Center of mass Definition and 916 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. R

    Center of Mass in Line Following Robot

    Hi Guys, I just want to know the effect of varying center of mass in a vehicle, specifically in a line following robot, I am going to put load on it, thus center of mass will vary, then i will study how will the varying center of mass affect the response of the line following robot, can anyone...
  2. J

    Find the Y component of the Center of Mass

    Homework Statement Find the Y component of the Center of Mass http://img607.imageshack.us/img607/7122/83370796.png Homework Equations ∫(r*dm)/M The Attempt at a Solution I keep coming up with (2/3)B but i know since that there is more mass near the origin axis, it should be...
  3. Z

    How Is the General Equation for Center of Mass Derived?

    Center of Mass Derivation..Doubt :( Homework Statement Guys,I don't seem to understand how the general equation for Center of mass was derived.I mean,yeah,i get that in a rod,or other symmetric structures,it should lie at the point of symmetry more or less,How waz the general equation...
  4. C

    Center of mass of a disc with a hole

    there is a similar case here :https://www.physicsforums.com/showthread.php?t=296966 so I've tried it and what i did was to equate both individual moment of inertia about z axis (and later applied parallel axis theorm) with negative mass for hole. added both these and found out Ix = M/4...
  5. A

    Center of mass of composite body

    Homework Statement In the image below, calculate the center of mass (x-bar,y-bar) of the composite body. Homework Equations I've used all of the equations for area for the shapes needed (triangle, square, circle). The Attempt at a Solution I have attached my calculations and...
  6. B

    Energy and Center of Mass Problem

    Homework Statement Energy, Center of Mass: As a project your team is given the task of designing a space station consisting of four different habitats. Each habitat is an enclosed sphere containing all necessary life support and laboratory facilities. The masses of these habitats are 10 x 105...
  7. S

    Solving the Stick of the Gods: Mass, Center of Mass & Moments of Inertia

    Homework Statement So the problem starts out like this Stick of the Gods! You hold one end of a stick, but it has no other end. It simply extends into infinity. Its one-dimensional density distribution is given by: λ=(λinitial)times(e^(-x/L)) λ is the density The problem doesn't state what...
  8. A

    Center of Mass Calculation: x-bar, y-bar

    Homework Statement I need to find the center of mass of the given figure attached. I am given the density (7850 kg/m^3) and a thickness of 0.3 cm (not sure where this goes) Homework Equations Just some things I need to verify: When finding the x-bar of the function y=x3 using the equation...
  9. A

    How do I calculate center of mass using integrals and density?

    Just some things I need to verify: When finding the x-bar of the function y=x3 using the equation: \bar{x} = \frac{∫\tilde{x}dm}{∫dm}, Is my \tilde{x} going to be the x distance(which will just be "x") times x3? Also, will x3 be in the denominator just before "dm"? The same process...
  10. L

    Find Center of Mass for an open box Confused about steps for this problem

    Homework Statement An appliance box has a square base with sides of length L, and has rectangular sides with a height of 3L. The top of the box is missing. The box is made from cardboard of uniform thickness and density. What is the height of the center of mass of this open box, with respect...
  11. L

    Solid hemisphere center of mass in spherical coordinates

    Hello, I am struggling with what was supposed to be the simplest calc problem in spherical coordinates. I am trying to fid the center of mass of a solid hemisphere with a constant density, and I get a weird result. First, I compute the mass, then apply the center of mass formula. I divide...
  12. T

    Calculating new moment of inertia matrix with a new center of mass

    Homework Statement The total length of the composite body is 4.5 feet. Before the propellant is burned, the projectile weighs 23 lbf. After the propellant burns, the remaining projectile weighs 16 lbf. Before the propellant is burned, the mass center is located 2.6 feet from the projectile...
  13. S

    Having trouble with a center of mass problem

    Homework Statement See the attachment below. The solution is stated to be number 4 (option a). I am having trouble understanding this. It seems that a majority of the sculpture's mass (in terms of its y-component) is well below the vertical component of point 4, and hence should lower the CM...
  14. I

