Six Ideas That Shaped Physics Notes

Unit C: Conservation Laws Constrain Interactions

Chapter 1: Introduction to Interactions

When an isolated set of objects interacts with one another, we find that (no matter how the objects interact) certain quantities (which we call the set's total energy, momentum, and angular momentum) are conserved (i.e., have values that do not change with time). The laws of physics are independent of time, place, and orientation.

Mechanics is the area of physics that deals with how the physical interactions between object affect their motion.

Newton's first law: Any object that does not interact with something else moves with a constant speed in a fixed direction. An interaction is a physical relationship between two objects that, in the absence of other interactions, changes the motion of each. Newton's model sees friction as being an interaction between a moving object and something else that actively slows down the object.

All matter consists of elementary particles known as quarks and leptons. Electrons are an example of the more general class leptons, while protons and neutrons are constructed of quarks.

There are four fundamental interactions:

  • string nuclear
  • electromagnetic
  • weak nuclear
  • gravitational

Between macroscopic objects, we see long-range interactions:

  • electrostatic
  • magnetic
  • gravitational

and contact interactions:

  • friction
  • compression
  • tension

Note that at the fundamental level, electrostatic, magnetic, and all contact interactions are different manifestations of the electromagnetic interaction.

SI Units

Type Unit Abbreviation
distance meter m
time second s
mass kilogram kg
temperature kelvin K
molecules mole mol
current ampere A
luminosity candela cd

Standard Prefixes

Power Prefix Symbol
10^12 tera T
10^9 giga G
10^6 mega M
10^3 kilo k
10^-2 centi c
10^-3 milli m
10^-6 micro ยต
10^-9 nano n
10^-12 pico p

Chapter 2: Vectors

A scalar quantity is any quantity (such as speed) that we can completely describe using a single numerical value. A vector is a quantity (such as velocity) that has direction in addition to a numerical magnitude.

Displacement is a shift in position from one point in space to another. See pages 27 and 28 for examples of vector addition (\(\vec{u_1}+\vec{u_2}\)), vector inverse (\(\vec{u_1}+\vec{u_2}=0\)), and vector difference (\(\vec{u_1}-\vec{u_2}\)). Note that \(\vec{u_1}-\vec{u_2}=\vec{u_1}+(-\vec{u_2})\).

A vector is a sum of three component vectors:

\begin{equation} \vec{u}=u_x\hat{x}+u_y\hat{y}+u_z\hat{z} \end{equation}

The \(\hat{x}\), \(\hat{y}\), and \(\hat{z}\) are directionals. We call \(u_x\), \(u_y\), \(u_z\) the x component, y component, and z component. A component vector is the combination of a component and a directional.

The magnitude of an arbitrary vector is as follows:

\begin{equation} mag(\vec{u})=\sqrt{u_x^2+u_y^2+u_z^2} \end{equation}

Pages 32 and 33 relate basic trigonometry to vectors. Note: the three angles of a triangle add to 180 degrees.

As expected, vector addition and difference can be calculated using components in vector notation.

A reference frame is an "imaginary cubical grid". We need to choose the origin, and the +x, +y, and +z directions. Note that we can attach a reference frame to any object that we please. We can also orient the frame in any way that we like.

We can calculate the displacement between two points with vector difference.

We can use our right hand to determine axis directions in a right-handed reference frame. See figure C2.11a on page 38 for an example. In standard orientation, +x=east, +y=north, +z=up.

Chapter 3: Interactions Transfer Momentum

A particle is a hypothetical object having zero volume, and thus its position is a mathematical point in space. The position \(\vec{r}\) of a point in space is its displacement from the reference frame origin to the point in question. The displacement between two arbitrary points is the difference of their positions: \(\Delta\vec{r}\equiv\vec{r_2}-\vec{r_1}\).

A particles velocity \(\vec{v}\) at a given instant of time t is a vector whose magnitude is the particle's speed \(v\) and whose direction is the particle's direction of motion at that instance of time. Let \(dt\) be the duration of a time interval containing the instant \(t\) at which we want to know \(\vec{v}\), and let \(d\vec{r}\) be the particle's tiny displacement during that tiny time interval. If \(dt\) is sufficiently short that neither the particle's speed nor its direction of motion changes appreciable during that time interval, then the particle's velocity is:

\begin{equation} \vec{v}\equiv\frac{d\vec{r}}{dt} \end{equation}

Speed is the magnitude of a particle's velocity:

\begin{equation} mag(\vec{v})\equiv\sqrt{v_x^2+v_y^2+v_z^2} \end{equation}

Remember from chapter 1 that an interaction between two objects is a physical relationship between them that allows each to affect the other's motion. Momentum, or the "quantity of motion", is defined as:

\begin{equation} \vec{p}\equiv m\vec{v} \end{equation}

The momentum-transfer principle: any interaction between two objects affects their motion by transferring momentum from one to the other.

Physicists call the amount of momentum that a particular interaction A between two particles transfers to either particle during a short interval of time the impulse \([d\vec{p}]_A\) that the interaction delievers to that particle during that interval. A financial analogy may help us understand some subtle distinctions in meaning between the terms a particle's momentum, impulse, and momentum in general. Think of the particle's momentum as being like a person's net financial worth. An interaction is like a financial transaction that increases on person's net worth at the expense of the other's. An impulse is like a check that a person writes or receives in the transaction. The term momentum is, like money, a general term for both what each particle (person) has and what is being transferred.

