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. D

    Center of mass system of two interacting identical Fermions

    Homework Statement Consider the center of mass system of two interacting fermions with spin 1/2. a) What is the consequence of the Pauli exclusion principle on the two-particle wave function? b)Let S1 and S2 be the spin operators of the two individual fermions. Show that the operators...
  2. kmm

    Magnetic field of a cylinder with current down the center

    I was just in a conversation with someone regarding the magnetic field resulting outside of a solid cylinder, with a current moving down the center of the cylinder, and then the resulting magnetic field after removing the current. Now I haven't thought about magnetic fields/magnets for a while...
  3. M

    MHB What caused the wrong answer for determining the center and radius of Circle 2?

    Determine the center and radius of circle.
  4. M

    MHB What is the Center and Radius of a Circle?

    A. Determine the center and radius of circle. B. Also, find the y-coordinates of the points (if any) where the circle intersects the y-axis.
  5. B

    I Center of Image - Brightness Distribution

    Hello, I am reading a review on weak gravitational lensing (https://arxiv.org/pdf/astro-ph/0509252.pdf) and they define the center of an image as follows: $$\vec \theta_c = \frac{\int d^2 \theta I(\vec \theta) q_I [I(\vec \theta)] \vec \theta}{\int d^2 \theta I(\theta) q_I[I(\vec \theta)]}$$...
  6. astrocytosis

    Magnetic field above the center of a square current loop

    Homework Statement Find the exact magnetic field a distance z above the center of a square loop of side w, carrying a current I. Verify that it reduces to the field of a dipole, with the appropriate dipole moment, when z >> w. Homework Equations (1) dB = μ0I/4πr2 dl × rhat (2) r =...
  7. H

    Displacement of Center of Mass Problem

    1. The problem statement, all variables, and given/known data A square of side 2 L is removed from one corner of a square sandwich that has sides of length L. The center of mass of the remainder of the sandwich moves from C to C’. The displacement of the y coordinate of the center of mass (from...
  8. C

    MHB Finding the Center of Mass of a Plywood Sheet: A Scientific Approach

    Dear Everybody, Shown below is a 4.00-ft by 8.00-ft sheet of plywood with the upper left quadrant removed. Assume the plywood is uniform and determine the x and y coordinates of the center of gravity. Hint: The Earth's gravitational field is also uniform for the entire sheet of plywood. I...
  9. F

    Biomechanics, center of mass of human body and balance

    Hello everyone, I know that the any object, human body included, when the vertical line from its center of mass to the floor falls outside the base of support, topples under the influence of gravity: the force of gravity produces a nonzero torque that rotates the body. Is the human body able to...
  10. M

    Center of Mass of a Sphere with uniform density

    Homework Statement Find the z -coordinate of the center of mass of the first octant of a sphere of radius R centered at the origin. Assume that the sphere has a uniform density. Homework Equations Mass = Integral of the density function Center of mass for z = Integral of density * z divided...
  11. A

    A Download full data from National Nuclear Data Center

    dear physicists is it possible to download full data of half lives (or another parameters) from Chart of Nuclides in http://www.nndc.bnl.gov/chart/ I can see data One by one for extract my Required data and it is too time consuming So, if there's any way I can download all the information, it...
  12. A

    Magnetic field of off center triangle - Biot Savart Law

    PICTURE INCLUDED 1. Homework Statement A piece of wire is bent into an isosceles right triangle whose shorter sides have length a The wire carries current I. Calculate the magnetic field for point P. Point P is located on the Y-axis ( 0, √2a). Two corners of the triangle are are located at...
  13. Draconifors

    Finding center of mass of solid

    Homework Statement A solid B occupies the region of space above ##z=0## and between the spheres ##x^2 + y^2 + z^2 = 16## and ##x^2+y^2+(z-1^2) = 1##. The density of B is equal to the distance from its base, which is ##z = 0##. The mass of the solid B is ##\frac{188\pi}{3}##. Find the...
  14. I

    How to center the bandwidth for carrier frequency?

    I have a baseband signal in IQ form. I have a method to calculate the carrier offset and estimate the carrier bin. I want to center the carrier to the middle of the bandwidth. How do I do so? Do I simply multiply the IQ data by the exponential with the carrier offset, but doesn't that shift the...
  15. karush

    MHB 232.15.6.31Find the center of mass

    $\tiny{232.15.6.31}$ $\textsf{Find the center of mass of the following solid, assuming a constant density.}$ $\textsf{Sketch the region indicate the location of the centriods}$ $\textsf{Use symetry when possible and choose a convient }$ $\textsf{coordinate system the sliced solid cylinder...
  16. CollinsArg

    B What does the Center of Mass equation mean?

    How can I understand logically this equations? it seems to me like an avarage of equation...but why each term is specifically in the way it is? Thank you
  17. C

    Force in Center of Mass frame equaling zero

    Homework Statement Suppose I have particles with masses m1 and m2 , both are in freefall: Then , looking at the center of mass coordinates , I know that the sum of momentums of masses m1 and m2 in the center of mass coordinates is equal to zero : P1c + P2c = 0 so now I know that the total...
  18. D

    What would the weight of a column of water be at the center of the Earth?

