PHYSUN1601 Physics I： Mechanics and Relativity

Physics 1600 Review Plan Physics 1600 Exam Preparation

Approach:

It seems better to listen while copying the equations (blu3mo)
- Handwritten notes help to focus on the details of the equations and geometry
- Maybe try writing in the style of kakeru

Final Summary:

I forgot about binomial expansion
I want to memorize the relationship between a=kx and parametric equations
I need to review polar coordinate acceleration
- $\dot{\overset{r}{˙}} = r \dot{θ}^{2}$
There are still some unclear points about energy integration in 3D
- Is line integral sufficient?
I can’t solve non-separable 3D mechanics
Classical turning point
- E=K+U, if E=U(r) then K=0, v=0
  - - I see, I understand now
I don’t understand inertial frame
- Ah, physics works in any frame, but in the same inertial frame, the value of force is the same
I need to review mass flow
- I need to review handling Δt and making it infinitesimal

Application of Energy:

It is helpful to derive the equation of motion from the force and velocity equations without actually solving the equation x=f(t). This gives a better understanding of the motion.
- It is useful to observe turning points to understand various aspects of the motion.

Energy:

When force is a function of position, the equation above holds
- = The situation where the force at a certain position is always the same
- Example: gravity, SHM, etc.
- If it is a force like a(x) = kx, then it applies
TODO
- Confirm the derivation of energy conservation
- Confirm the meaning of conservative force
- Confirm the meaning of the minus sign in F=-dU/dx
- Confirm the arbitrariness
  - It’s strange that E=mc2 holds even though it is arbitrary
  - There is an explanation for this in relativity, apparently

Applying Newtonian Mechanics:

When applying to a certain situation, constraints are common
- For example, an object placed on a table has the condition y > height of the table
- For example, a ball attached to a string has the condition that it cannot go further than the length of the string
And when there are constraints, there are always associated forces
- Normal force, centripetal force, etc.
- Example: Normal Force:
  - Assumption: When there is a surface, $v \cdot \overset{n}{^} = 0$
  - From this, $a \cdot \overset{n}{^} = 0$ , $Σ (F \cdot \overset{n}{^}) = 0$
    - So, the normal force compensates for the part that does not add up to zero
    - I see (blu3mo)
- Example: Tension Force
  - From the constraint of constant length, the existence of two tension forces can be deduced
    - $m_{ro p e} a = F_{T 1} + F_{T 2}$
    - Then, assuming that the rope is massless, $F_{T 1} + F_{T 2} = 0 ⟹ ∣ F_{T 1} ∣ = ∣ F_{T 2} ∣ = T$ can be derived
      - Oh, because the rope is massless, even if the rope is accelerating, the two forces can be zero (blu3mo)(blu3mo)(blu3mo)
        
        Otherwise, the force of the rope’s acceleration would affect something else, I see
- Example: Sprint
  - [- Is the constraint $\overset{x}{¨} = - k x$ ?]
  - No, $F_{s} = - k y$ is introduced like $F_{g} = - m g$
    - This is not a constraint force
    - As a result of transforming this, the equation $ma = - k y$ is obtained
  - This shows that the same behavior occurs even when gravity force is applied
- Constraint forces are deduced from observation
In practical terms, constraint forces often have limits
- Ropes can break and tables can break
- Friction explicitly deals with this
Procedure:
- Write down the constraints
- Draw a free-body diagram
- Write down the relationship between force and acceleration (=constraint) using Newton’s second law
- Choose a convenient coordinate system and organize in each direction
  - At this time, if constraint forces can be eliminated, do so

Galilean:

Principle of relativity
- Newton’s laws hold in all inertial frames (transformed galileanly)
  - The laws hold, but the values of velocity and momentum are different depending on the frame
  - However, Δv and Δp=Impulse are constant, so they can be uniquely derived from the values of F and a (blu3mo)(blu3mo)(blu3mo)
  - Counterexamples
    - $F = - mb v$ (drag force depends on v)
    - However, this is actually inaccurate
      - To make it hold in any frame, it becomes $F = - mb (v - v_{0})$ (v_0=velocity of air)- Assuming that air does not move with respect to the origin, the equation F = -mbv can be written.
If we want to write an equation that holds in any internal frame, the variable related to F is Δv, not v.
I see, I see. (blu3mo)(blu3mo)(blu3mo)

Impulse

Δp = I(t)
That’s right, the area of the force represents the difference in momentum.

Newton’s Law

As a premise,
- Inertial Frame: a frame where velocity remains constant when there is no interaction with other particles
  - This is true for each particle (even after Galilean transformations, acceleration does not change).
- Newton’s Laws only hold in an Inertial Frame.
  - So what I thought was Newton’s First Law is actually a condition for it to hold. (blu3mo)(blu3mo)(blu3mo)
  - This is why Newton’s laws do not hold in polar coordinates.
With that in mind, let’s consider the state variable called Force.
- It feels like physics is built from mathematics. (blu3mo)(blu3mo)
- F = ma
  - The state variables F and a are proportional, and the constant is the state variable m.
  - The F here is the net force resulting from interactions with other particles. (blu3mo)
  - Simply adding up the vectors of the four types of forces is incorrect. (blu3mo)
    - Really? (blu3mo)
    - In terms of relativity, only electromagnetic forces can be precisely added up.
      - Oh. (blu3mo)
    - However, in Newtonian physics, gravitational and electromagnetic forces can be added up.
      - We have been doing this so far, but we should know that it is an approximation. (blu3mo)
- Third Law: F_{21} = -F_{12}
These are hypotheses derived from real observations.
- Seems like it. (blu3mo)

Law 2

F = dp/dt holds even in relativity.
- Since the definition of p changes in relativity, F = ma is no longer valid.

