What is Center: Definition and 1000 Discussions

In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion.
In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid. The center of mass may be located outside the physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.
The center of mass is a useful reference point for calculations in mechanics that involve masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics. In orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass. The center of mass frame is an inertial frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system.

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  1. L

    Converting gravity into pressure inside the very center of a planet

    Gravity in the very center of the planet must be zero because all other atoms are pulling evenly around the center. We have a deap gold mine in South Africa and uranium waste storage Norway. Has the force of gravity shown change at these depths. Even at the bottom of the Pacific ocean gravity...
  2. P

    I Rolling Motion direction of Velocity center of mass

    Hi I have come across something confusing in rolling motion. If an object moves with a positive V_cm meaning to the right its angular velocity will be clockwise or negative. The formula is V_cm=wR but for a positive V_cm you get a negative w as it moves clockwise if V_cm is to the right...
  3. S

    Opening a portal to the center of the Sun

    Whilst perusing a D&D forum, I stopped to answer a question someone put out which was "what would happen if we used the gate spell to open a portal to the middle of the sun?" I replied (this was a while ago) and whilst I'm reviewing it, I am troubled by the fact that this uses such high forces...
  4. H

    Electric field at the center of the equilateral triangle

    I've found the distance from each point to the center, which is equal to r=20x1.732/3 = 11.55 cm. I find out that E2 and E3 due to -4µEyC on x-direction canceled each other. The E2y = E3Y = EY = E2Ycos60 = E2/2 = [(KQ2)/r^2]/2 So the net E-field: E = E1 +E2y+E3Y =kQ1/r^2 + [(KQ2)/r^2]/2 +...
  5. V

    Force between point charges at the center of two spherical shells

    If these point charges were placed in vacuum without any spherical shells in the picture, then the force between these charges would be ##F =\dfrac { k q_1 q_2} {d^2}##. But, I am unable to reason how spherical shells would alter the force between them. I do know that if charges were on the...
  6. V

    Why is electric field at the center of a charged disk not zero?

    The electric field strength at the center of a uniformly charged disk should be zero according to symmetry of concentric rings about the center, where each ring is contributing to the electric field at the center of the disk. For a thin ring of uniform charge distribution the formula is ##E =...
  7. Lotto

    B Can we say that a charged balloon has a center of charge?

    Doing so, we can consider the balloon to be a point charge (approximately). Can we do it in this case, when there are only electrons on its surface? Or is it stupid and we can't do it under any circumstances?
  8. shivajikobardan

    Bring the navbar text to the center of the screen in CSS

    https://demo.w3layouts.com/demos_new/template_demo/28-07-2021/biodata-liberty-demo_Free/2002651968/web/index.html This is the website that I am trying to build. This is my current navbar. This is what I want to build. My focus is on "Home" to "Contact". I want to put it at the center of the...
  9. greg_rack

    Engineering Calculating shear center for a "skewed" I beam

    Here is the diagram of the cs: As a premise I must say that this topic(shear center and shear flow distributions) is still very hectic in my mind; I aim to clarify it a bit by asking you guys this :) So, in order to identify the SC location, I must compute at what distance a point shear force...
  10. paulimerci

    Determine the net torque acting about the center of mass

    Net torque = 2gx20xcos30 - 2gx35xcos20 - 2gx15xcos60 + 2gx30xcos80 = -357Nm Have I done it right?
  11. A

    I Finding the center of area (centroid) of a right triangle

    To find the y value of the centroid of a right triangle we do $$\frac{\int_{0}^{h} ydA}{\int dA} = \frac{\int_{0}^{h} yxdy}{\int dA}$$ What is wrong with using $$\int_{0}^{h} ydA = \int_{0}^{b} y*ydx$$ as the numerator value instead especially since ydx and xdy are equal and where h is height of...
  12. kyphysics

    Computing for Dummies Q: What is difference between server and data center?