The change \(d\vec{p}\) in the momentum of a particle that participates in multiple interactions during a given interval is the vector sum of the impulses it receives during that interval

\begin{equation} d\vec{p}=[d\vec{p}]_A + [d\vec{p}]_B + ... \end{equation}

We define the force \(\vec{F}\) that an interaction exerts on a given object to be the rate at which momentum flows into the object because of that interaction:

\begin{equation} \vec{F}_A \equiv \frac{[d\vec{p}]_A}{dt} \end{equation}

The SI unit of force is the newton, where \(1 N = 1 kg*m/s^2\).

We can visualize an interaction between two particles as being like a hose that carries a flow of momentum from one to the other. Since any momentum that flows out of one must flow into the other, the magnitude of the rate of momentum flow must be the same for both particles. This statement and the definition of force directly imply that: A given interaction between particles A and B must exert a force on B that is equal in magnitude and opposite in direction the the force that it exerts on A: \(\vec{F}_{on B} = -\vec{F}_{on A}\). Physicists call this statement Newton's third law.

A particle's mass simply expresses the relationship between its momentum and its velocity, each of which we can define separately. Mass and weight are completely distinct concepts. An object's mass expresses its resistance to changes in its velocity. An object's weight \(\vec{F}_g\), however expresses the force that a gravitational interaction exerts on teh object as a certain point in space. Mass is a scalar measured in kilograms; weight is a vector measured in newtons.

\begin{equation} \vec{F}_g = m\vec{g} \end{equation}

Where \(\vec{g}\) is the gravitational field vector at a certain point in space. Near the earth's surface, \(\vec{g}\) points toward the earth's center, and the gravitational field strength \(g \equiv mag(\vec{g}) = 9.8 N/kg\).

There are two momentum flow and motion models discussed in the book: the multitap model and the three-reservoir model. See page 56. Also see page 57-58 for some common mistakes made when working with vectors.

Chapter 4: Particles and Systems

A system in physics is a set of interacting particles having a well-defined "boundary" that allows us to determine whether a given particle is inside or outside the system. An extended object is a material object with well-defined surface in space that defines its boundary and encloses a nonzero volume.

There are two categories of system interaction. If both particles involved in the interaction are inside the system, we call it an internal interaction; if one of the two particles involved in the interaction is inside the system and the other is outside, then we call it an external interaction.

We define a system's total momentum \(\vec{p}_{tot}\) to be the vector sum of the momenta of its constituent particles:

\begin{equation} \vec{p}_{tot} \equiv \vec{p}_1 + \vec{p}_2 + ... + \vec{p}_N \end{equation}

The momentum-transfer principal (from last chapter) means that a system's internal interactions only transfer momentum back and forth within the system and therefore cannot affect a system's total momentum. Only external interactions can transfer momentum into or out of a system and thus change its total momentum. If a system participates in no external interactions, we call it an isolated system. Since no momentum can flow either into or out of such a system, the following holds: The law of conservation of momentum: the total momentum \(\vec{p}_{tot}\) of an isolated system is conserved. That is, it does not change with time.

The center of mass of any system of particles is a mathematical point whose position we define as follows:

\begin{equation} \vec{r}_{CM} \equiv \frac{1}{M}(m_1\vec{r}_1 + m_2\vec{r}_2 + ... + ... m_N\vec{r}_N) \end{equation}

Where \(\vec{r}_1, \vec{r}_2, ..., \vec{r}_N\) are the positions of the system's N particles. \(m_1, m_2, ..., m_N\) are their masses; and \(M \equiv m_1 + ... + m_N\) is the system's mass.

In many systems, there's an outrageously huge number of elementary particles involved. A practical (but approximate) method is to break the object up into a moderate number of macroscopic chunks that are still small enough that we can consider the particles in each to have essentially the same position.

Note that we can calculate the position of the center of mass of any set of extended objects by treating each object as if it were a point particle with all its mass concentrated at its individual center of mass.

To find a systems total momentum, all that we need to do is to multiply its total mass M by the velocity of its center of mass \(\vec{v}_{CM}\), exactly as if the system were a single particle located at its center of mass!

\begin{equation} \vec{p}_{tot} = M\vec{v}_{CM} \end{equation}

In conjunction with the law of conservation of momentum, the equation above implies that an isolated system's \(\vec{v}_{CM}\) is a constant. This is simply Newton's first law, except we now see that it applies to arbitrary systems, not just to particles!

The particle model: A system's center of mass responds to its external interactions exactly as a point particle would respond to those interactions. This means that every idea of equation that we have formulated up to now for interacting particles also applies to any system of particles, as long as we substitute the system quantities \(\vec{r}_{CM}\), \(\vec{v}_{CM}\), and \(\vec{p}_{tot}\) for the particle quantities \(\vec{r}\), \(\vec{v}\), and \(\vec{p}\), respectively, and put the qualifier external in front of any references to interactions.

Inertial reference frames are reference frames in which we find Newton's first law to be obeyed; noninertial reference frames are frames in which we find it to be violated. The models of motion we have been developing work in inertial reference frames but fail in noninertial reference frames.

Chapter 5: Applying Momentum Conservation

A system does not have to be strictly isolated for its total momentum to be conserved. For example, the gravitation and compression interations a laptop experiences when it's sitting on a desk are negligible because they cancel each other out. In general, a system's momentum is conserved in any process where its external interactions do not change its total momentum significantly during that process. There are three distinct types of situations in which this is true:

  • Floats in space: only interacts gravitationally with distant objects
  • Functionally isolated: external interactions cancel
  • Momentarily isolated: strong external interactions that do not cancel, but we only look at the system's momentum just before and just after a very strong and brief interaction. If the collision process is sufficiently brief, external interactions simply do not have time to transfer significant momentum to the system.

