diff Section 7.5 in Matter and Interactions (4th edition) Heat Exchange: Energy Transfer due to a Temperature Difference Earlier, you read about work, which is the mechanical energy transfer of energy into or out of a system. This energy transfer is due to physical displacements of the system over macroscopic distances by the external forces acting on the system. When you place a hot object near a cold one, you observe an energy transfer from the hot object to the cold object, but this is not due … text/html 2026-01-04T19:29:15+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:impulsegraphs&rev=1767554955&do=diff Section 2.1, 2.2, 2.3 and 2.4 in Matter and Interactions (4th edition) Impulse Graphs As you read earlier, the Momentum Principle is used to explain and predict the motion of systems. These predictions and explanations can be represented mathematically, but it also possible to make use graphs to do so. In these notes, you will read about force vs time graphs and how they can be used to determine the change in momentum of a system.$$\Delta \vec{p} = \vec{F}_{net} \Delta t$$$$\Delta \vec{p} = \… text/html 2021-06-02T22:49:39+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:internal_energy&rev=1622674179&do=diff Section 7.4 in Matter and Interactions (4th edition) Internal Energy Up to now, you have read about systems that have no internal structure: point particle systems. Even when considering a multi-particle system, you have worked with uniquely identifiable objects. Now, you will read about the energy associated with systems that have some structure. $m$$k_s$$v$$K=\dfrac{1}{2} (M) v^2$$M$$M_{sys} = E_{sys}/c^2$$$\mathrm{Internal\:energy} = E_{thermal} + E_{rotational} + E_{vibrational} + E_{chem… text/html 2021-02-15T02:46:48+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:iterativepredict&rev=1613357208&do=diff Section 2.3, 2.4 and 2.7 in Matter and Interactions (4th edition) Predicting Motion Iteratively You read earlier how to predict the motion of a system that experiences a constant force. However, very few real systems can be approximately modeled using constant force motion. All systems can be modeled iteratively, that is, applying the motion prediction tools ($\vec{p}_f = \vec{p}_i + \vec{F}_{net}\Delta t$$\vec{r}_f = \vec{r}_i + \vec{v}_{avg}\Delta t$ text/html 2021-05-11T19:05:30+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:l_conservation&rev=1620759930&do=diff Section 11.7 in Matter and Interactions (4th edition) Conservation of Angular Momentum You have read about the angular momentum principle and how systems with no net torque about a point experience no change in angular momentum. This is the final conservation principle of mechanics: the conservation of angular momentum. $$\Delta \vec{L}_{sys} = 0 \longrightarrow \vec{L}_{sys,i} = \vec{L}_{sys,f}$$$$L_{sys,i} = m_{ball}v_{ball}r_{\perp}$$$$L_{sys,f} = I_{sys}\omega$$$$L_{sys,i} = m_{ball}v_{ba… text/html 2021-06-03T15:49:29+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:l_principle&rev=1622735369&do=diff Section 11.4, 11.5 and 11.6 in Matter and Interactions (4th edition) Net Torque & The Angular Momentum Principle You have read that torques can cause rotations, and that angular momentum is a measure of rotation. These two concepts are linked together in the last of 3 fundamental principles of mechanics: the angular momentum principle. $$\dfrac{\Delta \vec{L}_{sys}}{\Delta t} = \vec{\tau}_{ext}$$$$\vec{L}_{sys,f} = \vec{L}_{sys,i} + \vec{\tau}_{ext}\Delta t$$$\Delta t$$$\dfrac{d\vec{L}_{sys}}… text/html 2016-09-29T16:55:49+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:learning_goals_thursday_new.pdf&rev=1475168149&do=diff text/html 2016-09-06T15:48:26+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:learning_goals_week_1.pdf&rev=1473176906&do=diff Learning Goals text/html 2024-01-11T20:56:30+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:localg&rev=1705006590&do=diff Section 2.5 in Matter and Interactions (4th