    In reading Jules Verne's book A Journey to the Center of the Earth I came to a section where one of the characters questions if the massive weight of the air would be fatal at the center of the Earth. I lack the mathematical skills to derive an equation to graphically represent the results. My...
  19. C

    Shifting of the center of mass of a system

    Homework Statement A cubical block of ice of mass m and edge L is placed in a large tray of mass M. If the ice melts, how far does the center of mass of the system "ice plus tray" come down ? Homework EquationsThe Attempt at a Solution What I thought was that if no net force is acting on the...
  20. N

    Pressure at the center of an accelerating sphere

    Homework Statement Consider a glass sphere completely filled with water except for a small air bubble at 1 atmosphere. The sphere is accelerating to the right. What is the pressure at the center of the sphere? Let Pe = Atmospheric pressure r = radius of sphere pw = density of water a =...
  21. Ackbach

    MHB Simple Model of 9/11/2001 World Trade Center Collapse

    $\newcommand{\un}[1]{\,\text{#1}}$ https://www.youtube.com/watch?v=xSVHWiZu8NM One of my relatives sent me a link to a video (above) claiming that the WTC towers must have had internal demolition in order to fall as fast as they did. I think there are good reasons to oppose the theory, but I...
  22. N

    B Center of momentum and mass energy equivalence

    told so in another thread that two photons traveling in opposite directions "have mass". I understood the concept of centre of momentum, an inertial frame of reference and rest mass. My cynical reply wasn't conveyed and I got well thought out and accurate replies. the COM is an idealized point...
  23. E

    Angle between a Point on a Ellipse and its Center

    1. The problem statement, aall variables, and given/known data I need to find the angle between a point on the ellipse and the ellipse's center shown in the figure below: The known variables are 'd', the distance between point O (the center of the ellipse) and point A, and 'a', the angle...
  24. Daniel Petka

    Can violet lasers trick our eyes into seeing white?

    Violet itself is weird, because it lays at the other side of the spectrum than red, yet it can be easily mimicked with red and blue. The more powerful the violet light, the more it resembles white. My hypothesis: Violet tricks our eyes by activating not only blue, but also red and... with more...
  25. T

    Center of Mass Velocity in Multiple Dimmensions

    I was doing some practice problems to become more familiar with the Center of Mass Velocity and I came across this one from (Noted in the picture) Engineering at Illinois, that relates the velocity of the center of mass in both the x and y direction, however I don't quit understand the answer...
  26. J

    B How to debunk the idea we are the center of the universe

    My creationist grandpa says that we are at the center of the universe and that other galaxies are in spherical shells around us and claim that this means we are in a special place in the universe. Basically he believes in the "quantized redshifts" thing you may or may not have heard of, which...
  27. PhysicsKid0123

    Center of mass, Newton's third law, and rocket thrust

    Can someone clarify this for me-- how is the motion of the center of mass unaffected by internal forces, as is stated towards the end of the paragraph in the picture I've provided. It seems to me that motion of the center of mass directly affected by these internal forces. Specifically, the...
  28. SetepenSeth

    Electric field in the center of an arc

    Homework Statement Consider a conductor wire with a charge Q uniformly distributed, shaped in the form of an arc of radius R and amplitude 2A (were A is a given number between 0 and π). Find the value of the electric field in the center of the arc. Homework Equations ##E(P)= \int K_e...
  29. I

    Center of Mass (Subtraction over addition)

    Homework Statement Find the coordinates for the center of mass in the picture above. Plate is uniform.Homework Equations Since it's uniform, the lengths can be converted to the mass. So: coord x =(M1X1 +M2X2 + M3X3 + ... +MiXi)/(M1 +M2 + M3 + ... +Mi) coord y =(M1Y1 +M2Y2 + M3Y3 + ...
  30. J

    Off center fed dipole input impedance vs height above ground

    I have a question for the community. I am doing a further study and trying to locate any information regarding input impedance of an off center fed dipole at different heights above ground. A simple dipole targets 73.1 ohms for 1 wave length or higher above ground. I am not sure if the same...
  31. U

    Find the Radius and Center of a Sphere, Quadric Surfaces

    Homework Statement [/B] Find the radius and center of sphere ρ = 28 cos ϕ. Homework Equations Relevant equations would be the spherical and rectangular coordinate equations. The Attempt at a Solution I started off by multiplying both sides of the equation by ρ to get ρ^2 = 28 ρ cosϕ Then...
  32. MrDickinson

    Question about torque and center of mass

    Supposing that we have a hollow sphere of mass m and radius R. Furthermore, suppose that we attach the sphere to a rod of length L and mass M. Now, suppose we rest the rod-sphere contraption on a pyramid a distance (X+R) from the center of mass of the sphere and a distance [(L/2)-X)] from the...
  33. R

    B How to track the center of mass of a human in a video

    So a few months or years ago I learned that most of the energy from swinging comes from raising the center of mass at the lowest point of a swing. I also recently found a method of swinging that maximizes this energy gain. You squat on the swing, and stand up every time it gets to the lowest...
  34. K

    Acceleration of center of mass

    Homework Statement Three identical rectangular blocks are at rest on a level, frictionless surface. Forces of equal magnitude that act in the same direction are exerted on each of the three blocks. Each force is exerted at a different point on the block (indicated by the symbol "x"), as shown...
  35. F

    Force acting on the center of mass of a rolling disk

    Hello. The following situation I thought out confuses me so I am wondering where my mistake lies. A uniform disk of mass M and radius R sits on its edge. A string is attached to the highest point and pulled with a Force F in the x direction. The moment of inertia of the disk is MR^2/2 making...
  36. K

    B Does the plasma at the center of a star act like a solid?

