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. D

    Center of Mass involving three cubes

    Homework Statement Three cubes, of side l, 2l, and 3lo, are placed next to one another (in contact) with their centers along a straight line and the l = 2lo cube in the center. What is the position, along this line, of the CM of this system? Assume the cubes are made of the same uniform...
  2. V

    How Does the Dog's Movement Affect Its Distance from the Shore?

    Homework Statement A dog with mass 11 kg stands still in a boat with mass 45 kg. the dog is 20 meters from the shore. It then walks 8 meters in the boat towards the shore. how far is the dog from the hore aftter this? Homework Equations center of mass equation The Attempt at a...
  3. P

    Where Is the Center of Mass in a Right Triangle?

    Homework Statement In the figure, a 60 cm length of uniform wire, of 60 g mass, is bent into a right triangle. The x- and y-coordinates of the center of mass, in cm, are closest to... I am going to explain the picture in my textbook. The origin is at (0,0). It goes up 10 cm and to the...
  4. I

    Center of mass: mass's on a right triangle system

    A system consists of three particles located at the corners of a right triangle as shown below. Mass 1 is 7.0 kg, mass 2 is 6.4 kg, and mass 3 is 9.9 kg. If the distances a equals 15.8 m and b equals 2.0 m, then what is the center of mass (in m) in the vertical direction? Y center of...
  5. I

    Center of mass: 2 objects in triangle system

    Three objects of different masses are located at the corners of the triangle shown in the picture. If distance a equals 4.21 m, b equals 1.55 m, and c equals 11.93 m, then what is the center of the mass (in m) of the system in the horizontal with respect to the origin? X center of...
  6. X

    Finding the Center of Mass of a Meter Stick with Variable Mass Density

    Homework Statement A meter stick has constant thickness and width , but the material that the stick is constructed from is very strange ... it has a variable mass density that is given by, ρ(x) = 0.800(1 + 0.00250x) grams/cm3 where x is measured in cm. Find the center of mass of the meter...
  7. E

    Medium level calculation of center of mass

    Homework Statement ~ Homework Equations integration The Attempt at a Solution for a) i use linear density and calculate the mass of circumference and integrate it bottom to top. is it riight?any other methods? for b) i use surface density and calcuate the mass of disk and...
  8. E

    Calculation of center of mass difficult

    Homework Statement ~ Homework Equations integration The Attempt at a Solution a) i use surface density and calculate the mass of a disk and integrate it from bottom to top. Is it right? But the cone is hollow so i only need to use the mass of the circuference? b) use "negative...
  9. B

    Acceleration of Cylinder's Center of Mass (Due 9AM)

    Homework Statement A 4.02 kg hollow cylinder with inner radius 0.19 m and outer radius 0.42 m rolls with- out slipping when it is pulled by a horizontal string with a force of 47 N, as shown in the diagram below. What is the acceleration of the cylinder’s center of mass? Its moment of inertia...
  10. V

    Finding center of mass of a right triangle

    Homework Statement Find the center of mass for the following triange: http://www.ehow.com/how_5132033_calculate-hypotenuse.html above is a picture of a triangle, it is a right triangle the side labeled a in the picture is a length of 1 and side b is length of 2, the total weight of...
  11. C

    Center of Mass of Truck with weights added

    distance between front and rear wheels of the truck is 3.50m. total upward force on front wheels is 12300N and rear wheels is 9180N. (I have already calculated that the center of mass is 1.28 m behind the front wheels). Now a 4850N object is added to the tailgate 4.23m from the front wheels...
  12. I

    Why Is My Center of Mass Calculation for a Uniform Square Offset by Width/4?

    Homework Statement To get used to finding the center of mass of an object, I have decided to start with a uniform density square. The square is centered at the origin. The center of mass should be at the center of the square, and thus at the origin. When I tried to solve this however, my...
  13. H

    Calculate the moments Mx, My and the center of mass

    Homework Statement 2. The attempt at a solution I'm working on #49 ONLY. I know the top line equation is f(x) = x but I need to find the g(x), so how can I find with the point is given (4,3). Please help!
  14. J

    How to Calculate the Center of Mass of a Semicircular Disk from an Arc?

    i have calculated the C.M. of semi-circular arc of radius r and mass m. How can i use this answer to calculate the C.M of semi-circular disk of radius r and mass m? thanks:) ps. how about the converse?
  15. G

    Flat plate, circular hole center of mass

    Homework Statement A uniform flat plate of metal with a circular hole, where it is a rectangle starting from -6 to 7 from the length, and the height is -3 to 3. The circle has a radius of 2, and it starts at 1 to 5. Calculate the x-coordinate of the center of mass xcm of the metal plate...
  16. G

    Magnitude of Momentum, relative to the center of mass

    Homework Statement A 4.39 g particle is moving at 4.15 m/s toward a stationary 9.64 g particle. (a)With what speed does the heavier particle approach the center of mass of the two particles? Answer in units of m/s. (b) What is the magnitude of the momentum of the lighter particle...
  17. H

    Moment of Inertia of an object rotating about its center of mass?

