Center Definition and 1000 Threads

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. 0

    Center (in y direction) of Mass of 3D pyramid

    Homework Statement A monument is made from stone blocks of density 3800kg/m^3. The monument is 15.7m high, 64.8m wide at the base, and 3.6m thick from front th back. How much work was required to build the monument? (Hint: find ycm). Homework Equations ycm = (1/M) * ∫ydm, M = mass total...
  2. B

    Rotation around center of mass

    I understand that the center of mass is a point which can be considered to contain all of an object's mass, for the purpose of calculations involving universal gravitation. I also understand that the center of mass of an object of uniform density is located at the centroid. In this case, I...
  3. JK423

    Coupling only to the center of mass degrees of freedom?

    I am interested in the following scenario. Suppose that you have a composite system, e.g. ion, atom or whatever you want. This system will have internal degrees of freedom (d.o.f.), i.e. relative positions and momenta, and also center of mass d.o.f..What i am looking for is to find an...
  4. ShayanJ

    Center of mass of a uniform sphere

    We know,by symmetry,that the center of mass of a uniform sphere is at its center.So we expect the formula r_{com}=\frac{\int r \rho d\tau}{\int \rho d\tau} to give us zero for this case.So let's see: r_{com}=\frac{\int_0^{R} \int_0^{\pi}\int_0^{2\pi} r^3 \sin{\theta} d\phi d\theta...
  5. icesalmon

    Velocity in Elastic Collision involving Center of Mass

    Homework Statement part #1: A wooden block with m1 = 2.9kg is sliding across a frictionless surface in the −x direction at 4.5 m/s. A smaller wooden block with m2 = 1.1 kg block is traveling in the +x direction at 2.9 m/s. part #2: Now they make a head-on elastic collision. (This means that in...
  6. N

    Center of mass with 2 people, a medicine ball, and a beam

    Homework Statement A person with mass m1 = 50 kg stands at the left end of a uniform beam with mass m2 = 95 kg and a length L = 2.4 m. Another person with mass m3 = 60 kg stands on the far right end of the beam and holds a medicine ball with mass m4 = 14 kg (assume that the medicine ball is at...
  7. C

    Potential at center of polyhedron = average of potential on surface

    Homework Statement A closed volume is bounded by conducting surfaces that are the n sides of a regular polyhedron (n = 4,6,8,...). The n surfaces are at different potentials Vi, i = 1,2,3,...,n. Prove that the potential at the center of the polyhedron is the average of the potential on the n...
  8. D

    What is the center of mass in a triangular beam?

    Hello,, I'm reading about beams in mechanics of matrials. But I'm a bit stuck because of one thing that I either don't understand or just don't comply with! So, here's a picture from the book: picture 003 It says that the center of mass should be at 2/3 of the length.. I tried to...
  9. B

    Find Center of Mass for Uniform Log: 2.33m, 94.9kg, 72.7kg, 26.1kg

    A uniform log of length 2.33 m has a mass of 94.9 kg and is floating in water. Standing on this log is a 72.7-kg man, located 18.9 cm from one end. On the other end is his daughter (m = 26.1 kg), standing 1.15 m from the end. Find center of mass [b]2. m1r1+m2r2/(m1+m2)=R [b]3. I used...
  10. M

    A disc of radius r is spinning about its center

    A disc of radius r is spinning about its center in the horizontal plane with a constant angular speed w (omega). A bug walks along the radius of the spinning disc traveling from the center of the disc toward the edge. The bug maintains a constant speed v relative to the disc. (In other words...
  11. V

    ChetFinding the Center of Mass of a Figure with a Circular Cutout

    Homework Statement http://img9.imageshack.us/img9/2866/siyx.png Uploaded with ImageShack.us Homework Equations Where is the COM of the figure? (width = 8, height = 8) The Attempt at a Solution area of triangle: 0.5(8*8) = 32 32 - (pi(0.52) - (2.5*1) - (2.5*0.5) = 27.4646 I...
  12. N