    I've been looking this up and don't seem to have a great understanding. Can someone confirm or correct that my understanding is accurate. Is a data center simply a large collection of individual servers? If not, how do they differ? Thanks.
  13. sachin

    Motion of center of mass under gravity

    While solving this question I could not figure out the concept of two blocks sticking together. the question is, Two particles A and B of masses 1 kg and 2 kg respectively are projected in the directions shown in figure with speed uA =200m/s and uB =50m/s. Initially they were 90m apart. They...
  14. C

    I Off center torque applied to a rotating body

    Hello everyone! So I've been studying gyroscopes, and see that a torque about the shaft alters the momentum, we can find the new momentum vector by finding the torque, multiplying by a small amount of time, and finally adding that vector to the momentum vector. This will create a precession for...
  15. C

    I Center of mass calculation

    Hello everyone! I've been reading Mr. McMullen's book and took some curiosity in an equation on the cover art, it is as follows:$$y_{cm} = \frac \rho m \int_{r=0}^R\int_{\theta=0}^\pi (r\sin \theta)rdrd\theta$$I'm trying to understand what it means, firstly the limits of integration for the...
  16. Ahmed1029

    Griffith's problem 4.37 -- A point dipole at the center of a sphere of linear dielectric material

    I tried to solve it for some time and then looked at the solution manual, which got me completely lost. Those are the first lines of the solution : I'm not so sure how equation 4.39: makes him conclude that the same relation holds for dipole moments. My second concern is that I'm not sure how...
  17. N

    I Solving for the trajectory of the center of mass

    I'm working on the physics engine component of a game engine I'm building, and I need some guidance with this particular situation. Consider a square with mass M that is free to translate in the xy plane and free to rotate about any axis perpendicular to the page (Fig. 1) If a linear impulse J...
  18. qnt200

    I The gravitational center of the Earth

    We know that gravitational forces are nullified near the center of the Earth, so the gravitational field's influence is not felt. Is it because of the Moon's gravitational field that the area of zero gravity has shifted away from the center of the Earth? If this is the case, this eccentric area...
  19. V

    Angular momentum of a disk about an axis parallel to center of mass axis

    I am using the following formula to solve this problem. $$ L_a= L_c + \text { (angular momentum of a particle at C of mass M)}$$ Because the point C is at rest relative to point A, so the second term in RHS of above equation is zero. Hence, the angular momentum about A is same as angular...
  20. S

    Volume of pyramid formed by center of 5 spheres inside a hemisphere

    Let the radius of the small sphere = r 3r = 1 → r = 1/3 ##x=\sqrt{4r^2-r^2}=r\sqrt{3}## Volume of pyramid: $$=\frac{1}{3} \times (2r\sqrt{3})^2 \times r$$ $$=\frac{4}{27}$$ So m + n = 31, but the answer is 29. I guess my mistake is assuming line AB is tangent to the top sphere. How to do...
  21. J

    Find magnetic field at center of rotating sphere

    if a sphere rotates, it's like multiple currents going around in a circle. I can find the magnetic field of each of those currents at the center point of the circle and add them together. We can integrate with respect to y and R. y ranges from 0 to 5 cm away from the center of the loop and the...
  22. V

    Finding tension as a function of distance from the center of rotation

    I'm not too sure how to account for both the mass and the rope at once. I think the following are true for the two individually: For the mass at the end, ## T = m ω^2 L ##, following from ##a = v^2/r##and ##v=ωr##. For the rope, ##dT = ω^2 r dM##, where ##dM = λ dr## and λ is the mass per unit...
  23. A

    I Electric field is zero in the center of a spherical conductor

    Electric field is 0 in the center of a spherical conductor. At a point P (black dot), I do not understand how the electric field cancels and becomes 0. Electric field is in blue.
  24. P

    I Polygon Coordinates given the Area and Center point

    I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following - Center point is known - area is known - Point A is known - Points B, C, and D are UNKNOWN I am NOT a math pro - this is for a puzzle I’m trying to solve and I can’t remember if this...
  25. A

    Find the inertia of a sphere radius R with rotating axis through the center

    $$I = \int{r^2dm}$$ $$dm = \sigma dV$$ $$dV = 4\pi r^2dr$$ $$\sigma = \frac{M}{\frac{4}{3}\pi*R^3}$$ $$I = \sigma 4 \pi \int_0^R{r^4 dr} = \frac{3*MR^2}{5},$$ which is not the correct moment of inertia of a sphere
  26. A