    Finding the Center of Mass of a Man-Boat System

    Homework Statement A man with mass m1 = 69.0 kg stands at the left end of a uniform boat with mass m2 = 163.0 kg and a length L = 2.8 m. Let the origin of our coordinate system be the man’s original location as shown in the drawing. Assume there is no friction or drag between the boat and...
  15. B

    Integration when transform to center of mass frame

    Hi, I am having some difficulty doing the integral ∫d^{3}v1d^{3}v2 | \overline{v1}-\overline{v2}|, where u1\leq|v1|,|v2|\lequ2, and \overline{v1} means vectors. It seems better to evaluate it in the center of mass frame, by substitution \overline{v1}+\overline{v2}=\overline{V}, and...
  16. I

    What is the displacement of the beam after the medicine ball is thrown?

    Homework Statement A person with mass m1 = 67.0 kg stands at the left end of a uniform beam with mass m2 = 91.0 kg and a length L = 3.3 m. Another person with mass m3 = 56.0 kg stands on the far right end of the beam and holds a medicine ball with mass m4 = 12.0 kg (assume that the medicine...
  17. F

    How Do You Calculate the Center of Mass of a Human Leg?

    Homework Statement To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 92.0cm long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a...
  18. V

    Center of Mass: xcom, ycom Calculations

    What are (a) the x coordinate and (b) the y coordinate of the center of mass for the uniform plate? Since the plate is uniform, we can split it up into three rectangular pieces, with the mass of each piece being proportional to its area and its center of mass being at its geometric center. We’ll...
  19. N

    Rotational problem another time (this time not the center of mass)

    Homework Statement A chimney (length L = 82.6 m, mass M = 2280 kg) cracks at the base, and topples. Assume: - the chimney behaves like a thin rod, and it does not break apart as it falls - only gravity (no friction) acts on the chimney as if falls - the bottom of the chimney tilts but...
  20. E

    What is the Center of Mass of an NH3 Molecule?

    Homework Statement NH3 (ammonia) that is shown in the picture. These three hydrogen-atoms is formed as triangle. Center of the triangle has a distance d = 0.940 from each hydrogen atom. The nitrogen atom located in the vertex of a pyramid in which the three hydrogens defines "the base"...
  21. Y

    Calculating Center of Mass: Xi & Yi for Statics

    Ive been given this table in a lecture (attached) and I'm not sure what they're using to fill in the columns of Xi an Yi, it looks like it should just be half of each section in Yi, but that is not the case in Xi, can anyone please explain this to me. Thanks =)
  22. B

    Finding Center of Mass of Nonuniform Rod: Integrating Over dm

    When looking at example 3 in this pdf: http://www.physics.isu.edu/~hackmart/centerofmass.pdf where it shows how to find the Center of Mass of a Nonuniform Rod. I was wondering when or what you have to know in order to integrate over dm rather than changing the variables in terms of dx.
  23. C

    How Is the Center of Mass Calculated for a Solid Cone?

    I am using the textbook called Classical Mechanics by John R. Taylor. Z = 1/M ∫ ρ z dV = ρ/M ∫ z dx dy dz On page 89, example 3.2, it says: "For any given z, the integral over x and y runs over a circle of radius r = Rz / h, giving a factor of πr2 = πR2z2 / h2." I wish the book would...
  24. M

    Need help finding center of mass

    A solid is formed by rotating the region bounded by the curve y=e^(-6x/2) and the x-axis between x=0 and x=1 , around the -axis. The volume of this solid is (pi/6)(1-e^(-6)). Assuming the solid has constant density, find the center masses of x and y. center of mass (x or y)=...
  25. M

    Application of force not acting through center of mass.

