Axes Definition and 178 Threads

I have very little clue as to how to imagine (visualise) the rotation and how the axes will look. All I can do is to draw the image of what I mean by the task. Of course, this is before the rotation takes place. In the diagram, ##\mathbf{OP}## is the axis of rotation. For simplicity, we may...

Alright, I've been wondering this for a while now. Say you have an infinite grid of squares in hyperbolic geometry, such that the curvature makes it so each angle of each square is 72° (5 squares at each corner). At the very 'center' of the grid, or the origin, there would be 5 straight rays...

Why are the axes taken as perpendicular to each other rather than at some other angle? Is it just a matter of convention? Is there any mathematical reason behind it? Is there some other reason?

I tried posting this on the physicsstackexchange, but wasn't making any progress in understanding what's going on. Suppose the axes in two coordinate systems S, S' are parallel. Now, suppose I rotate S through some angle ##\theta## and also rotate S' through the same angle ##\theta## It's not...

Statement figure: My attempt at a solution: FIGURE 1 ##\rightarrow A=a^2## CG ##\rightarrow \overline{x}=-a/2, y=-a/2## $$\overline{Ix}_1=\dfrac{bh^3}{12}=\overline{Iy}_1=\dfrac{a^4}{12}$$ $$Ix_1=\overline{Ix}_1+\overline{y}^2A=\dfrac{a^4}{12}+\dfrac{a^4}{4}=\dfrac13 a^4\...

Hi, it's about the task e) Since the density is homogeneous, I have assumed the following for ##\rho=\frac{M}{V}##. I then started the proof of ##I_{23}##, the integral looks like this: $$ I_{23}=\int_{}^{} -\frac{M}{V}r'_2r'_3 d^3r$$ Now I apply the transformation $$ I_{23}=\int_{}^{}...

Hi, unfortunately, I am not getting anywhere with the following task The inertia tensor is as follows $$\left( \begin{array}{rrr} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \\ \end{array}\right)$$ I had now thought that I could simply rotate the...

Suppose $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$ is a subset of $$\mathbb{R}^2 $$ with subspace topology. Can this be a 1d or 2d manifold? Thank you!

Using the equation above I get Xcm = 0.022 m. I set the origin be at the left of the vertical rod parallel to its centre of mass as in the diagram. But I’m not sure if the equation is correct for 3d. for the moments of inertia I am using I = Icm + md^2 = (mr^2)/2 + md^2 where d is the...

I have been trying to determine an expression for a unit vector in the direction of F for hours now. I think the expression is supposed to look something kind of like this, But I don't understand at all how to arrive at this expression. Any help?

The world building thread about a derelict spaceship got me wondering. An object can rotate on two axes simultaneously, yes? Is that stable in flat space? If so, what would occupants experience as gravity? Would it change over time?

I am confused by the drawing of the door with hinges A and B attached. I do not understand why -Ax = Bx. I would have thought that Ax = Bx

Been studying Special Relativity in Uni. and I've noticed that all examples of relativistic motion provided are motions only along a single axis, like the one below: The particle's Reference Frame is moving only along the X axis in the example above. In this case the Lorentz Transformation for...

The answer in the textbook are options A, C and D. I understand why it is option A, because it is a scalar, I also get that option D is correct because the magnitude of a vector doesn't depend on the coordinate axes. I don't get how option C could be correct. If option C is correct why not D as...

I do disagree. How accurately a variable can be measured is not the significant issue. The head/tail result of a coin toss can be measured with great accuracy but that does not make that result the independent variable. The decision of whether to model Y=aX+b+##\epsilon## versus...

I've a body having initial angular velocity at ## t=0 ## as shown. The axis shown are fixed in inertial space and initially match with the principal axis. I want to find the infinitesimal change at ##t+\Delta t## in the angular momentum along the ##z## axis. I've seen the following approach...

I've a disc which can rotate freely about two perpendicular axis (fixed to the body) If I simultaneous try to rotate it about the two axis, what will happen?