Problem-Solving Framework

  • Translation -> draw a picture
  • Conceptual model -> applicable theories/principles
  • Algebraic solution -> solve equations symbolically
  • Evaluation -> check results make sense

Draw an interaction diagram as follows: draw a large circle to represent the system, and draw one rectangular box inside that circle for each object inside the system, labeling the box with the object's name. If the object inside the system interact with object outside the system, draw a box outside the circle for each relevant object outside the system, and label these boxes as well. Then draw lines connecting the boxes to represent the internal and external interactions involved.

Chapter 6: Introduction to Energy


Total energy E is defined as:

\begin{equation} E = K_1 + K_2 + V(r_{12}) \end{equation}

Conservation of energy implies that:

\begin{equation} \Delta E = \Delta K_1 + \Delta K_2 + \Delta V(r_{12}) = 0 \end{equation}

For a system involving two objects.

In the context of the newtonian model, kinetic energy is defined as:

\begin{equation} K \equiv \frac{1}{2}mv^2 \end{equation}

Experimentally, we find that for the gravitational interaction,

\begin{equation} V(z) = mgz \end{equation}

Note that this equation only applies when the object is close to the earth's surface.

Ambiguity in the value of \(V\) allows us to choose \(V\) to be zero at a convenient reference separation. If we set up the reference separation to not be \(z = 0\), but instead \(z_0\), then \(V(z) = mgz - mgz_0\).

Chapter 7: Some Potential Energy Functions


The potential energy function for the interaction between two charged particles is:

\begin{equation} V(r) = +k\frac{q_1q_2}{r} \end{equation}

Reference separation: \(V(R) \equiv 0\) when \(r = \infty\). This equation describes the electrostatic potential energy function \(V(r)\) for an electromagnetic interaction between two charged particles. In the equation, \(q_1\) and $q_2$d are the two particles' charges, \(r\) is their separation, and \(k\) is the Coulomb constant = \(8.99\:x\:10^9 J \cdot m/C^2\). The SI unit of charge is the coulomb.

The potential energy function for a gravitational interaction between two particles is:

\begin{equation} V(r) = -G\frac{m_1m_2}{r} \end{equation}

Reference separation: \(V(R) \equiv 0\) when \(r = \infty\). This equation describes the potential energy function \(V(r)\) for a gravitational interaction between two particles separated by a distance \(r\). The symbols \(m_1\) and \(m_2\) are the particles' masses, and \(G\) is the universal gravitational constant (empirically, G = \(6.67\:x\:10^11 J \cdot m/kg^2\).

We can treat two object connected by a spring as if they were participating in a long-range interaction whose potential energy is very nearly:

\begin{equation} V(r) = \frac{1}{2}k_s(r - r_0)^2 \end{equation}

Reference separation: \(V(r) \equiv 0\) when \(r = r_0\). In the equation, \(r_0\) is the separation of the objects' centers when the spring is relaxed, and \(k_x\) is a spring constant characterizing the spring's stiffness.

Chapter 8: Force and Energy


We define the dot product \(\vec{u} \cdot \vec{w}\) of two arbitrary vectors \(\vec{u}\) and \(\vec{w}\) to be the following scalar quantity:

\begin{equation} \vec{u} \cdot \vec{w} \equiv uw\cos{\theta} = u_xw_x + u_yw_y + u_zw_z \end{equation}

We can therefore express the change in a particle's kinetic energy as being \(dK = \vec{v} \cdot d\vec{p}\). This then implies that:

\begin{equation} [dK] \equiv \vec{F} \cdot d\vec{r} \end{equation} \begin{equation} dK = [dK]_A + [dK]_B + ... \end{equation}

This equation describes how to compute the tiny change \(dK\) in a particle's kinetic energy from the tiny amount of k-work \([dK]\) that each interaction contributes during a tiny time interval \(dt\).

Chapter 9: Rotational Energy


The radians can be defined as:

\begin{equation} \left|\theta\right| \equiv \frac{s}{r} \end{equation}

Where \(s\) is the arclength subtended by the angle along the circumference of a circle of radius \(r\).


The radian is a "unitless unit". We usually display it when we want to make it clear that the quantity involves measuring an angle or if we want to convert to or from other angular units, such as the degree or revolution.

We define an object's angular velocity \(\vec{\omega}\) to be:

\begin{equation} \vec{\omega} \equiv \left|\frac{d\theta}{dt}\right|\hat{\omega} \end{equation}

This equation only makes sense when applied to a rigid object rotating around a well-defined axis or to a particle moving in a plane around some origin.

An object's moment of inertia \(I\) and its rotational kinetic energy \(K^{rot}\) for rotations around a given axis are given by:

\begin{equation} I \equiv m_1r^2_1 + m_2r^2_2 + ... + m_Nr^2_N \equiv \sum_{i=1}^{N} m_ir^2_i \end{equation}

Where \(m_i\) is the mass of the $i$th particle in the object, and \(r_i\) is the particle's distance from the rotation axis.

\begin{equation} K^{rot} = \frac{1}{2}I\omega^2 \end{equation}

(compare with \(K = \frac{1}{2}mv^2\))

How can we calculate the value of \(I\) for a given object? The actual number of elementary particles in any extended object is so large that a literal application of the equation above is impractical. But we can group the particles in an object into a reasonable number of pieces (each small enough that all its particles are approximately the same distance from the axis of rotation) and then sum over the pieces instead of the particles.