Translation

Galilean space: a space where the origin moves at a constant velocity
- As the equation above shows, the acceleration in that space does not change.
Here, it is assumed that t does not change for any origin.
- This assumption is violated in the context of special relativity. I see. (blu3mo)(blu3mo)

Plane Polar Coordinates

x = rcosΘ, y = rsinΘ
Since we have only discussed the description of position and not Newton’s laws, it seems that the choice of coordinates does not affect the discussion.
- If we start talking about Newton’s laws, we can only say things like “objects continue to move at a constant velocity” in Cartesian coordinates.
Let’s differentiate it as usual and find $v$ .
- $v = \overset{r}{˙} \overset{r}{^} + r \dot{θ} \hat{θ}$
Let’s differentiate this to find the acceleration.
- Assuming r = R is constant, we obtain a constant magnitude acceleration in the opposite direction of $\overset{r}{^}$ .
  - $a = - R ω^{2} \overset{r}{^}$
  - Oh, I know this one. (blu3mo)(blu3mo)
- Generalizing, we have
  - I want to have an intuitive understanding of each term. (blu3mo)
- For example, how can we explain the absence of acceleration when a rotating ball is let go?
- I want to review this whole process and be able to explain it myself. (blu3mo)(blu3mo)(blu3mo)

Vector Acceleration

Actually, it is impossible to solve the acceleration analytically.
- Even if we have $a_{x} (x, y, t)$ and $a_{y} (x, y, t)$ , we cannot solve for x.
- If we have $a_{x} (x, t)$ and $a_{y} (y, t)$ , we can separate them and solve.
  - Well, it’s a limited case.
  - On the other hand, maybe the reason we have been able to decompose into x and y in physics is that it has been limited to such cases. (blu3mo)(blu3mo)
  - In this case, we can take x = f(t) and y = f(t), so we can rearrange them to get y = f(x).
    - We can determine the function of the trajectory.
- Can we separate $a_{r} (r, θ, t)$ and $a_{θ} (r, θ, t)$ ?

Metrics

In fact, the distance can only be obtained using the Pythagorean theorem in special cases (in this world).
- Based on this, the dot product of vectors is also defined, so if it were a different world, the cross product would be different.
I couldn’t follow the equation transformation here, so I need to review it. (blu3mo)

Curve Distance

Same as what we did in APMAE2000 Multivariable.- $d s = ∣ d r ∣ = ∣ v ∣ d t$
Therefore, $s = \int_{t_{a}}^{t_{b}} ∣ v ∣ d t$
- This can be written as $s = ⨐_{t_{b}}^{t_{a}} d s$

Position

As a primitive definition, an affine space
- Simply a state where there are multiple points
- Since there is no reference point, addition of positions cannot be done
- However, the difference between points, or distance, can be measured
Define an arbitrary origin
- Then, a position vector (the difference between position and origin) can be created
- Finally, vectors can be added and manipulated

Plane Normal Vector Derivatives

The change in angle Θ between a vector and another reference vector
- Important: Depending on the reference, the value may change, but an increase in Θ always means a left turn
  - (in a right-handed coordinate system)
Writing it component-wise (x, y, z) and taking the derivative is not sufficient as it lacks important distance and angle information
- Therefore, we want to differentiate $v = ∣ v ∣ \overset{v}{^}$
  - $\frac{d}{d t} v = \dot{∣ v ∣} \overset{v}{^} + ∣ v ∣ \dot{\overset{v}{^}}$
    - This is easier to understand when considering the case where only the direction or magnitude changes
      - The term with a constant value disappears because its derivative is 0
  - About $\dot{\overset{v}{^}}$ :
    - $\dot{\overset{v}{^}} = \dot{θ} \cdot (\overset{n}{^} \times \overset{v}{^})$
    - The derivative of a unit vector is not a unit vector (blu3mo)(blu3mo)(blu3mo)
      - This is evident when considering dimensions, as the unit of d/dt unit vector is not 1 but 1/time
    - As a result, the derivative of $\overset{v}{^}$ only changes its direction, or perpendicular to $\overset{v}{^}$
      - This is easier to understand