    Magnitude of magnetic field at the center of a semi-circle

    Homework Statement A very long wire carrying a current I = 5.0 A is bent into a semi-circle with a diameter of 10 cm as shown in the figure. What is the magnitude of the magnetic field at the point P, halfway between the ends of the straight sections of wire and at the center of curvature of...
  13. P

    Work to remove point charge from center of spherical conducting shell

    Homework Statement A point charge q is at the center of a spherical conducting shell of inner radius a and outer radius b. How much work would it take to remove the charge out to infinity? Homework Equations Potential, W = 1/2qV The Attempt at a Solution I am going at this in...
  14. S

    Torque as measured from center of mass.

    The angular momentum of a system of particles α = 1,2,3,...n can be written as the sum of the two angular momenta: \vec{L} = \vec{R} \times \vec{P} + \sum_{\alpha} \vec{r_{\alpha}}' x \vec{\rho_{\alpha}}' where the first term is the angular momentum of the center of mass with all mass M = Ʃ mα...
  15. B

    Center of Mass (Boxcar water example)

    A boxcar of length 8.00 m and height 2.40 m is at rest on frictionless rails. The empty boxcar has a mass of 3750 kg. Inside the boxcar, located at the left end, is a tank containing 2450 kg of water. The tank is 2.00 m long and 2.40 m tall. The tank starts to leak, and the water fills the...
  16. N

    Physics Center of Mass, driving me nuts

    Homework Statement A stone is dropped at t = 0 s. A second stone, with a mass 2.00 times that of the first, is dropped from the same point at t = 0.220 s. Assuming that neither stone has yet reached the ground, how far from the release point is the centre-of-mass of the two stones at t...
  17. J

    Light that is directed towards the center of a black hole

    If a beam of light is on a trajectory that is perfectly tangent to the event horizon of a black hole (or a little below tangent, allowing it to "dip" into the EH for a moment), is it possible that that photon could be pulled into a never ending orbit around the black hole? Or is the gravity too...
  18. P

    0N/C Electric Field at center of equilateral Triangle

    Homework Statement I got an answer as you can see in both pictures, but apparently it is wrong. What did I do? Homework Equations The Attempt at a Solution In this picture I found the y components of the electric field at C. Since the bottom 2 are the same i just doubled one of...
  19. D

    Star is expected to be torn apart in the center of our Galaxy

    Based on http://en.wikipedia.org/wiki/Sagittarius_A* So "mid to late 2013" - almost the time, are there any new observations yet?
  20. PhizKid

    Electric field on a charge some height above the center of a disc

    Homework Statement Given a disc of radius 'R' with a charge 'h' above the center of the disc, find the electric field on the charge.Homework Equations ##E = \frac{kQ}{d^2}##The Attempt at a Solution I can find the charge density from a ring then integrate over R: ##\sigma = \frac{Q}{A}##...
  21. B

    Proving the scalar matrices are the center of the matrix ring

    I read that scalar matrices are the center of the ring of matrices. How would I prove this? Tips are appreciated. It is already obvious that scalar matrices commute with all matrices, but the converse seems tricky. BiP
  22. PsychonautQQ

    Finding center of circle with Polar Coordinates

    Homework Statement r=7sin(∅) find the center of the circle in Cartesian coordinates and the radius of the circle The Attempt at a Solution My math teacher is impossible to understand >.< and then the stupid homework is online and crap blah this class but I REALLY want to understand the material...
  23. E

    Center of Mass Formula: Understanding the Intricacies

    Hello, I'm reading Mathematical Methods in the physical sciences by Mary Boas and in it, she defines the center of mass of a body in 3 dimensions \int \overline {x}dM=\int xdM \int \overline {y}dM=\int ydM \int \overline {z}dM=\int zdM In standard undergraduate textbooks...
  24. nearc

    Center of magnetic potential vs center of mass

    first i might be using magnetic potential incorrectly so please be nice when correcting me. i'm attempting to model the behavior of a magnetic compass when it is placed next to a ferrous piece of metal. basically the compass will point to the metal rather than north. my first simplification...
  25. P

    How Far Does m1 Travel Before Colliding with m2?