The total kinetic energy of an object that is simultaneously moving and rotating turns out to be simply:

\begin{equation} K = K^{cm} + K^{rot} = \frac{1}{2}Mv^2_{CM} + \frac{1}{2}I\omega^2 \end{equation}

The angular speed \(\omega\) of an object of radius R that rolls without slipping is linked to its center-of-mass speed \(v_CM\) by:

\begin{equation} \omega = \frac{v_{CM}}{R} \end{equation}

fRolling without splipping is causes by the friction part of a contact interaction between the object and a surface. Therefore, we can handle such an interaction in a conservation of energy problem by including a rotational kinetic energy term in our conservation of energy equation.

Chapter 10: Thermal Energy

Thermal physics focuses on physical processes involving heat and temperature change. In 1843, James Prescott Joule showed a connection between heat and energy. Object are made up of molecules that are in ceaseless random motion. An object's temperature \(T\) is a measure of the intensity of this motion. The SI unit for temperature is kelvins.

For gases and simple solids under everyday conditions, temperature is directly proportional to the average kinetic energy per molecule of its molecules' random motions in the substance (treating the molecule as a point particle):

\begin{equation} K_{avg} \:(per\:molecule)\: = \frac{3}{2}k_BT \end{equation}

Where \(k_B = 1.38 x 10^{-23} J/K\) and is called Boltzmann's constant.

An object whose molecules are all at rest has a temperature of zero. Since molecules cannot have negative kinetic energy, this is the lowest possible temperature an object can have: we therefore call this temperature absolute zero.

When two surfaces are in contact and one moves past the other their surface molecules often become entangled. The entangled molecules are stretched away from their normal positions. Their interaction transforms a tiny bit of kinetic energy to potential energy in the bonds between molecules. At a certain point, the molecules suddenly become unstuck and snap back toward their initial positions, where they oscillate wildly. The kinetic energy apparently lost in friction and collisions interacts actually is transformed to thermal energy.

As is the case for potential energy, it is the change in thermal energy that is usually important. Internal energy \(U\) is the total energy that an object contains within the boundary defining its surface. When a hot object is placed in contact with a cold object, energy spontaneously flows across the boundary between them from the hot object to the cold object until both objects have the same temperature. In physics, heat \(Q\) is any energy that crosses the boundary between the two objects because of the temperature difference between them. We define work \(W\) to be any other kind of energy flowing across an object's boundary. For example, if I stir a cup of water vigorously and it gets warm as a result, I have not "heated" the water, I have done work on it; the mechanical energy that flows across the water's boundary flows not because of a temperature difference but because of my stirring effort.

Conservation of energy implies that in a given process:

\begin{equation} \Delta U = Q + W \end{equation}

Formally, an object's thermal energy \(U^{th}\) is the part of its internal energy that increases with temperature. Note heat is an energy flow across an object boundary, whereas thermal energy is energy enclosed by that boundary. An object can absorb heat, but it never contains heat!

An object's thermal energy is generally a complicated function of temperature, but for relatively small temperature changes not too close to the temperatures where an object changes phase from solid to liquid or liquid to gas, we have:

\begin{equation} dU^{th} = McdT \end{equation}

Where \(M\) is the object's mass, and c is its specific heat. Note that for this equation to work, the change in temperature \(dT\) must be small enough that c is mostly constant over the temperature range.

Keeping track of thermal energies enables us to solve conservation of energy problems involving:

  1. Impact or friction interactions that do work on objects in the system or
  2. Thermal contact interactions that transfer heat between objects.

To represent each such interaction, we add a \(\Delta U^{th}\) term to our conservation of energy equation for each object involved.

Chapter 11: Energy in Bonds

Note: only sections one and two are included below.

A potential energy diagram for a two-objevt system is simply a graph of its total interaction potential energy \(V(r)\) versus the objects' separation \(r\). We represent the system's fixed total energy \(E\) as a horizontal line on a potential energy diagram. The light object's kinetic energy \(K\) at any given \(x\) is \(K = E - V(x)\). Since \(K\) cannot be negative, regions of the x axis where \(V(x) > E\) are forbidden regions, while regions where \(E > V(x)\) are allowed regions. The points where $E = V(x) are turning points: the velocity reverses direction at such points to keep from entering a forbidden region.

We can read from these diagrams the x component \(F_x\) that the interaction exerts on the light object at any position. This value is given \(F_x = \frac{-dV}{dx}\). Points along the x axis where the slope of \(V(x)\) is zero (implying \(F_x = 0\)) are equilibrium positions where the light object could in principle remain at rest. If such a point corresponds to the bottom of a valley in the potential energy function, it is a stable equilibrium position; otherwise it is an unstable equilibrium position.

TODO: bonds. See page 197

Chapter 12: Power, Collisions, and Impacts

Power expresses the magnitude of the rate at which energy flows into or out of a given form in the course of some interaction or process.

\begin{equation} P \equiv \left|\frac{dK}{dt}\right| \quad or \quad \left|\frac{dV}{dt}\right| \quad or \quad \left|\frac{dU}{dt}\right| \end{equation}

The SI unit of power is the watt, where \(1W \equiv 1J/s\).

If a specific interaction exerts a force on a moving object, the rate at which energy is transformed by that particular interaction is:

\begin{equation} P = \left|\frac{[dK]}{dt}\right| = |\vec{F}\cdot\vec{v}| \end{equation}

Unit N: The Laws of Physics are Universal

Chapter 1: Newton's Laws

Newton's theories of motion were the first that were able to explain both terrestial and celestial phenomena using the same theoretical framework. This unification is called the newtonian synthesis.