3D Kinematics

It is not ideal to consider vectors as just a collection of numbers
- Oh, I was thinking of it that way (blu3mo)
- Why..?
Important: There is no starting point, only differences are represented
- That’s true (blu3mo)
Arrows, for example, are one representation of vectors
$\hat{V}$ is the unit vector of V $= \frac{V}{∣ V ∣}$
- The important thing is that this is dimensionless
- Therefore, when expressing $V = ∣ V ∣ \hat{V}$ , the dimension of |V| is the same as $V$ (blu3mo)
  - This may not be obvious
Cross product
- Consider $n = v_{1} \times v_{2}$
  - If we imagine the angle of v1 to v2 as positive, then the angle of the normal vector to the plane is also positive (meaning it comes out of the plane), we can understand it without the right-hand rule
- If $n = \overset{v}{^}_{1} \times \overset{v}{^}_{2}$ , then $∣ n ∣ = sin θ$
  - If Θ=90, then $n = \overset{v}{^}_{1} \times \overset{v}{^}_{2} = 1$ , and a new unit vector is created (blu3mo)(blu3mo)
    - Taking the cross product of the unit vectors in the x and y directions produces the unit vector in the z direction

1D Kinematics

Let’s consider a one-dimensional state variable called x for now
- [x] = length
The choice of the position of zero and the coordinate system is arbitrary, and physics does not change depending on it
- Well, that’s true, but it’s important to prove it mathematically (blu3mo)
From there, differentiate to get v, and differentiate twice to get a
- That’s right
- $f (x) = 3 x$
- So, $x (t) = x (t_{0}) + \int_{t_{0}}^{t} d t^{'} v (t^{'})$ and $v (t) = v (t_{0}) + \int_{t_{0}}^{t} d t^{'} a (t^{'})$
- That’s right, basic stuff
- In high school physics, instead of using the integral, we write $x = v (t - t_{0})$
  - ==However, this is only valid when v is constant and x(t_0) = 0==(blu3mo)
  - Alternatively, we can define the average v as $〈 v 〉 = \frac{1}{Δ t} \int_{t_{0}}^{t} d t^{'} v (t^{'})$ and say $x = 〈 v 〉 Δ t$
- So, it’s important to properly consider the limits of definite integrals (blu3mo)(blu3mo)
  - Actually, I may have been doing math ambiguously before
  - In fact, if you do it carelessly in a first-order differential equation, you will make mistakes
Drag Acceleration
- If we have $a = - k v$ or $a = - g - k v$ , the velocity converges to a terminal velocity
  - Solving it as a first-order differential equation, we get $v = v_{0} e^{- b t}$
- Review (blu3mo)
  - I should review the discussion on quadratic drag, which was not covered in class
  - Calculation of drag acceleration when there is -g
- Calculations show that regardless of the initial velocity, it converges to the same terminal velocity

2nd Derivative- $a = \frac{d ^{2} x}{d t ^{2}} = f (x)$ (not $f (t)$ ).

When we want to express $a$ in terms of $t$ , we cannot integrate the right side because it is not a function of $t$ .
However, this can be solved easily using the energy method, but that will be done later.
Let’s consider the special cases of the 2nd derivative: $a = + K x$ and $a = - K x$ .
For $a = - K x$ :
- Reviewing (blu3mo)
- If we consider the only equation that fits this, it becomes $x = a s in (t) + b cos (t)$ .
  - If we transform it, it becomes simple harmonic motion (blu3mo)(blu3mo)(blu3mo).
  - So, it seems like SHM was derived backwards.
For $a = + K x$ :
- This takes the form of $e^{x}$ .
  - In fact, if we express it using hyperbolic sine/cosine, it becomes a similar form to $- K x$ .
    - Amazing! (blu3mo)(blu3mo)

20220913

It seems like it would be good to organize the content of this class in a high degree of freedom note using GoodNotes…?
- Well, the original notes are already strong, so maybe it’s okay.

20220908

20220906

Before the first class
- Read Feynman Physics.
It seems like coordinate transformations and symmetries, as well as approximations, are important.
They seem to emphasize understanding the essence.
While I am very familiar with the complexities of being an undergraduate in college, I beg you, please do not allow yourself to evolve into a mode where you are starting and trying to complete reading or problem sets at the last minute. If you allocate time each day to read and work through the material in the notes and start the homework before recitations so you can take maximal advantage of the interaction with the TAs, I have every confidence that you can succeed in the class. However, if you do not dedicate the required time and attention, no matter how effective I am and/or the TAs are, it will be difficult for you to succeed.
- Yes…
Homework
- Due next Sunday, 9 days from now.
Textbooks
- There is Kleppner and Kolenker’s book.
- There are also Berkeley Physics and MIT’s materials.
- The course notes will be treated as the main material for now.
Computers
- Not needed for class/exams.
- Might be needed for homework?
Point particles
- $N_{D OF}$
- Although things like chalk and protons are not point particles, it is natural to assume them as such.
- Electrons, on the other hand, are truly point particles.
  - Ohh… (blu3mo)
Quantities dealt with in physics
- Physical constants
  - Constants that are determined for some reason.
- State variables
  - Physics mostly deals with the relationships between these variables.
- I don’t understand how the other quantities are different from state variables (blu3mo).
Terminology
- Coordinate vs Position
  - Coordinate is varied, while position is a specific one.
  - coordinate vs position

Shutaro Aoyama

Explorer

PHYSUN1601 Physics I： Mechanics and Relativity

Graph View

Backlinks