    Homework Statement Two small particles of mass m1 and mass m2 attract each other with a force that varies with the inverse cube of their separation. At time t0, m1 has velocity v directed towards m2, which is at rest a distance d away. At time t1, the particles collide. How far does m1...
  26. C

    Veritasium - Firing bullet in block - along center and away from cente

    This question is about this video on YouTube, in which a bullet is fired vertically into the center of a wooden block from below, sending the block up into the air. Next, a bullet is fired vertically but off-center into a similar block from below, again causing the block to rise into the air...
  27. F

    Time at the center of a black hole?

    Sorry if I am woefully uninformed, but I am really curious. What happens to time at the center of a black hole according to general relativity?
  28. P

    Binary star revolving arround center of mass

    why the binary star revolving around center of mass ? could type 2 supernova ever had a stellar wind after explosion ?
  29. A

    Finding Center of Mass of 800g Steel Plate Triangle

    Homework Statement An 800 g steel plate has the shape of the isosceles triangle shown in the figure. What are the x and y coordinates of the center of mass? https://www.physicsforums.com/attachment.php?attachmentid=13559&d=1208292919 Homework Equations x=1/M ∫ x dm [b]3. The...
  30. E

    Moment of Inertia through center of mass

    Homework Statement A rod is 10 ft long and has a density that goes from 4 to 24 lb/ft. a) Find the mass. b) Find the center of mass. c) Find the moment of inertia through the center of mass. Homework Equations I = Ml^2 I = I(cm) + Md^2 (parallel axis theorem) The Attempt at...
  31. PsychonautQQ

    Why Is Total Mass Used as the Denominator in the Center of Mass Equation?

    so for the center of mass of an object.. x = (m1x1 + m2x2 + ...) / M total y = (m1y1 + m2y2 + ...) / M total how come in this equation the denominator is the total mass of the system? It seems like it would make more sense to have the denominator the total mass in whatever component you...
  32. W

    Center of the galaxy and the great attractor

    I want to know about the center of our galaxy, and every other galaxy by the way, I heard that the condensed matter and light comes from a Super massive black hole that 'aspires' everything around it & I want to know if it's true ( If we have proved it ) or is it just a suposition, & by the way...
  33. B

    Finding The Coordinates of The Center Of Curvature

    Homework Statement Let C be a curve given by y = f(x). Let K be the curvature (K \ne 0) and let z = \frac{1+ f'(x_0)^2}{f''(x_0)}. Show that the coordinates ( \alpha , \beta ) of the center of curvature at P are ( \alpha , \beta ) = (x_0 -f'(x_0)z , y_0 + z) Homework Equations The...
  34. B

    What is the definition of center of mass?

    Hello, I am trying to describe the concept of center of mass, and need some help with particulars: The center of mass can be thought of the balancing point of a body; it is the point where mass is "evenly distributed" with respect to this point, such that if you were to place the body in a...
  35. B

    How do different forces affect the acceleration of a spinning uniform disk?

    Hello, can someone explain to me what is the difference between center mass acceleration and angular acceleration? If i have an uniform disk laying still, then i give it a force to make it spin while it stays put. The angular acceleration obviously has a number. What about the acceleration of...
  36. B

    Gravity moving toward the center of the Sun?

    Gravity moving toward the center of the Sun?? Ok can someone tell me if I am wrong in my understanding here (in a heated debate with a physics professor) : As you move from the surface of the sun to the center, does the force of gravity increase or decrease and why? The point I am trying to...
  37. H

    How does a motocylist move his center of mass?