Conservation of momentum implies that the total momentum of an isolated system is conserved. Newton's first law directly follows from this: "In the absence of external interactions, an object's (or system's) center of mass moves at a constant velocity."

A tiny impulse \([d\vec{p}]\) flowing out of one object A is the same as \(-[d\vec{p}]\) flowing into A. Force \(\vec{F}\) exerted by an interaction on an object is defined to be the rate at which the interaction transfers momentum into the object. The rate at which a given interaction transfers momentum into object B is the negative of the rate at which it transfers momentum into object A. Newton's third law says that "When object A and B interact, the force the interaction exerts on A is equal in magnitude and opposite in direction to the force that it exerts on B." Or mathematically:

\begin{equation} \vec{F}_B \equiv -\vec{F}_A \end{equation}

Consider a 10,000-kg truck traveling at 65 mi/h. Imagine that it hits a parked 500-kg Volkswagen Beetle. Which vehicle exerts the stronger force on the other during the collision? According to the third law, the magnitudes of the forces that the contact interation exerts on either vehicle must be the same: any momentum that the interaction transfers out of the truck must go into the Beetle at the same rate. This seems counter intuitive. Note that the fact that the forces are the same on each doesn't mean that each has to respond in the same way. Because the Beetle is 20 times less massive than the truck, a given amount of momentum flowing into the Beetle will lead to a change in its velocity whose magnitude is 20 times larger than the change in the truck's velocity from the same momentum flow.

For point particles, Newton's second law looks like:

\begin{equation} \vec{F}_{net} = \frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} \equiv m\vec{a} \end{equation}

Where a particle's acceleration \(\vec{a} \equiv d\vec{v}/dt\). Now let's consider a system of particles that interacts with its surroundings. Internal interactions only transfer momentum from particle to particle inside the system, so they will not change the system's total momentum.

\begin{equation} \vec{F}_{net,ext} = \frac{d\vec{p}_{tot}}{dt} = M\frac{d\vec{v}_{CM}}{dt} \equiv M\vec{a}_{CM} \end{equation}

Note that the object must have a fixed mass. This equation is Newton's second law for systems of particles. To understand how an object's center of mass moves, we only need to be able to quantify the net exernal force acting on that object.

Forces do not cause motion, they cause acceleration. A force does not overcome another force to cause an object to move in a certain way: rather all forces acting on the object act in concert to direct the object's acceleration.

Classification of Forces

See pages 11-12 of the the Unit N book for more specific information on these forces.

  1. Long-Range Interactions
    Symbol Name
    \(\vec{F}_g\) Gravitational force
    \(\vec{F}_e\) Electrostatic force
    \(\vec{F}_m\) Magnetic force
  2. Contact Interactions
    Symbol Name
    \(\vec{F}_{sp}\) Spring force
    \(\vec{F}_T\) Tension force
    \(\vec{F}_N\) Normal force
    \(\vec{F}_{SF}\) Static friction
    \(\vec{F}_{KF}\) Kinetic friction
    \(\vec{F}_D\) Drag force
    \(\vec{F}_L\) Lift force
    \(\vec{F}_{Th}\) Thrust force
    \(\vec{F}_p\) Pressure force
    \(\vec{F}_B\) Buoyant force

TODO Free-Body Diagrams

See page 13 of the Unit N book!

Chapter 2: Vector Calculus

An object's average velocity:

\begin{equation} \vec{v}_{\Delta t} \equiv \frac{\Delta\vec{r}}{\Delta t} \end{equation}

With a nonzero \(\Delta t\), this equation most closely approximates the object's instantaneous velocity at an instant \(t\) halfway throught the interval \(\Delta t\).

An object's average acceleration:

\begin{equation} \vec{a}_{\Delta t} \equiv \frac{\Delta\vec{v}}{\Delta t} \end{equation}

Again, this equation provides a good approximation for instantaneous acceleration midway through the interval.

TODO Motion Diagrams

See page 26 of the Unit N book! Note that motion diagrams display both velocity and acceleration.

Uniform Circular Motion

Uniform circular motion is when an object is moving at a constant speed around a circle of radius \(R\). The acceleration of an object moving at a constant speed around a circle points directly toward the center of the circle at every instant. Note that this only applies if the object is moving at a constant speed. Page 32 gives a superb diagram of the relation between velocity and acceleration. A mathematical representation of this acceleration can be seen below:

\begin{equation} mag(\vec{a}) = \frac{mag(\vec{v})^2}{R} = \frac{v^2}{R} \end{equation}

Chapter 3: Forces from Motion

The kinematic chain connects position, velocity, and acceleration by derivative relationships:

\begin{equation} \vec{r}(t) -time\:derivative\rightarrow \vec{v}(t) -time\:derivative\rightarrow \vec{a}(t) \end{equation}

Net-Force Diagrams

A net-force diagram is similar to a free-body diagram but more vividly displays the quantitative relationship between these forces. Net force is the vector sum of all external forces acting on an object. Here are the steps to creating a net-force diagram:

  1. Draw a free-body diagram first
  2. Copy the force arrows and arrange them in sequence by drawing each new force arrow with its tail end starting at the tip of the previously drawn arrow
  3. Draw the arrow that represents the vector sum of these force arrows.

Third-Law and Second-Law Pairs

A pair of forces fitting the following conditions are third-law partners:

  1. Each force in the pair must act on a different object
  2. Both forces must reflect the same interaction between the objects.