edition) Constant Force: Gravitational Force near Earth You've read that the net force acting on an systems will change the system's momentum, but until now you haven't considered any particular forces. The first force that you will consider is the one that results from the interaction between objects with mass: $m$$\vec{g}$$y$$g$$$\vec{F}_{Earth} = m\vec{g}$$$y$$$\vec{g} = \langle 0, -g, 0\rangle \approx \langle 0, -9.81, 0\rangle \dfrac{m}{s^2}$$$… text/html 2021-03-13T19:40:18+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:model_of_a_wire&rev=1615664418&do=diff Section 4.4 and 4.5 in Matter and Interactions (4th edition) Modeling a Solid Wire with Springs To understand how solids exert different forces, you must learn how the microscopic, ball and spring model relates to more macroscopic measures such as elongation/compression and force. To do this, we will need to model the interatomic bond between two atoms in a cubic lattice a spring. $d$$6.02\times10^{23}$$195.08 g$$21.45 g/cm^3$$$\rho = 21.45 \dfrac{g}{cm^3} \left(\dfrac{1kg}{10^3g}\right)\left… text/html 2021-03-13T19:29:40+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:model_of_solids&rev=1615663780&do=diff Section 4.1, 4.2 and 4.3 in Matter and Interactions (4th edition) Matter & Models of Solids Until now, you have read (primarily) about forces that result from the gravitational interaction in both its exact and approximate forms. The exception thus far has been the $\sim1\times10^{-15}m$$\sim1\times10^{-10}m$$\Delta t$$$\Delta \vec{p} = \langle 0,0,0 \rangle = \vec{F}_{net} \Delta t\:\:\:\mathrm{implies}\:\:\:\vec{F}_{net} = \langle 0,0,0\rangle$$$$\vec{F}_{Earth} + \vec{F}_{wire} = \vec{F}_{… text/html 2022-12-01T19:39:59+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:modeling_with_vpython&rev=1669923599&do=diff Section 1.7 and 1.11 in Matter and Interactions (4th edition) Modeling Motion with VPython There is a restricted class of motion that can be modeled or explained with analytical tools (i.e., algebra and calculus). Most modern scientific research (and, indeed, engineering work) uses computational modeling as a significant part of the scientific endeavor. VPython is a Python-based programming language that allows you to create short programs that model the motion of physical systems. text/html 2014-11-03T00:08:31+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:moment_of_inertia_ex&rev=1414973311&do=diff Calculating the Moment of Inertia You have read that the moment of inertia for a system of discrete particles (that rotates about its center of mass) can be calculated by adding up the product of each mass and the square of the distance from the center of mass.$$I = \sum_i m_i r_{\perp i}^2 = m_1 r_{\perp 1}^2 + m_2 r_{\perp 2}^2 + m_3 r_{\perp 3}^2 + \dots$$$$I = \int r^2\,dm$$$r^2$$L$$M$$$I = \int_{\mathrm{whole\:rod}} r^2\,dm$$$$I = \int_{-L/2}^{+L/2} x^2\,dm$$$dm$$x$$dx$$$dm = \dfrac{M}{L}d… text/html 2021-09-06T13:34:28+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:momentum_principle&rev=1630935268&do=diff Section 2.1 and 2.2 in Matter and Interactions (4th edition) The Momentum Principle The motion of a system is governed by the Momentum Principle. This principle describes how a system changes its motion when it experiences a net force. We observe that when objects move in a straight line at constant speed they experience no net force.$$\Delta \vec{p} = \vec{p}_f - \vec{p}_i = \vec{F}_{net,avg} \Delta t$$$\Delta t$$\Delta t \rightarrow 0$$$\lim_{\Delta t \rightarrow 0} \dfrac{\Delta \vec{p}}{\D… text/html 2021-02-04T23:07:46+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:momentum&rev=1612480066&do=diff Section 1.8, 1.9, 1.10 and 1.12 in Matter and Interactions (4th edition) Momentum A central principle of mechanics involves the relationship between momentum and force. In these notes, you will learn about the concept of momentum, and when it is ok to use the approximate form of the momentum vector.