    Moment of Inertia of an object rotating about its center of mass?? Homework Statement An object is formed by attaching a uniform, thin rod with a mass of m = 6 kg and length L = 4 m to a uniform sphere with mass M = 30 kg and radius R = 1 m. What is the moment of inertia of the object about...
  18. J

    Physics- center of mass problem ?

    Homework Statement The coordinates of the center of mass for the extended object shown in the figure are (L/4, −L/5). What are the coordinates of m1? (Assume m1 = 4 kg, m2 = 2 kg, and m3 = 3 kg. Use any variable or symbol stated above as necessary.) Here is the diagram ...
  19. F

    Physics Problem Center Of mass HELP

    Homework Statement The problem is asking to find the Acceleration vector before jack reaches the less steep section. the Ans from back of book says : (3.95)i + ( -2.29 ) j My answer does not match at all : ANY HELP PLZZZZ anyone know how to get this ? Two skiers, Annie and Jack...
  20. U

    How Far Does the Canoe Recoil When I Walk from One End to the Other?

    I am floating in a canoe in still water. I carefully walk from one end of the canoe to the other. If my mass is M and the canoe's is Mc, how far does the canoe recoil? I've solved more complicated problems but this one confuses me. Isn't recoil the velocity of a body after collision? I would...
  21. T

    What Defines the Total Momentum in the Center of Mass Frame?

    Homework Statement A key concept about the center-of-mass frame is that A. the total momentum in the center of mass frame is always zero. B. both the initial and final momentum vectors form pairs of equal magnitude and opposite direction C. this frame always moves at the same velocity...
  22. A

    Locating center of mass for hollowed out sphere

    Homework Statement A uniform sphere of radius R has a spherical hole of radius r removed. The center of the hole is at a distance d from the center of the original sphere. Locate the CM relative to the center of the sphere. Homework Equations The Attempt at a Solution Ok, so...
  23. M

    How Does the Position of the Man Change as He Walks to the Edge of the Boat?

    Ok this one is messing me up. I already got number one (1.28). The question now is: I thought it would just be simple-the distance across the boat. But apparently as the man moves, so does the boat. This is confusing me. How do I begin to solve this one?
  24. M

    What is the Correct Center of Mass Calculation for a Man and a Boat?

    This should be easy enough, but the fact that the guy is standing at x=0 is confusing me. The center of mass should be m[guy]*x[guy]+m[boat]*x[boat] / total mass But since x[guy] is zero, then its just 162*3.4/(163+53) which comes out to 2.55, which is incorrect, since it has to be less than...
  25. 2

    Seesaw physics and center of mass

    I want to write a program that basically mimics the physics of seesaws. Part of a project I'm working on. I understand how equilibrium works and everything, but let's say that I have a mass on each end of the seesaw, and both masses are equal distance from the pivot. If both masses have the...
  26. M

    Closed System and Center of Mass

    Hello, Can a closed system change the position of its center of mass if no external force is exerted on it?
  27. N

    Rotation about the Center of Mass

    Why is it that when an object moves both transitionally and rotationally the center of mass follows the path as if there was no rotational energy? My theory is that objects naturally rotate around the center of mass and since the center of mass is a point, rotation doesn't affect its...
  28. M

    Where is the Center of Mass of Stacked Pizza Boxes?

    Homework Statement im trying to find the center of mass of two identical strong square pizza boxes of dimension one inch deep by 18 inches wide one is placed on top of the other with the edge of the top box in line with the center of mass of the bottom box , Homework Equations n/a ...
  29. C

    Potential energy in center of mass frame.

    Homework Statement I am trying to find the total energy of a twobody system (two planets orbiting each other) in the center of mass frame. And I was so wondering how to think about this problem.. I know that the kinetic energy in the center of mass is just the sum of the kinetic energies...
  30. M

    Center of mass equation for an ellipsoid?

    Asuming a uniform distribution, how can I find the center of mass of a planet such as Earth?
  31. e2m2a

    The Acceleration of the Center of Mass and External Force

    Suppose I perform an experiment, set up as follows: There is a rotating body which rotates around a vertical axis. The axis is at one end of the rotating body (denoted as the rotator) so that the axis is not through the center of mass of the rotator. The vertical axis is attached to a second...
  32. S

    Center of mass and a bit vector calculus

    Hi guys, recently I am reading Kleppner's Mechanics about center of mass, well, I am always a fairly fast learner, but I really got stick here. you see, in the page 119 example 3.3 (in short, a rod with nonuniform density ) let Q(x) be the function density of location vector it said...
  33. S

    How Do You Calculate Net Torque Without Knowing the Center of Mass?