Two forces that act on the same object cannot be third-law partners.

Forces are called second-law partners if:

  1. Both forces act on the same object
  2. The forces oppose each other along a certain axis
  3. No other forces have any component along that axis
  4. The component of the object's acceleration along that axis is zero.

Graphs of One-Dimensional Motion

When constructing a graph of \(a_x(t)\) from one of \(v_x(t)\), the value of the object's x-acceleration at every instant of time will correspond to the slope of \(v_x(t)\) when the latter is plotted versus \(t\) on the graph.

When constructing a graph of \(v_x(t)\) from one of \(x(t)\), the value of \(v_x\) at every instant of time will correspond to the slope of a graph of \(x(t)\) versus \(t\).

Always use the symbols \(v_x\), and \(a_x\) to refer to signed numbers and always use \(r\), \(v\), and \(a\) to refer to the (nonnegative) magnitudes of the corresponding vectors.

Chapter 4: Motion from Forces

The kinematic chain of relationships link an object's position to its velocity and its acceleration. We can reverse the chain by taking antiderivatives. This is called the reverse kinematic chain:

\begin{equation} \vec{a}(t) -time\:antiderivative\rightarrow \vec{v}(t) -time\:antiderivative\rightarrow \vec{r}(t) \end{equation}

There are lots of different approaches to finding the antiderivatives of acceleration.

  1. Graphical Antiderivatives

    Use the reverse of the slope method seen in N3. Note that there is a certain ambiguity in this process from constants.

  2. Integrals for One-Dimensional Motion

    \(v_x(t) = \int a_x(t) dt + C_1 \qquad and \qquad x(t) = \int v_x(t) dt + C_2\)

    Note that we can determine the \(C_1\) if we know the object's initial velocity \(v_x(t)\). We need only substitute t = 0 into the expression for $v_x(t) and choose the value of \(C_2\) that gives the correct initial velocity \(v_x(0)\). We can similarly determine \(C_2\) by choosing it so that we get the correct value of \(x(0)\).

  3. Constant x-acceleration equations

    \(v_x(t) = a_xt + v_0x \qquad and \qquad x(t) = \frac{1}{2}a_xt^2 + v_{0x}t + x_0\)

When functions are really vector functions, we can find antiderivativesd by considering each component separately. This means that we can treat vector functions (symbolically) as if they were scalar functions:

\begin{equation} \vec{v}(t) - \vec{v}(0) = \int_{0}^{t}\vec{a}(t)dt \qquad and \qquad \vec{r}(t) - \vec{r}(0) = \int_{0}^{t}\vec{v}(t)dt \end{equation}

Constructing Trajectory Diagrams

To draw a trajectory diagram, first center the object's initial velocity arrow on its initial position. Then use the known acceleration vector to construct future position points from the initial arrow's two endpoints.

Chapter 6: Linearly Constrained Motion

A free-particle diagram is like a free-body diagram, but all of the force arrow's tails start at the center of mass. Also, the diagram should contain a set of reference frame axes.

If an object moves in a straight line at a constant velocity, its acceleration is zero, implying that the net force on the object must be zero.

TODO Static and Kinetic Friction Forces

\(F_{KF}\) is usually less than the maximum value of \(F_{SF}\).

\begin{equation} mag(\vec{F}_{SF}) \leq mag(\vec{F}_{SF,max}) \approx \mu_s mag(\vec{F}_N) \end{equation} \begin{equation} mag(\vec{F}_{KF}) \approx \mu_k mag(\vec{F}_N) \end{equation}

TODO Drag Forces

\begin{equation} mag(\vec{F}_D) \approx \frac{1}{2}CpAv^2 \end{equation}

Where \(C\) is the drag coefficient, \(p\) the density, and \(A\) the cross section. Note that the density of air is \(1.2\:kg/m^3\).

Chapter 7: Coupled Objects

A pair of coupled objects consists of two objects that are constrained by some kind of connection so that their motions are linked in some well-defined manner.

For cases where forces are acting on more than one object, we have a notation convention for force symbols. We add to the basic symbol for the type of force involved a pair of superscripts, the first indicating the object on which this force is exerted and the second (in parentheses) indicating the other object involved in the interaction that gives rise to this force. For example: \(\vec{F}^{A(B)}\). Note that this really helps recognize third-law partners.

TODO Strings, Real, and Ideal

See page 118. Talk about working with real strings that have a mass and ideal strings that don't.

TODO Pulleys

see page 122. Talk about ideal pulleys that have a very small mass and are nearly frictionless.

TODO Chapter 10: Projectile Motion

In physics we define an object's weight \(\vec{F}_g\) to be the force that gravity exerts on that object. The weight force itself is given by \(\vec{F}_g = m\vec{g}\). If the only force on an object is its weight \(\vec{F}_g\), we say that it is freely falling.

As long as (1) an object remains "sufficiently close" to the surface of the earth and (2) its trajectory is "sufficiently short" that the curvature of the earth is not significant, the magnitude and direction of \(\vec{g}\) (and thus of the object's weight \(\vec{F}_g\)) will be approximately constant. If in addition (3) the object is "not significantly affected" by other forces except possibly air drag, then we call it a projectile and its motion projectile motion. When (4) we also neglect drag, we say that the object's motion is simple projectile motion.