    I was passing time by Googling the properties of the sun (temperature, mass, etc) and got to wondering what lies at the center of stars. I found out its plasma under roughly 340 billion atmospheres of pressure. I realize Boyles law may not be strictly meant for plasmas, but using it I...
  37. harpazo

    MHB Mass, Center of Mass, Triangle

    Find the mass and center of mass given the following region R. R: triangle with vertices (0,0), (b/2, h), (b,0) Let p = rho p = k The letters b and h as given in the points make it hard to find the equation of the lines needed for the inner upper and lower limits. I have worked this out...
  38. harpazo

    MHB Find Mass & Center of Mass

    Find the mass and center of mass of the lamina bounded by the graphs of the equations for the indicated density. y = sqrt{a^2 - x^2}, y = 0 Rho = k 1. How do I find the inner and outer limits of integration? 2. Is k the density function? 3. Is the letter a in the radicand constant? 4. Is...
  39. harpazo

    MHB Mass and Center of Mass for a Lamina with Variable Density and Given Points

    Find the mass and center of mass of the lamina for the indicated density. R: (0, 0), (a, 0), (0, b), (a, b) ρ = k I know the formula to find the mass and center of mass. My problem is twofold: 1. Finding the limits of integration for the inner and outer integrals considering the fact that...
  40. harpazo

    MHB What is the center of mass and how is it related to balancing an object?

    In simple terms, what exactly is the center of mass as taught in Calculus 3?
  41. W

    Torque with 2D center of mass

    Homework Statement A cube of mass M rests tilted against the wall as shown (see below). There is no friction between the wall and the cube, but the friction between the cube and the floor is just sufficient to keep the cube from slipping. When ##0\lt\theta\lt 45^\circ## find the minimum...
  42. jophys

    Center of mass oblique collisions

    Hi, in terms of oblique collisions, COM frames and laboratory frames, is this statements ALWAYS true? "The scattering angle of the first particle in the laboratory frame is half of the scattering angle in the center of mass frame." Please elaborate if you know, thanks in advance!
  43. I

    Acceleration of center of mass

    Homework Statement "Represent (linear) acceleration of center of mass of link OA in terms of variables shown." OmegaOA is given to be some constant value, I have assigned it 'omegaOA'. Homework Equations Already derived equations to previous parts of the problem, fairly certain these are...
  44. I

    Acceleration of Center of Mass

    I have a model of two slender rods (call them OA and AB) connected with a pin at A, anchored at O with a pin, and attached to horizontal slider at B. I am supposed to show graphs of different variables in Matlab as the angle theta changes. One of these steps is to show the acceleration of the...
  45. E

    Instantaneous center of rotation

    <Mentor's note: ^Moved from a technical forum and therefore no template.> at the instant shown during deceleration, the velocity of the tire is 40 ft/s to the right and the velocity of point A is 5ft/s to the right. locate the instantenous center of rotation. Can the instantaneous center of...
  46. Jewelz

    Moment of Inertia of a Spinning Record with a Hole in Center

    Homework Statement A 45 RPM record is a disk with a wide hole in the center. The mass of such record is 45g. It is 7 inches in diameter, and the hole in the center has a 1.5 inch diameter. (Sorry for the odd units, this was the way it was given). Find the moment of inertia of the 45 RPM...
  47. S

    Angular momentum and when center of rotation is changed

    Hello. The problem is this, what happens to angular momentum, tangential velocity and centripetal force when you change the center of rotation. For example, if we have rotating hinged arm, weight at the end, with certain angular momentum and tangential speed etc. which then gets stopped at...
  48. M

    MHB Find Center and Radius of Circle

    Determine the center and the radius of circle. x^2 + y^2 - 10x + 2y + 17 = 0 Can this be done using completing the square?
  49. nomadreid

    I Clusters at center and periphery

    An acquaintance has told me that the Sloan Digital Sky Survey found clusters of galaxies in which most of the galaxies were divided between the center and the periphery, with a smaller density of galaxies in between. Is this case, and if so, why?
  50. R

    Center of mass on plank problem

    Problem A 60kg woman walking at a speed of 1.0 m/s steps onto a long plank that has a mass of 60kg. Upon stepping on the plank, both she and the plank begin to slide with speed v and spin with angular rate ω on a frictionless surface. 1. How far from the woman is the center of mass of the woman...
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