    In the following question, I'm unable to find the center of mass to do the Torque calculation: Two 59.0 N forces are applied in opposite directions to the 13.6 cm long figure shown. If the angle θ=37.0°, what is the net torque about the center of mass? Image...
  34. M

    Where is Earth's center of mass?

    As a first-order approximation I can simply calculate the location and radii of an ellipsoid that contains the center of mass, but that's not good enough. Doing a Google search for "center of mass earth" doesn't pull up anything.
  35. J

    Increase of velocity of center of mass of a certain system

    Will the velocity V2 be greater than or less than velocity V1 in the attached sketch? Yes and why or No and why not.
  36. M

    Find mass and center of mass of ice cream cone

    Homework Statement A toy manufacturer wants to create a toy ice cream cone by fitting a sphere of radius 4 cm inside a cone with a height of 8 cm and radius of the base of 3 cm. The base of the cone is concave, but the rest of the cone is solid plastic so that with the sphere attached...
  37. R

    How Do You Calculate the Center of Mass for a 2D Plate with Variable Density?

    Homework Statement Find the center of mass of the 2-dimensional plate which occupies the region inside the circle x^2 + y^2 = 2y, but outside the circle x^2 + y^2 = 1, and for which the density is inversely proportional to its distance from the origin. Homework Equations m =...
  38. F

    General question about center of mass

    I hope this is the right section. consider the CM of a uniform ring. Suspend that ring by light cables above the earth, by a crane maybe? Where is the rings center of mass now? If this system was in space (crane, cables, ring), the CM would be somewhere else right? If the ring was...
  39. C

    Conceptual center of mass question - No calculations

    Homework Statement In using a ladder where the wall is smooth but the floor has a coefficient Us, why is it necessary to make the angle large? Homework Equations The Attempt at a Solution I'm a little confused by this, because I would think that you would want to make the angle...
  40. P

    Gravity towards Center of Mass?

    Gravity towards Center of Mass?? Conceptual question: Does gravity always act from the center of mass, regardless of shape of source of field, for finite mass distributions?
  41. M

    Initial Acceleration of a Center of Mass

    Homework Statement A uniform rod of length 1.25 m is attached to a frictionless pivot at one end. It is released from rest from an angle θ = 23.0° above the horizontal. Find the magnitude of the initial acceleration of the rod's center of mass. I keep getting the wrong answer...
  42. S

    Calculating the Center of Mass of a Boat with 3 People

    Homework Statement Lets image the following a boat has three people in it A has a mass of 120lb B has a mass of 180lb C has a mass of 160lb and the boat has a mass of 120lb also distance from person A to the front of the boat is 2 feet and distance from person A to person B is...
  43. S

    Center of mass and Moment of Intertia

    Homework Statement The thickness of the object is 2 cm. The density of the material (pronounced row) is 7800. The angle (theta) is pi/6. The radius is .75 meters. What is the moment of inertia and the center of mass? Here is what the object looks like...
  44. O

    Center of mass: What would make the truck tipover?

    Homework Statement http://screencast.com/t/ZTk3NDUxNjU I need help with understanding number 18. Homework Equations Center of mase formula: m1x1+m2x2/(all masses) The Attempt at a Solution How do you tell if adding the weight will make the truck tipover. I calculated that...
  45. C

    Center of Mass and Surface Area (multivariable)

    Homework Statement 1. Verify the given moments of inertia and find the center of mass. Assume each lamina has a density of p=1. The problem gives a circle with a radius a. 2. Find the area of the surface of the portion of the sphere x^2 + y^2 + z^2 = 25 inside the cylinder x^2 + y^2 = 9...
  46. H

    Center of Mass via Scalar Line Integrals

    Homework Statement A thin wire has the shape of the first quadrant part of the circle with center at the origin and radius a. If the density function is rho(x,y)=kxy, find the mass and center of mass of the wire. Homework Equations My parametric equation of the circle was x=a*cos(t) and...
  47. P

    Find the center of mass of a lamina

    The boundary of a lamina consists of the semicircles y=\sqrt{1-x^2} and y=\sqrt{4-x^2} together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is proportional to its distance from the origin. I drew a graph that looks like...
  48. O

    How do you calculate Center of mass

    Homework Statement I'm confused about how to find the center of mass http://www.screencast.com/users/ntrinh3/folders/Jing/media/a9773d23-38f1-4d28-a6f8-3b681d42d4c2 and http://www.screencast.com/users/ntrinh3/folders/Jing/media/bfb40c40-0068-41e8-ac83-2146e1df3330 Homework...
  49. Y

    Falling chain center of mass dynamics

    Not sure if this is in the right subforum but: A chain of uniform mass density, length b, and mass M hangs in a loop from the ceiling (both its ends are adjacent to each other) At time t = 0, one end, end B is released. Find the tension in the chain at point A after end B has fallen a distance...
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