Hello, This question is with regards to the discussion around page 56 (1971 Edition) in Anthony French's Newtonian Mechanics. He is discussing the choice of a coordinate system where the axes are not necessarily perpendicular to each other. Here is the summary of what I read (as applied to...

The problem: Visualising the problem (My question is with regards to this): Why is the above set-up correct? In the above diagram, S would be moving at velocity -v relative to S', instead of v. Is this because the question says "speed v", and so we can set the direction as such? Why would the...

We know that for a non-rigid body, the most stable type of rotation of it is the rotation about the axis with the maximum momentum of inertia and thus the lowest kinetic energy. However, for this question involving a rigid body, the most stable axis is the one with the lowest moment of inertia...

$$B = \frac {\mu_0 I}{2 \pi r} $$ By Right-hand Grip Rule, the direction of the magnetic field by wire in y-axis is into the paper (z) while the direction of the magnetic field by wire in X-axis is upwards (+i) The answer state the Magnetic field is in the (i - y) direction though. Next...

I reslove Ma and Mb into y and z component. Ma=1200sin(20)j+1200cos(20)k Mb=900sin(20)j+1200cos(20)k Mc=-840i I looked at the solution and it states that the y component of Ma is negative (-1200sin(20)j+1200cos(20)k). I understand that Mc is -840i because it is Clockwise. How do you determine...

I attached a file with some explanations of the variables in the code and the plot that I should get. I don't know what is wrong. Any help will appreciated. from scipy.integrate import quad import numpy as np from scipy.special import gamma as gamma_function from scipy.constants import e...

I was reading through a set of notes and found something a little odd. The aim is to solve the beam structure shown below, which is massless and of length ##l##. By considering the beam as a whole, we obtain ##A_y = P##, ##A_x = N## and by taking moments about A we see ##M_A = Pl##. However...

I am including a link to a B level discussion of this I found on-line to try and anchor my question, not because I think the below article is good or poor - I am not able to assess that. https://www.einstein-online.info/en/spotlight/changing_places/#The_analogy In particular I am asking about...

Disclaimer: I am a physics student and I have very little knowledge of topology or differential geometry. I don't necessarily expect a complete answer to this question, but I haven't really found any reference that approaches what I'm trying to ask, so I'd be quite happy to simply be pointed in...

Summary:: I'm quite stuck on this problem i don't know what I am going to use formula to solve this one This is the given I am not sure if this is a resolution problem or it involve parallelogram law

As I understand it, dimension is a way of describing direction, with the first three spatial dimensions being straight lines which extend infinitely in one direction, perpendicular to each other. In string theories, several additional dimensions are required, sometimes up to nine or 10, I...

Therefore, if someone were to ask what the magnitude of centripetal acceleration is at the top of the wheel at a given instant (relative to the ground): ##v_{cm} = v_{translational, center-of-mass/wheel}## ##ω = ω_{point-of-contact}## ##v_{top} = 2(v_{cm}) = 2(rω)## ##a_{c(top)} =...

Summary: I need to translate points on coordinate axes as part of a calculation process Hello everyone, I've created the diagram below to try and explain what I am trying to do as part of an existing software app that's used to generate profiles and programs to drive a CNC machine to grind...

Which could be the first step to find the V(t) ?

(Throughout all my post, I will refer to “gravitational potential energy” just as “potential energy”) Hi! I have this confusion about when is potential energy positive/negative and how it is related to how we define our axes. I think it is easier to understand my confusion with the following...

Determine Force from F2 along u and v axes. CompF= F*Cos(angle); CompF onto U = 8kN*Cos(30) = 6.93kN Angle between F2 and V: 180-75 = 105; 105-30 = 75 degrees CompF onto V = 8kN*Cos(75) = 2.07 kN. Since in the -V direction; -2.07kN. I just would like for somebody to verify these answers. I've...

1.To shift the graph of a function : Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative. Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is...

I am exploring the behaviors of complex integers (Gaussian and Eisenstein integers). My understanding is that when a complex integer z with norm >1 is multiplied by itself repeatedly, it creates a series of perfect powers. For instance, the Gaussian integer 1+i generates the series 2i, -2+2i...