Simple Projective Motion

Newton's second law implies that object's in simple projectile motion will have constant acceleration \(\vec{a} = \vec{g}\).

\begin{equation} \left[ \begin{array}{ c c } v_x(t) \\ v_y(t) \\ v_z(t) \end{array} \right] = \left[ \begin{array}{ c c } v_{0x} \\ v_{0y} \\ -gt+v_{0z} \end{array} \right] \end{equation} \begin{equation} \left[ \begin{array}{ c c } x(t) \\ y(t) \\ z(t) \end{array} \right] = \left[ \begin{array}{ c c } v_{0x}t + x_0 \\ v_{0y}t + y_0 \\ -\frac{1}{2}gt^2 + v_{0z}t + z_0 \end{array} \right] \end{equation}

The object must be in simple projectile motion for these equations to apply.

How do we determine when the object has reached its maximum altitude and when it reaches the ground? Well the peak is when \(v_z(t)=0\) and the object hits the ground at \(z(t) = 0\).

Drag and Terminal Speed

Since the force from drag \(\vec{F}_D\) is proportional, as a falling object's downward speed increases, the grag force also increases, decreasing the net force on the object and thus its downward acceleration. When the net force is essentially zero, the object no longer accelerates, but continues to fall with constant downward velocity. We call the magnitude of this final constant downward velocity the object's terminal speed \(v_Tf\).

\begin{equation} v_T = \sqrt{\frac{2mg}{CpA}} \end{equation}

Note that this equation is created by the equation for the force from drag solved for v. The \(F_D\) has been substituted for \(mg\) since at terminal velocity drag is equal to weight (\(F_g=F_D\)).

An object falling from rest at time \(t = 0\) will behave as a simple projectile for times \(t << t_T \equiv v_T / g\), but will essentially reach its terminal spped after a time \(t \approx 2t_T\).

Unit R: The Laws of Physics are Frame-Independent

Chapter 1: The Principle of Relativity

Think of being on a smooth sailing plane. If you drop something it acts as if the plane is sitting in the loading dock. The laws of physics are the same inside a laboratory moving at a constant velocity as they are in a laboratory at rest. This is an unpolished statement of what we call the principle of relativity. This idea is the foundation for Einstein's special theory of relativity.

An event is any physical occurrence that we can consider to happen at a definite place in space and at a definite instant in time. We can quantify when and where the event occurs by four numbers: three numbers that specify the location of the event in some three-dimensional spatial coordinate system and one number that specifies what time the event occurred. We call these four numbers the event's spacetime coordinates. The motion of any particle can be mathematically described to arbitrary accuracy by specifying the spacetime coordinates of a sufficiently large number of events suitably distributed along its path.

We need to add the time of an event to our definition of reference frames. Imagine that we attach a clock to every lattice intersection. We can then define the time coordinate of an event to be the time displayed on the lattice clock nearest the event (relative to some specified time \(t = 0\)) and the event's three spatial coordinates to be the lattice coordinates of that nearest clock, specified in the usual way by stating the distances along the lattice directions that one has to travel (from some specified spatial origin) to the clock. We can determine these four numbers to whatever precision we want by sufficiently decreasing the lattice spacing and the time between clock ticks.

Why is it important to have a clock at every lattice intersection? The point is to make sure that there is a clock essentially at the location of any event to be measured. If we attempt to read the time of an event by using a clock located a substantial distance away, we need to make assumptions about how long it took the information that the event has occurred to reach that distant clock. For example, if we read the time when the sound from an event reaches the distant clock, we should correct that value by subtracting the time that it takes sound to travel from the event to the clock; byt to do this, we have to know the speed of sound in our lattice. We can avoid this problem if we require that an event's time coordinate be measured by a clock essentially present at the event.

We must synchronize the clocks. This will be discussed later.

An operational definition of a physical quantity defines that quantity by describing how the quantity can be measured. A reference frame is defined to be a rigid cubical lattice of appropriately synchronized clocks or its functional equivalent. The spacetime coordinates of an event in a given refgerence frame are defined to be an ordered set of four numbers, the first specifying the time of the event as registed by the nearest clock in the lattice, followed by three that specify the spatial coordinates of that clock in the lattice. An observer is defined to be a (possibly hypothetical) person who interprets measurements made in a reference frame.

A reference frame may be moving or at rest, accelerating, or even rotating about some axis. The beauty of the definition of spacetime coordinates given above is that measurements of the coordinates of events can be carried out in a reference frame no matter how it is moving. However, not all reference frames are equally useful for doing physics. We can divide reference frames into two general classes: inertial frames and noninertial frames. An inertial frame is one in which an isolated object is always and everywhere observed to move at a constant velocity. In a noninertial frame such an object is observed to move with a nonconstant velocity in at least some situations.

Any inertial reference frame will be observed to move at a constant velocity relative to any other inertial reference frame. Conversely, a rigid, nonrotating reference frame that moves at a constant with respect to nay other inertial reference frame must itself by inertial.

Consider two inertial reference frames, which we will call the Home Frame and the Other Frame. An isolated object at rest in the Other Frame must move at a constant velocity with respect to the Home Frame, so the whole Other Frame must move at the same constant velocity relative to the Home Frame.

We cannot physically distinguish a reference frame "moving at a constant veloicty" from one "at rest". The principle of relativity specifically states that a reference frame moving at a constant velocity is physically equivalent to a frame at rest. Our final polished statement of The Principle of Relativity is a follows: The laws of physics are the same in all inertial reference frames.