    Hello PF! I've got a lab on rigid body motion tomorrow, and I need help completing one of the prep questions: A rigid body is acted on by a force F through its center of mass, and also experiences a torque caused by a similar force F at radius R. At time t, what are the linear and angular...
  26. H

    Center of Mass of a Triangle (uniform)

    Homework Statement Where exactly is the center of mass of a triangle ? the file attached shows a diagram of a triangle which is equilateral, and the blue spots are the mid-points of each side. Homework Equations a distance = 2/3 from the vertex (along the median) The Attempt...
  27. R

    Center of mass velocity greater than 1c?

    center of mass velocity greater than 1c? Basically I have two electrons moving in opposite directions towards each other. One is moving 2.5x10^8 m/s and the other is 2.0x10^8 m/s and I'm trying to use the following equation for finding the center of mass velocity: V_cm = ((p1 +...
  28. D

    Center of Mass involving Dumbbell With Uneven Weigths

    Homework Statement You work in a plant that manufactures heavy dumbbells. Due to a manufacturing error, one side of a 100 kg dumbbell was found to be 60 kg (M1) while the other was only 40 kg (M2). The mass of the bar itself is negligible. The factory has provided you and your colleague...
  29. H

    Center of mass and center of area

    what is the difference between center of mass and center of area ?? i know that they are the same for an constant density body but why ?and why they are different for nonconst density body?? thank u ,,
  30. mnb96

    Question on center of mass (centroid)

    Hello, I was thinking of the following question: suppose we have a body in two-dimensional space and its density is described by the function f(x,y) (Note! in Cartesian coordinates!). The x-coordinate of the center of mass is given by C_x= \frac{\int\int xf(x,y)dxdy}{\int\int f(x,y)dxdy}...
  31. A

    Center of mass of two extended bodies

    Homework Statement Consider a system comprising two extended bodies, which have masses M1 and M2 and centers of mass at R1 and R2. Prove that the CM of the whole system is at [M1R1+M2R2] / [M1+M2 ] Homework Equations Definition of CM The Attempt at a Solution First I...
  32. H

    Center of mass of fluid in rotating cylinder

    Homework Statement A closed cylindrical canister with central axis coincident with the Z axis has a height H and a radius R. It is suspended by a rod coincident with the Y axis that passes through the canister, transecting its central axis at a height h above the bottom surface of the canister...
  33. M

    Center of mass of right triangle

    I've been reading that the center of mass of a right triangle - the coordinates of the COM, is (1/3b,1/3h)- I can't for the life of me figure out why this is. Is there some sort of clear proof I can take a look at? I don't really know what to integrate..
  34. A

    Rotation around center of mass question

    I had quite a few posts about this some weeks ago, but I am still not sure about it. My question is about why you can always view the motion of an object as a traslation of the cm and a rotation around it. It makes sense in the light of the cm being the point which moves as a point particle only...
  35. A

    Understanding the Center of Mass of a Spring: A Conceptual Explanation

    A week ago I posted a thread about my conceptual understanding of the center of mass of a body. I have however not yet gained the intuition that I want, so let me ask a question about the center of mass of a spring. Consider a spring which is elongated in outer space and left to oscillate. As...
  36. R

    Finding the Center of Mass for a Hemisphere and Right Cone

    Suppose there's a hemisphere of radius R (say) and a right cone of same radius R but ht. R/2 is scooped out of it then i have to find the center of mass of the remaining part. Here's how i approached... clearly by symmetry, Xcm = 0 Now, Let M be the mass of the hemisphere so...
  37. R

    Center of mass of a solid hemisphere (where's the error)?

    On finding the center of mass of a solid hemisphere i came up with some different result. Here's what i did... consider a small ring at a distance r from the center of the hemisphere and one more ring at a distance of r+dr from center of the ring. let, mass of the small element formed...
  38. R

    Solving Center of Mass Doubt: What is 'y' in Integration?