Let's consider two inertial frame S and S'. S' moves with speed v w.r.t S along x-axis towards the right. Now we can draw the two co-ordinates system. The t' axes will make an angle arctan(v/c) with t axes rotated towards x-axis and similarly the x' axis will be tilted towards the t axes...

Hi, I'm stuck on a homework problem in my Calculus III class. I solved 3a really easily, but 3b is giving me a lot of trouble. I know that to find the tangent line, I first have to find the slope, which is represented by the vector: <3cos^2(t)(-sin(t)), 3sin^2(t)(cos(t))>. I know the formula...

Is there a method to calculate inertia tensor form principal axes moment of inertia? Like now we have moment of inertia: (Ix,Iy,Iz)=(20,18,25), and hot to calculate the inertia tensor like (Ixx,Ixy,Ixz Iyx,Iyy,Iyz, Izx,Izy,Izz)? I have read about this page several times, but still have no idea.

EDIT: I realize now that I have fundamentally misunderstood a crucial aspect of deriving the Bell inequality for this case which is the existence of the third axis. The setup of the problem did state that the axes were chosen at random. Therefore I can't just look at the possibility of choosing...

Homework Statement A cube balanced with one edge in contact with a table top and with its center of gravity directly above the edge is in _______ equilibrium with respect to rotation about the edge and in _________ equilibrium with respect to rotation about a horizontal axis that is...

I am currently reading Leonard Susskind's "Quantum Mechanics the Theoretical Minimum". In chapter 2.3 and 2.4, he defines |r>, |l> and|i>, |o>, for r and l along the x-axis and i and o along the y axis. The equations are: $$|r> = \frac{1}{\sqrt{2}}|u>+\frac{1}{\sqrt{2}}|d>$$ Since...

See the screenshot for the full problem. 1. Homework Statement We're given a pulley with one side hanging (A) and the other on an inclined plane (B). We are also given that the mass of A is 15 kg and the mass of B is 17 kg. The angle of elevation of the inclined plane is 32 degrees. The...

Homework Statement : [/B] r vector = 3t i + (4t-5t2)j. Find the angle made by the vector with respect to the x-axis and the y-axis. 2. Homework Equations : A.B=AxBx+AyBy 3. The Attempt at a Solution : I tried to take the dot product of the unit vector along x-axis and r. I did the same...

Suppose i have an equilateral triangle and i want to find the principal axes of rotation passing through one of the vertex. How can i do that? I am thinking along the following lines but I'm not too sure: 1)Since the equilateral triangle has symmetry about a median, that definitely is one...

Homework Statement Hello, I know about the inertia tensor about one axis, but how about a body that rotates around 3 axis x,y and z such as a spacecraft with changes in the attitude. Thanks for you help. Homework EquationsThe Attempt at a Solution

A car traveling north makes a right turn, to head east. How do the tires completely offset py and simultaneously create px? (If the tires merely acted like guidance jets, it might turn to face east but continue traveling north.

Homework Statement So, I have this problem here that's pretty basic, but the solution manual sets different axes, and I'm having a bit of trouble understanding the geometry part, meaning how he applies the given forces to the new axes. A model airplane of mass 0.750 kg fl ies with a speed...

Homework Statement Just doing some practice problems from past finals and I needed some help on this one. Sorry if my question doesn't exactly fit the template. 2) Relevant Equations / Information For part a) and for M_1, I drew the axes such that the x-axis points to the top right, in...

A simple method to find the potential of a conservative vector field defined on a domain ##D## is to calculate the integral $$U(x,y,z)=\int_{\gamma} F \cdot ds$$ On a curve ##\gamma## that is made of segments parallel to the coordinate axes, that start from a chosen point ##(x_0,y_0,z_0)##. I...

Dear All, I have a monoclinic cell with a=18.7,b=3.55,c=9.069 and beta=134 degrees, angle between a and c. Now I need to make the c axis parallel to z axis for surface calculations, ie to introduce a vacuum. What is the rotation matrix that would do this?? Thank you.