    Hi all:we all know that a motorcyclist need to tilt his/her center of mass with the motorcycle when he is turning in order to balance the torque exerted on him.When he is tilted, gravity exert a torque on him that allows him to balance torque by centrifugal force about the point of contact with...
  38. A

    Magnetic Field at the center of a square

    Homework Statement Homework Equations not sure The Attempt at a Solution I am not sure how to start this problem.
  39. andyrk

    Center of Mass of Uniform Symmetric Bodies

    Why are centre of mass of all uniform symmetric bodies at there geometric centre? We know the following result: "Consider a system of point masses m1,m2,m3... located at co-ordinates (x1,y1,z1), (x2,y2,z2)...respectively. The centre of mass of this system of masses is a point whose co-ordinates...
  40. M

    Gravitational center of the Local SuperCluster

    I spent days trying to find this information on the internet, but i think i will never find it, so i came here looking someone who knows it. The Mystery is : Where is the Gravitational center of the Local SuperCluster (Also known as Virgo SuperCluster) ? Some Usefull information that i...
  41. L

    Compare Velocity of Center of Mass - Two Hollow Cylinders

    Homework Statement Homework Equations mgh=\frac{1}{2}mv^2+\frac{1}{2}Iω^2 The Attempt at a Solution I used conservation of energy, and found that the final velocity of the center of mass is independent of the object's mass, so in both cases: v=\sqrt{gh} . Therefore, the ratio is 1. However...
  42. D

    Center of Mass: Definition, 3D Calculation & Frame

    What exactly is center of mass, and how do I calculate in three dimensions? Also, I need to know how to calculate it in three dimensions at relativistically high speeds? Also, what do they mean by 'center of mass frame', as in the question "What are the energy and momentum in the center of...
  43. S

    Type of curvature of gradient force from edge to center of a sphere

    I was doing some simple physics with a ball resting on a table and I made this curve (0,0) (25, 6.8) (50, 27.51) (75, 63.4) (100, 112.34) (125, 175.7) (150, 253.3) (175, 345.4) I was wondering if anyone could identify what sort of curve it is? I am just curious. This is not a homework...
  44. C

    News Turkish crowds riot over proposed shopping center

    Nowadays, Turkish police use water cannons and tear gas to disperse protesters. Local media is almost completely silent to this action. Thousands of people just using their democratic rights of protesting the destruction of a park in Taksim square. However, they are experiencing barbarous...
  45. F

    Instantaneous Center of Rotation for a Differential Drive Robot

    Hi, I want to find the instantaneous center of rotation of a differential drive robot. Assuming I know that the robot will travel with a particular linear and angular velocity (v,w) I can use the equations (given here http://rossum.sourceforge.net/papers/CalculationsForRobotics/CirclePath.htm...
  46. E

    Rotation induced by moving center of mass

    Two identical uniform beams are connected together by a set of servo motors. When the motors activate the two beams will come together uniformly and the center of mass of the system will move outward along the arrow shown. I would expect that the torque exerted by the motors would only cause the...
  47. J

    Finding the center of mass of an incomplete circle

    Homework Statement Locate the x coordinate of the center of mass of the homogeneous rod bent into the shape of a circular arc. Take r = 170 . The arc goes from (-5/6) to (5/6)pi (counterclockwise). It has a radius of 170mm.Homework Equations x=rcosθ, y=rsinθ, dL=r*dθThe Attempt at a Solution...
  48. J

    Fidning the center of mass of an incomplete circle

    Homework Statement Locate the x coordinate of the center of mass of the homogeneous rod bent into the shape of a circular arc. Take r = 170 . The arc goes from (-5/6) to (5/6)pi (counterclockwise). It has a radius of 170mm.Homework Equations x=rcosθ, y=rsinθ, dL=r*dθThe Attempt at a Solution...
  49. M

    Instantaneous Center of Velocity

    Hello, I am trying to find the angular velocity of a link in a 4 link mechanism problem but I have reached a scenario which is new to me. Please look at the attached image. I already know the velocity of the slider which is 3m/s. I have indicated in the diagram a line parallel to AB to indicate...
  50. J

    Center of mass of a slender rod with variable density

    Hi to everybody Homework Statement I´ll show the problem with a picture: Homework Equations L\overline{x}=∫xc ρ dl The Attempt at a Solution Well the total length of the rod is 1 feet, I only need to calculate the integral. The moment xc of a differential element of mass of the rod is...
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