$$\vec{p} = \gamma m \vec{v}$$$\gamma$$$\gamma = \dfrac{1}{\sqrt{1-\left(\dfrac{|\vec{v}|}{c}\right)^2}}$$$\gamma$$\mathbf{|\vec{v}|}$$\mathbf{|\vec{v}|/c}$$\mathbf{\gamma}$$$\vec{p} = m\vec{v}$$ text/html 2021-02-04T23:25:04+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:motionpredict&rev=1612481104&do=diff Section 2.3 in Matter and Interactions (4th edition) Applying the Momentum Principle Your primary job in mechanics is to be able to predict or explain the motion of systems. Previously, you read about the position update formula, which allows you to predict the future location of a system given information about its current location and its velocity (or $$\Delta \vec{p} = \vec{p}_f - \vec{p}_i = \vec{F}_{net} \Delta t$$$$\vec{p}_f = \vec{p}_i + \vec{F}_{net} \Delta t$$$$p_{fx} = p_{ix} + F_{ne… text/html 2021-04-01T01:50:49+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:mp_multi&rev=1617241849&do=diff Section 3.10 and 3.11 in Matter and Interactions (4th edition) The Momentum Principle in Multi-Particle Systems Until now, you've only considered systems of a single particle. This greatly simplifies the concept of a system, but doesn't really communicate why the concept of a system is so essential to physics. When you have several objects in a system, we refer to these as $m_i$$\vec v_i$$m_i$$\vec{v}_i$$\vec{p}=m\vec{v}$$$\vec{p}_i = m_i \vec{v}_i$$$$\vec{p}_{sys} = \sum_i \vec{p}_i = \vec{p… text/html 2021-04-01T12:54:13+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:newton_grav_pe&rev=1617281653&do=diff Section 6.8 in Matter and Interactions (4th edition) Gravitational Potential Energy You have read about the gravitational potential energy associated with a system consisting of an object and the Earth. This form of the gravitational potential energy turns about to be the approximate form of the potential energy for two objects that interact gravitationally. The gravitational potential energy is a powerful tool for modeling the motion of objects that interact through the gravitational force. … text/html 2021-05-06T20:42:03+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:point_particle&rev=1620333723&do=diff Section 6.2 in Matter and Interactions (4th edition) The Simplest System: A Single Particle The energy principle is widely applicable and helps to explain or to predict the motion of systems by considering how the system exchanges energy with its surroundings. For now, you will read about the simplest of systems, that of a single particle. $$E_{tot} = \gamma m c^2$$$m$$c$$\times10^8$$\gamma$$v=0$$$E_{tot} = \gamma m c^2 = \dfrac{1}{\sqrt{1-(v^2/c^2)}}mc^2 = \dfrac{1}{\sqrt{1-(0^2/c^2)}} mc^2 … text/html 2021-03-12T02:43:52+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:potential_energy&rev=1615517032&do=diff Section 6.7 in Matter and Interactions (4th edition) Potential Energy For multi-particles systems, you will have to keep track of the energy changes associated with the internal forces. That is, the work done by objects in the system on other objects in the system. As you will read, we can often associate an energy with pairs of interacting of objects, which we call $\vec{f}_{1,2}$$\vec{F}_{1,surr}$$$\Delta E_1 = \left(\vec{f}_{1,2} + \vec{f}_{1,3}\right)\cdot\Delta \vec{r}_1 + \vec{F}_{1,surr… text/html 2021-04-15T17:17:05+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:power&rev=1618507025&do=diff Section 7.6 in Matter and Interactions (4th edition) Power: The Rate of Energy Change Until now, you have only considered that energy changes from one state to another, but not the rate at which that change can occur. The rate at which energy changes is called $$P_{avg} = \dfrac{W}{\Delta t} = \dfrac{\vec{F}\cdot\Delta \vec{r}}{\Delta t}$$$$P = \dfrac{\vec{F}\cdot d\vec{r}}{dt} = \vec{F}\cdot\dfrac{d\vec{r}}{dt} = \vec{F}\cdot\vec{v}$$ text/html 2024-11-07T14:23:15+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:pp_vs_real&rev=1730989395&do=diff Section 9.3 in Matter and Interactions (4th edition) 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 