What is Center of mass: Definition and 914 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. T

    Center of mass of a disk with a hole

    Homework Statement We have a circle of radius 4 with center at the origin of a referential, and a circle hole in it of radius 1 and center at (-2,0). We're supposed to calculate the center of mass. 2.MY QUESTION. I know, by the usual formulas for calculating center of mass I get 2/15 as the x...
  2. Hamza Qayyum

    I Force applied to a sphere off center of mass

    I'm trying to model a sphere having force applied at position P in the following diagram: I know that this applied force will have an x and y component; the y component will propel it upwards, but what I am confused about is the x component of the force. I know that the x component will propel...
  3. karush

    MHB 244.15.6.1 center of mass

    $\textsf{Find the center of mass of a thin plate of density}$ $\textsf{ $\delta=3$ bounded by the lines $x=0, y=x$, and the parabola $y=2-x^2$ in the $Q1$}$ $\begin{array}{llcr}\displaystyle &\textit{Mass}\\ &&\displaystyle M=\iint\limits_{R}\delta \, dA\\ &\textit{First Moments}\\...
  4. Hiero

    I Center of Mass Motion: Velocity & Momentum

    If we have a system of masses in motion, will the velocity of the center of mass always be given by the net momentum divided by (1/c^2 times) the total energy of the system?
  5. J

    Center of mass of a uniform wire

    Homework Statement Homework EquationsThe Attempt at a Solution I am having trouble understanding the setup as well as how to approach this problem . I am not understanding the orientation of the two pieces of the wire . They make angle 60° with each other , but then what does it mean when...
  6. Isaac0427

    Why center of mass? Why not center of charge?

    Hi, In electrostatics, we have Coulomb's law for the electric field of a point charge. We can generalize to the electric field of a continuous charge distribution, using $$E=k\int r^{-2}dq.$$ It would seem, using the same argument that gets us to the previous equation, we could define the...
  7. Julian Erickson

    Translational + Angular Acceleration of Free Body (not fixed)

    Imagine a long brick in outer space. You apply a force tangential to the center of mass. The brick accelerates in a transitional and angular fashion. There are no constraints or fixed axis. How would I calculate the translational and angular acceleration? I would like to run some simulations...
  8. X

    Velocity of the center of mass for MD simulations

    When calculating the center of mass for an MD simulation, I've seen a few codes that subtract the average velocity from each atoms velocity. I think the reasoning behind this is that you want the velocity of center of mass for the simulation to be zero, which I understand, but I don't understand...
  9. Likith D

    The solution of a problem regarding center of mass

    A rod of mass M (=4m) and length L is placed over a smooth horizontal table. One end of rod is attached with a block of mass m through a light string as shown in the figure. The pulley is friction less and light. the acceleration of com of rod at the given instant is (g = acceleration due to...
  10. A

    Computing angular speed wrt CM I get a contradiction

    Hello! I have been brushing up my Rigid Body Dynamics. I tried computing the angular speed with respect the Center of Mass (CM) using the usual split of kinetic energy and also the split of Angular momentum using the CM. First, a simple case: Two particles of mass M each separated by a distance...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. 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 + ...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. R

    B Physics Behind Weebles: Can You Build One?

    Weebles wobble but they don't fall. I came to know that they wobble and come back to position since the center of mass lies close to the bottom. Can I apply the formula for center of mass and do a human weeble as in In that case if I consider a hemisphere of radius 60 cm and the average...
  29. 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...
  30. 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...
  31. 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...
  32. 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?
  33. 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...
  34. 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!
  35. 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...
  36. 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...
  37. 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...
  38. Seth Newman

    Changing the Seasons by Shifting Center of Mass

    Homework Statement On Earth, seasons occur due to the tilt of our planet. In the SSDS (A Dyson Sphere built around our sun), we can cause seasons to occur by having the Dyson sphere be not perfectly centered around the sun. If you want the sun to be 10% closer to the one side of the Dyson...
  39. V