In standard orientation we have the Home Frame's \(x\), \(y\), and \(z\) axes point in the same directions as the corresponding axes in the Other Frame. We conventionally distinguish the Home Frame and the Other Frame axes by referring to the Home Frame axes as \(x\), \(y\), and \(z\) and the Other Frame axes as \(x'\), \(y'\), and \(z'\). It also is conventional to define the origin event (the event that defines \(t = 0\) in both frames) to be the instant at which the spatial origin of one frame passes the origin of the other. Finally, we conventionally choose the common \(x\) axis so that the Other Frame moves in the \(+x\) direction with respect to the Home Frame.

Suppose the Other Frame moves at a constant velocity \(\vec{\beta}\). The relationship between the object's position vectors in the two frames a given time is as follows:

\begin{equation} t' = t \qquad x' = x - \beta t \qquad y' = y \qquad z' = z \end{equation}

Physicists call these four equations the galilean transformation equations. They allow us to find the position of the object at a given time \(t'\) in the Other Frame if we know its position at time \(t = t'\) in the Home Frame (assuming, or course, that time is universal and absolute).

If we take the time derivative of both sides of each of the last three equations, we get:

\begin{equation} v_x' = v_x - \beta \qquad v_y' = v_y \qquad v_z' = v_z \end{equation}

These equations tell us how to find the velocity of an object in the Other Frame, given its velocity in the Home Frame: we call these equations the galilean velocity transformation equations.

If we take the time derivative again, we get:

\begin{equation} a_x' = a_x \qquad a_y' = a_y \qquad a_z' = a_z \end{equation}

This tells us that observers in both inertial frames agree about an object's acceleration at a given time, even thought they may well disagree about the object position and velocity components at that time!

When we say that the "laws of physics are the same" in different inertial frames we realize that observers in different inertial frames may disagree abut the values of various quantities (particularly positions and velocities) by each oberver will agree that the same basic equations will be found to describe the laws of physics in all inertial reference frames.

Chapter 2: Synchronizing Clocks

In 1878, James Clerk Maxwell published a set of equations (now called Maxwell's equations) that summarized the laws of electromagnetism in compact and elegant form. These equations predicted that light waves must travel at a specific speed \(c = 3.00 \times 10^8 m/s\). Einstein argued, to make Maxwell's equations consistent with the principle of relativity, that the speed of light is a frame-independent quantity.

If time is not "universal and absolute", what is it? Einstein said that we must define what we mean by "time" operationally within each inertial frame by specifying a concrete and specific procedure for synchronizing that frame's clocks that is consistent with both the principle of relativity and the laws of electromagnetism. Since the speed of light has to be \(c\) in every inertial frame anyway, let us in face synchronize the clocks in our inertial reference frame by assuming that light always has the same speed of \(c\)! Imagine that we have a master clock at the spatial origin of our reference frame. At exactly \(t = 0\), we send a light flash from that clock that ripples out to the other clocks in the frame.

SR Units

In this unit distance will be measured in a new unit, called a light-second or just second. A light-second or second is defined to be the distance that light travels in 1s of time. There is exactly 299,792,458 m in 1 light-second by definition.

Agreeing to measure distance in seconds allows us to state the definition of clock synchronization in an inertial frame in a particularly nice and concise manner: Two clocks in an inertial reference frame are defined to be synchronized if the time interval (in seconds) registered by the clocks for a light flash to travel between them is equal to their separation (in light-seconds).

In this chapter we will use a slightly modified version of SI units, called SR units, where distance is measured in seconds (i.e., light-seconds) instead of in meters.

In SR units, the light-second is considered to be equivalent to the second of time, and both units are simply referred to as seconeds. This means that these units can be canceled if one appears in the numerator of an expression and the other in the denominator. For example, in SI units, velocity has units of meters per second; but in SR units it has units of seconds per second = unitless(!). Thus an object that travels 0.5 light seconds in 1.0 s has a speed in SR units of 0.5s/1.0s = 0.5 (no units!).

Spacetime Diagrams

We can conveniently depict the coordinates of events by using a special kind of graph called a spacetime diagram.


Note that the point marked \(O\) in the figure also represents an event. This event occurs at time \(t = 0\) and at position \(x = 0\). We call this event the origin event of the diagram.

If we need to draw a spacetime diagram of an event \(A\) that occurs in space somewhere in the \(xy\) place, we must add another axis to the spacetime diagram (seen in the image to the right).

On a spacetime diagram, an event is represented by a point. The motion of any object is represented by an infinite set of infinitesimally separated points, which is a curve.

In drawing spacetime diagrams, it is also convenient and conventional to use the same-size scale on both axes. If this is done, a flash of light always has a slope of either 1 (if the flash is moving in the \(+x\) direction) or -1 (if the flash is moving in the \(-x\) direction), since light travels 1.0s of distance in 1.0s of time by definition in every inertial reference frame. It is also conventional to draw the worldline of a flash of light with a dashed line instead of a solid line.

The Radar Method

If we are willing to confine our attention to events occurring only along the x axis (and thus to objects moving only along that axis), it is possible to determine the spacetime coordinates of an event with a single master clock and some light flashes: we don't need ton construct a lattice at all! The method is analogous to locating an airplace by using radar.

After sending a light flash out, the values of the emission and reception times \(t_A\) and \(t_B\) are sufficient to determine both the location and the time that event \(E\) occurred!

\begin{equation} t_E = \frac{1}{2}(t_B + t_A) \qquad x_E = \frac{1}{2}(t_B - t_A) \end{equation}

This equation expresses an event \(E\)'s spacetime coordinates \(t_E\) and \(x_E\) in terms of the time \(t_A\) at which a light flash leaves a given frame's origin and the time \(t_B\) when it returns after being reflected by the event.