be able to predict and explain how these systems behave. Examples of these kinds of deformable systems are everywhere includin… text/html 2014-11-04T14:00:31+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:proof_of_pp&rev=1415109631&do=diff Proof of the Point Particle Energy Principle You can start this derivation from the momentum principle for a multi-particle system, $$ \dfrac{d\vec{p}_{sys}}{dt} = \vec{F}_{ext}$$ As you might remember, the momentum of the system is directly related to the total mass of the system ($m$) and the velocity of the center of mass ($\vec{v}_{cm}$$$\vec{p}_{sys} = m\vec{v}_{cm}$$$i \rightarrow f$$$\int_i^f \dfrac{d\vec{p}_{sys}}{dt}\cdot d\vec{r}_{cm} = \int_i^f \vec{F}_{ext} \cdot d\vec{r}_{cm}$$$$… text/html 2023-01-15T16:24:41+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:relative_motion&rev=1673799881&do=diff Relative Motion All motion requires a frame of reference, an origin from which to make measurements of displacement, and thus velocity, and so on. In many cases, you can take the origin as fixed and make all measurements from that origin. But what happens if you are in a plane, in car, or on a train? Below is a video made to demonstrate what happens when you compare measurements in fixed and moving reference frames. $\vec{v}_{T/G}$$\vec{v}_{B/T}$$\vec{v}_{B/G}$$\vec{v}_{B/G} = \vec{v}_{B/T} +… text/html 2021-05-06T20:02:42+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:rest_mass&rev=1620331362&do=diff Change of Rest Mass Energy Until now, you have dealt with particles that do not change their identity. Changing the identity of a particle occurs when a particle decays to another particle (or, typically, set of particles), or when two or more particles fuse together. In these notes, you will read about a new unit of energy (the $$E_{rest} = mc^2 = (1.6749\times10^{-27}kg)(3\times10^{8} m/s)^2 = 1.51\times10^{-10}J$$$$1eV = 1.6\times10^{-19}J$$$$E_{rest} = 1.51\times10^{-10}J \dfrac{1eV}{1.6\ti… text/html 2022-01-21T10:37:53+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:review_video_2&rev=1642761473&do=diff Week 2 Learning Issue Review Video text/html 2022-01-28T06:50:17+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:review_video_3&rev=1643352617&do=diff Week 3 Learning Issue Review Video text/html 2022-01-14T10:27:02+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:review_video&rev=1642156022&do=diff Week 1 Learning Issue Review Video text/html 2023-11-07T16:42:39+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:rot_ke&rev=1699375359&do=diff Section 9.2 in Matter and Interactions (4th edition) Rotational Kinetic Energy Earlier, you read about how to separate the different forms of kinetic energy (translation, vibrational, and rotational). In this set of notes, you will read about the kinetic energy that is due to rotation about the center of mass. In these notes, you will also be introduced to the moment of inertia $^{\circ}$$\pi$$T$$\pi$$T$$$\omega = \dfrac{2\pi}{T}$$$\pi$$T$$r$$2r$$r$$r$$2\pi r$$2r$$2\pi 2r$$$v(r) = \dfrac{2 \p… text/html 2024-01-30T13:50:05+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:scalars_and_vectors&rev=1706622605&do=diff Section 1.4 in Matter and Interactions (4th edition) Scalars and Vectors We often use mathematics to describe physical situations. Two types of quantities that are particularly important for describing physical systems are scalars and vectors. In the notes below, you will read about those quantities (in general) and their properties.$\vec{r}$$$ \mathbf{r} = \vec{r} = \langle r_x, r_y, r_z \rangle $$$\vec{r}$$\vec{r}$$$r = | \vec{r} | = \sqrt{r_x^2+r_y^2+r_z^2}$$$$ \hat{r} = \dfrac{\vec{r}}{|\v… text/html 2014-11-01T14:40:25+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:sep_k_ex&rev=1414852825&do=diff Formal derivation of $K_{trans}$ and $K_{rel}$ In the notes, you will read about the formal derivation to separate the total kinetic energy into a translational component that tracks the energy associated with the motion of the center of mass, and the relative component that that tracks the energy of due to the motion relative to the center of mass.