    Find the center of mass of a uniform semicircular plate of radius R

    $$rcm = \frac{1}{M}\int_0^\pi(rdm)$$ $$dm = \sigma{dA}$$ $$dA = (\pi*R^2)*\frac{d\theta}{2\pi}$$ $$\sigma = \frac{M}{\frac{\pi*R^2}{2}}$$ $$dm = M*\frac{d\theta}{\pi}$$ $$r = R(cos(\theta)\vec i + sin(\theta)\vec j)$$ $$rcm = \int_0^\pi{\frac{R}{\pi}(cos(\theta)\vec i + sin(\theta)\vec j)} =...
  27. K

    Analyzing applied torque to one of lug nuts instead of wheel center

    We all know that torque consists of force and distance. If we apply torque to the center of a car wheel, the force that the tire exerts to the ground can be calculated by dividing the torque by tire radius but what about applying torque to one of lug nuts which is off center? In the above...
  28. L

    Disk with rod attached rotating about the center of the disk

    1) Since the rod is uniform, with mass m and length l, it has a linear mass density of ##\lambda=\frac{m}{l}##, so ##I_{rod_O}=\int_{x=r}^{x=r+l}x^2 \lambda dx=\frac{\lambda}{3}[(r+l)^3-r^3]=\frac{\lambda r^3}{3}[(1+\frac{l}{r})^3-1]=\frac{1}{3}mr^2[3+\frac{3l}{r}+\frac{l^2}{r^2}].##...
  29. tivva

    Tipping force and Center of gravity for wall fitted furniture

    Hi All I was wondering if anyone can assist with a task of calculating whether an MDF unit will tip over if fixed only to the wall behind it with mechanical fixings as shown below. And what force will be required to do so. I've given it a try. Let me know your thoughts, would be much...
  30. I

    B Calculate needed dimensions of table - Center of mass, levers etc

    My high school physics days are long ago ;) This is not homework, well, other than it is work, at home. For a real application: Very space constrained "workshop", got a bench drill press, and want to build a table on wheels for it, to be able to move it into a corner when not needed. Those...
  31. L

    Work to bring a charge to the center of two quarter circles

    By measuring angle \theta from the positive ##x## axis counterclockwise as usual, I get ##d\vec{E}=k( (\lambda_2-\lambda_1)\cos(\theta)d\theta, (\lambda_2-\lambda_1)\sin(\theta)d\theta )## and by integrating from ##\theta=0## to ##\theta=\frac{\pi}{2}## I get...
  32. A

    I If the center of gravity changes on a die, how do the odds change?

    If the center of gravity changes on a die, how do the odds on each of the eyes change?
  33. A

    Calculating the center of mass as an astronaut moves on a shuttle

    I had solved this question but it didn't seem to be appropriate to post in the classical physics problem as my question is still homework-based. Originally I had thought this might be a conservation of momentum problem. But since we don't have any initial conditions it leaves too much to guess...
  34. T

    B T: "Exploring the Possibility of a Theoretical Center in a Finite Universe

    Unless the universe is infinite, and I don’t see how if the Big Bang is true… There must be a finite point beyond which the universe has yet to expand. It seems if measured from side to side we could determine it’s general diameter, allowing for undulations that result in a possibly...
  35. N

    I Find the center manifold of a 2D system with double zero eigenvalues

    I have to find the center manifold of the following system \begin{align} \dot{x}_1&=x_2 \\ \dot{x}_2&=-\frac{1}{2}x_1^2 \end{align} which has a critical point at ##x_0=\begin{bmatrix}0 & 0\end{bmatrix}##. Its linearization at that point is \begin{align} D\mathbf {f}(\mathbf {x_0}) =...
  36. S

    Engineering Center of gravity and stability of a system

    Suppose I have an object consisting of a hemisphere of radius r and a cone of radius r and height h. The shapes are glued to each other on their faces and the object is set standing on its hemisphere side. Depending on the value of h, the center of gravity for the system will change. I have...
  37. ektov_konstantin

    I Moving center of coordinates in the polar graph

    I have a function in polar coordinates: t (rho, phi) = H^2 / (H^2 + rho^2) (1) I have moved the center to the right and want to get the new formulae. I use cartesian coordinates to simplify the transformation (L =...
  38. lela

    Is the force exerted by a pivot always towards the center of mass?