    Understanding the Center of Mass Energy in Fixed Target vs. Collider Experiments

    Homework Statement While not explicitly a homework question, I am having some trouble with deriving expressions for the center of mass energy in a fixed target experiment versus a collider experiment. The question is effectively, "Derive an expression for the center of mass energy in a fixed...
  40. J

    Finding the center of mass of an arbitrary uniform triangle

    Homework Statement 1. Show that for an arbitrary uniform triangle ABC, with A at (x1, y1), B at (x2,y2), C at (x3, y3), the CM (xcm, ycm), is simply defined by xcm=(x1+x2+x3)/3, and ycm =(y1+y2+y3)/3 Homework Equations xcm = 1/M * ∫xdm ycm = 1/M * ∫ydm M = ∫dm = ∫δdA where δ = M/A = dm/dA...
  41. Y

    Potential in center of mass for Hydrogen atom

    Homework Statement A Hydrogen atom is interacting with an EM plane wave with vector potential $$\bar A(r,t)=A_0\hat e e^{i(\bar k \cdot \bar r -\omega t)} + c.c.$$ The perurbation to the Hamiltonian can be written considering the proton and electron separately as...
  42. P

    B Is Center of Mass a vector or scalar quantity?

    I am a little bit confused on Center of Mass and Center of Gravity about what are they vectors or scalars . As I may think they are neither because they are simply two points . Am I saying right or wrong .
  43. A

    How Does Splitting a Mass in Midair Affect the Center of Mass's Trajectory?

    Hi all, I've been stuck on thinking about a situation where a mass splits in two in midair. The mass originally undergoes a projectile motion with no air resistance, and it splits in two in midair but one of the piece reaches the ground before the other. In this case, I know that the trajectory...
  44. RavenBlackwolf

    Center of Mass and Moment of Inertia

    Homework Statement A small pine tree has a mass of 15kg. Its center of mass is located at .72m from the ground. Its trunk is sawed through at ground level, causing the tree to fall, with the severed trunk acting as the pivot point. At the instant the falling tree makes a 17° angle with the...
  45. Delta2

    Rigid body with center of rotation other than CoM

    Suppose we have a rigid body that rotates around a point P other than its center of mass, with point P being a point of the rigid body. This implies that there are external forces to the rigid body. If the external forces cease to exist(like we have an axis passing through P and we suddenly...
  46. toforfiltum

    Find center of mass of planar quadrilateral

    Homework Statement Consider the planar quadrilateral with vertices (0, 0), (2, 0), (1, 1) and (0, 1). Suppose that it has constant density. What is its center of mass? Homework EquationsThe Attempt at a Solution Since it has constant density, could I assume that the center of mass would be the...
  47. E

    What is the kinetic energy of the center of mass

    Homework Statement Two 0.50-kg carts are pushed toward each other from starting positions at either end of a 6.0-m low-friction track. Each cart is pushed with a force of 2.9 N , and that force is exerted for a distance of 1.0 m. What is the kinetic energy of the center of mass of the system...
  48. merricksdad

    Find the center of mass of a cone with variable density....

    Homework Statement Find the center of mass of an inverted cone of height 1.5 m, if the cone's density at the point (x, y) is ρ(y)=y2 kg/m. Homework Equations The formula given for this problem is rcm=1/M * ∫rdm, where M is total mass, r is position, and m is mass. The Attempt at a Solution...
  49. 1

    Center of mass of a sphere with cavity removed

    Homework Statement A solid sphere of density ##ρ## and radius ##R## is centered at the origin. It has a spherical cavity in it that is of radius ##R/4## and which is centered at ##(R/2, 0, 0)##, i.e. a small sphere of material has been removed from the large sphere. What is the the center of...
  50. E

    Determine the location of the center of mass of spheres

    Homework Statement All three disks are made of sheet metal of the same material, and the diameters are 1.0 m , 2.0 m , and3.0 m . Assume that the x-axis has its origin at the left-most point of the left-most object and it points to the right. Part A) Determine the location of the center of mass...
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