$m_1$$\vec{r}_1$$m_2$$\vec{r}_2$$$\vec{r}_{cm} = \dfrac{m_1\,\vec{r}_1 + m_2\,\vec{r}_2}{m_1+m_2}$$$\vec{r}_{1,cm}$$\vec{r}_{2,cm}$$$\vec{r}_1 = \ve… text/html 2021-05-25T16:17:37+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:spring_pe&rev=1621959457&do=diff Section 6.2 in Matter and Interactions (4th edition) Spring Potential Energy Earlier you read about springs and the motion of spring-like systems. This provided the foundation to model solid materials using the ball and spring model. You were able to predict the stretching of materials as well as model contact interactions. In these notes, you will revisit the energy associated with spring interactions. Spring Potential Energy $$\vec{F}_{spring} = -k\vec{s}$$$$F_s = -ks$$$$F_s = -ks = -\dfra… text/html 2026-01-04T20:19:30+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:springmotion&rev=1767557970&do=diff Section 4.10, 4.11 and 4.12 in Matter and Interactions (4th edition) Non-constant Force: Springs & Spring-like Interactions In all real-world interactions, the forces acting on a system change with time. This complication is often ignored in physics courses. We often model the motion of systems using constant forces (e.g., the gravitational force near the surface of the Earth) without additional complications (e.g., velocity dependent drag forces). $$x(t) = A \sin \left(\dfrac{2\pi}{T} t \rig… text/html 2021-11-15T17:25:15+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:static_eq&rev=1636997115&do=diff Section 5.4 and 11.5 in Matter and Interactions (4th edition) Static Equilibrium While you are beginning to learn about how objects rotate, it's worth taking an aside to discuss how objects remain still. You have already begun this work, when you read about Free Body Diagrams and worked with $$\vec{F}_{net} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \dots = 0$$$$\sum F_x = 0 \qquad \sum F_y = 0$$$$\vec{\tau}_{net,A} = \vec{\tau}_{1,A} + \vec{\tau}_{2,A} + \vec{\tau}_{3,A} + \dots = 0$$ text/html 2023-10-30T16:04:48+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:supplemental&rev=1698681888&do=diff Boar Tigers After the “reckoning day” a number of mutant species began to develop due to the contamination of food and water supplies with radiation. One of the most vicious of these species was the boar-tiger. Stories of the new species became prevalent on the west coast with the working theory that they spawned from boars and tigers which had escaped from the San Diego Zoo. Others argued that a Dr. Moreau, a once famous geneticist had become crazed after the devastation of the reckoning and… text/html 2021-04-15T17:19:55+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:system_choice&rev=1618507195&do=diff Section 7.7, 7.8 and 7.9 in Matter and Interactions (4th edition) Choosing a System Matters Earlier, you read about a puzzle involving a ball falling to the Earth. We resolved this puzzle by introducing the concept of potential energy. In these notes, you will look an example of a person lifting a box using different choices of system. The goal here is to demonstrate that the choice of a system matters.$m$$h$$v$$F$$E_{person}$$h$$v$$$\Delta E_{sys} = W_{surr}$$$$\Delta K_{box} + \Delta U_{gra… text/html 2021-05-08T18:56:22+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:torque&rev=1620500182&do=diff Section 5.4 and 11.4 in Matter and Interactions (4th edition) Torques Cause Changes in Rotation Until now, you have worked with forces and work to explain and predict the motion of objects. You've even used work and energy to begin to explain that objects can rotate, but you haven't yet unpacked how that occurs -- only that a system can share energy between translation and rotation. $\vec{r}_A$$$\vec{\tau}_{A} = \vec{r}_A \times \vec{F}$$$$F_{\perp} = F\sin\theta$$$$\left|\vec{\tau}_{A}\right… text/html 