    A silly doubt regarding center of mass... As we know for bodies having continuous distribution of mass we can know their center of mass by the method of integration... like, Xcm = 1/M∫x.dm but what is x here? in many cases... like in finding the COM of a ring Xcm = 0 and Ycm = 2r/∏...
  39. K

    Rotational Dynamics about an axis parellel to the axis at the center of mass

    Homework Statement A uniform sign is nailed into a wall at its top two corners. The nail on the upper-right corner breaks, and the sign begins to rotate about the remaining nail in the upper-left corner. The mass of the sign is 69 kg, L (length) = 1.6 m, H (height) = 2.2 m, and the moment of...
  40. A

    Understanding Center of Mass: Properties & Physically Representing

    I don't feel I have a good understanding of what the center of mass of an object it, and what its properties are. I know it's the position of all mass elements weighted by their mass and divided by the total mass. I have learned that the center of mass moves as if it was only subject to...
  41. J

    Center of Mass constant with near-correct attempt

    Homework Statement A wagon wheel is made entirely of wood. Its components consist of a rim, 16 spokes, and a hub. The rim has mass 5.1 kg, outer radius 0.90 m, and inner radius 0.86 m. The hub is a solid cylinder with mass 3.1 kg and radius 0.12 m. The spokes are thin rods of mass 1.1 kg...
  42. phosgene

    Center of mass of infinite cylinder of air

    Homework Statement The density of air at height z above the Earth’s surface is proportional to e^(−az) , where a is a constant > 0. Find the centre of mass of an infinite cylinder of air above a small flat area on the Earth’s surface. Hint : Consider line density and the identities...
  43. A

    Understanding Why a Rod Doesn't Rotate Around Its Center of Mass

    Consider a free rod lying horizontically in the air. Gravity produces no torque around the center of mass. Now let the rod be attached to a hinge like of that one the picture. Now the hinge provides an upwards force on the left end of the rod, whilst there is a torque effectively acting at the...
  44. I

    What is the Center of Mass for a Donut and an Empty Box within a Box?

    1. FIgures in this link: ( a donut, and a Box with and empty box inside) http://postimage.org/image/smdut3cxb/ 2.Find the CoM of the two figures 3. as i attempted for the circle: Homework Statement Homework Equations The Attempt at a Solution
  45. F

    Center of Mass of a Solid Enclosed by a Spherical Coordinate Surface?

    Homework Statement Find the center of mass of a solid of density \delta = 1 enclosed by the spherical coordinate surface \rho = 1-cos\phi.Homework Equations The Attempt at a Solution I'm a bit confused about how to start here, mainly because the surface is defined by spherical coordinates...
  46. A

    Rigid body kinematics problem: finding the velocity of the center of mass

    I have to deal with the problem of finding an expression for the kinetic energy of a rigid body. One of its point is pivoted to a point that moves arbitrarily. So in order to find an expression for the kinetic energy I use König's theorem, but I need the velocity of the center of mass. I use...
  47. R

    Center of mass and rigidity and torsions

    I've been researching about this for hours at internet reading dozens of pdfs.. but can't seem to understand the concept. What does it mean if the center of mass and center of rigidity are not coincident, torsions would be produced.. can you give an example of it in more intuitive or using...
  48. L

    Having trouble figuring center of mass between Sun and Jupiter

    I am taking an astronomy class because I am interested in it and wanted to know more. I love all I am learning unfortunately my math skills are holding me back from getting all that I can out of this class. I realize this might not be the toughest math problem but can someone explain help me out...
  49. X

    Applying forces to a body, outside it's center of mass.

    Hello. I'm currently coding on a physics engine. To do so, I looked up some code from open-source engines. I noted that there is a function where you can apply a force to a rigid body, outside it's center of mass. It goes something like this: ApplyForceAPosition(vector force, vector...
  50. S

    Center of mass of half square without a half circle

    Homework Statement Find the position of the center of mass for a thin sheet and homogeneous, with sides R and 2R ,from which has been subtracted a half circle of radius R. [Xcm=(2/3)*R*(4-pi)]Homework Equations Rcm=(1/M)*∫rdm The Attempt at a Solution By symmetry we know Ycm=0. For de...
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