    I thought that the force by the pivot A on the pole AB would be the reaction force to the x-component of the gravitational force on AB. This would mean that the force by the pivot would be parallel to the pole, but in my notes from class the force vector seems to be more along the bisector of...
  39. guyvsdcsniper

    Help finding the Electric field at the center of charged arc

    I am having trouble understand where area circled in red. I get that lamda is Q/L. The charge is +Q. Length is pi/R/2. I am having trouble understanding why the length is pi/R/2? Is it because the circumference of a circle is 2*pi*R and since we have broken this problem down to just...
  40. W

    B Is it the nucleus or a proton at the center?

    Which one is closer to reality, is it this picture https://en.wikipedia.org/wiki/Hydrogen#/media/File:Hydrogen_atom.svg or this https://www.naturphilosophie.co.uk/heart-hydrogen-atom/? The reason why I asked the question is according to the picture of hydrogen atom at Wikipedia, which is the...
  41. fresh_42

    A Center of a linear algebraic group

    Let ##G\leq GL(n)## be a linear algebraic group of dimension ##m,## and ##C## its ##c##-dimensional center. What do we know about lower and upper bounds of ##c=c(m)\,\text{?}## Clearly ##c(0)=0, c(1)=1## and ##n^2\geq c(m)\geq 1## for ##m\neq 0.## By Schur's Lemma we also know ##c(n^2)=1##. Did...
  42. Dan Rozenfarb

    B Please help in explaining 'we are not the center of the Universe'

    When I discuss about the Big Bang, the expansion of the Universe and the fact that on average every galaxy is receeding from us, I get "oh, so then we are at the 'center' of the Universe." I know that's not the case, we are not in a special place, etc. But, is this a proven fact, or is just a...
  43. Istiak

    Find the center of mass of an arc

    In my mind, I had cut half of B and, thought it's semi-circle. Then, I was trying to find Center of Mass by taking it as semi-circle. But, I get an answer which is approximately, close to main answer. Someone else had solved it another way This way I can get the accurate answer. But, the...
  44. karush

    MHB -2.4.27 find center and radius of circle

    Determine the graph of $x^2+y^2+6x+8y+9=0$ $\begin{array}{rll} \textsf{rewrite} &(x^2+6x )+(y^2+8y)=-9\\ \textsf{complete square} &(x^2+6x+9)+(y^2+8y+16)=-9+9+16\\ \textsf{simplify equation} &(x+3)^2+(y+4)^2=16=4^2\\ \textsf{observation} &C(-3,-4), \quad R=4...
  45. D

    I Questions about center of gravity

    Hello, I am very new to the concept of center of gravity and I have a question. I wanted to know if the center of gravity of an object is always in the same location in 3-D space. For example, if I was able to find the center of gravity for cylinder/rectangle when its lying flat on a horizontal...
  46. R

    How to interpret the center of aero forces on a sports car?

    Hello. I am working on a physics project for a simulation title and have stumbled upon on an interesting challenge. Below is the example from wind tunnel data of a Dodge Viper GTS sports car. Wheelbase: 2,44m Lift front axle: 54kg Negative lift rear axle: 26kg Can somebody please explain to...
  47. G

    Find the center of gravity of a combined cone and cylinder

    Could I please ask for help with the following: Given: The centre of gravity of a uniform solid right circular cone of vertical height h and base radius a is at a distance 3h/4 from the vertex of the cone. Such a cone is joined to a uniform solid right circular cylinder of the same material...
  48. O

    MHB Finding the Center of a Circle

    Hello! I have an application where I need to find the center of a circle where I am having trouble coming up with a simple way to do this. The diameter of the circle is known and i want to be able to determine the location of it where only a portion of the circle is known. (see the image...
  49. A

    A What are the probabilities of a die with an offset center of gravity?

    With a pure die, all odds are equal. With a pure die, the center of gravity is exactly in the middle of the die. But what if the center of gravity is not in the center? How are the odds then. For example, how do the odds become if the center of gravity is exactly on the line that runs through...