2021-11-15T17:25:36+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:torquediagram&rev=1636997136&do=diff Torque Diagrams To investigate situations in static equilibrium more thoroughly, you can make use of an extended free-body diagram that shows the “point of application” of the force. That is, you won't crush the system down to a point particle and treat all the forces as acting at the center of mass. Instead, you will consider where the forces are applied because doing so will be necessary for determining $m_p$$3d$$m_2$$m_1$$m_1$$m_1$$m_2$$m_p$$$\vec{F}_{net} = 0 \longrightarrow F_{net,x} = 0\;… text/html 2021-02-18T21:12:08+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:ucm&rev=1613682728&do=diff Uniform Circular Motion There are times when you will observe systems that move around some central axis in a very regular fashion. For example, the Moon revolves around the Earth in an orbit that is nearly circular. In doing so, it moves with nearly the same speed (not velocity!) at every location in its orbit. A system whose motion can be modeled as moving in a circular orbit at constant speed is said to execute $v$$R$$v$$R$$\Delta t$$\theta$$$ \vec{F}_{net} = \dfrac{\Delta \vec{p}}{\Delta t}… text/html 2021-09-26T05:30:57+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:vpython_furtherresources&rev=1632634257&do=diff Here is a link to the first of a series of videos that were created to walk people through the use of Vpython (which is pretty much the exact same thing as Glowscript). We did not make these videos and have not watched them and so cannot endorse their validity for this class but it is an additional coding resource that could be helpful. text/html 2022-11-14T15:52:36+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:vpython_resources&rev=1668441156&do=diff Computation in P-Cubed In class, you will make use of Glowscript to model the motion of different physical systems. Glowscript is an online coding platform that means that you can work on our computational activities without having to install anything on your computer. Glowscript has extensive help documentation which can be found text/html 2021-03-12T02:34:45+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:work_by_nc_forces&rev=1615516485&do=diff Section 6.3 in Matter and Interactions (4th edition) Work Done by Non-Constant Forces Until now, the definition of work that has been used is for forces that are constant vectors (constant in magnitude and direction). In these notes, you will read about how to determine the work done by forces that change (either in their magnitude or direction).$\vec{F}_{1}$$$W_{1} = \vec{F}_{1}\cdot\Delta \vec{r}_{1}$$$$W_{total} = W_{1} + W_{2} + W_{3} + \dots = \vec{F}_{1}\cdot\Delta \vec{r}_{1} + \vec{F}… text/html 2026-01-04T20:24:57+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:work&rev=1767558297&do=diff Section 6.3 and 6.4 in Matter and Interactions (4th edition) Work: Mechanical Energy Transfer As you read earlier, the change in the total energy of a system is equal to the work done on that system by its surroundings. In these notes, you will read about the formal definition of work, which is the transfer of mechanical energy, and a mathematical idea that underpins work - the dot product.$$W = \vec{F}\cdot\Delta\vec{r} = F_x dx + F_y dy + F_z dz$$$d$$F$$\theta$$\Delta \vec{r}$$\vec{F}$$\the… text/html 2021-02-18T20:41:48+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:youngs_modulus&rev=1613680908&do=diff Section 4.5 and 4.6 in Matter and Interactions (4th edition) Young's Modulus Earlier, you read how to add springs in parallel and in series. In these notes, you will read about how the microscopic measurements of bond length and interatomic spring stiffness relate to macroscopic measures like $L = 2m$$S = 1mm$$s = 1.166mm$$$\vec{F}_{net} = \vec{F}_{grav} + \vec{F}_{wire} = 0$$$$\vec{F}_{wire} = -\vec{F}_{grav} = \langle 0, k_{s,wire}s \rangle = -\langle 0,-mg \rangle$$$$k_{s,wire} = \dfrac{mg…