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. M

    Adjusting Center of Mass for Equal Weight Objects

    [The subject should say "adjusting center of mass"] Hello, I am trying to make a machine that "pairs" two object's center of mass and weight. I was trying to figure this out myself but with barely primary school completed, I obviously could not figure it out. Any help is very much appreciated...
  2. S

    Solving for Center of Mass in a Bent Wire: xcm and ycm Values Explained

    A wire of uniform density is bent into the shape shown in the figure below. The value of L is 0.740 cm. Answer the following questions assuming that the origin is at the bottom left corner. What is the value of xcm? cm What is the value of ycm? cm Attempt xcm = 2*.74 = 1.48...
  3. M

    Finding Center of mass and Moment of inertia Experimentally.

    Hi, I am doing a science experiment. For that, I have made a small structure (sort of, almost pyramid like) using metal strips. I have calculated mathematically: its center of mass/gravity, moments of inertia matrix and principle MI axis ( usual formulae). My job first is to find these same...
  4. K

    Having trouble finding the center of mass for multiple objects

    Homework Statement Find the x-coordinate of the center of mass of the composite object shown in the figure. The sphere, cylinder, and rectangular solid all have a uniform composition. Their masses and dimensions are: sphere: 200 g, diameter = 10 cm; cylinder: 450 g, length = 11 cm, radius =...
  5. P

    Find the center of mass with tripl-integral

    A being is on a sphere like planet. Sudently hey grabs a GPS and starts diging straight on a diemeter, heading to the center. We can theoretically say tha he is carrying a level with him. Then tha sphere is theoretically cut in two section , A and B. I have(well...with help) found the volume of...
  6. T

    Momentof inertia and center of mass

    momentof inertia and center of mass! in book of serway : he says moment of inertia of a body is m(r^2).---->(1) is mass of the body and r is the distance of the body. but for a rigid body we will divide into particles of very small masses so i = E(Mi * (Ri)^2) E() is submission function to...
  7. edpell

    Center of mass of the universe

    Isn't the inertial frame defined by the center of mass of the universe somehow special? To get to any other frame an object must experience a force and be accelerated to a new velocity. In this case the CMB shows us the inertial frame of the CM of the universe. I understand that velocity is...
  8. Y

    Conservation of linear momentum & center of mass

    Homework Statement Two cars are driving along the road at constant speeds V1, V2. At t=0 they begin to accelerate with constant acceleration a1, a2. Under what circumstances is the center of mass frame of reference inertial?Homework Equations Fext=0 ==> change in momentum=0The Attempt at a...
  9. M

    Conservation of linear or angular momentum + center of mass.

    So I have been reading about conservation of linear momentum and conservation of angular momentum of a rotating object. For example a thin rod resting horizontally on the a flat surface. If this rod is being hit by a small object(collision), the rod will rotate about it's center of mass and will...
  10. A

    Center of Mass and Mass Moment of Inertia

    Homework Statement Using masses, springs a stop watch and ruler, determine the location of the center of mass and mass moment of inertia about an axis passing through the center of mass of a bicycle crank. The Attempt at a Solution I've been staring at this problem for a while now...
  11. X

    Find Center of Mass: Calculate x,y Coordinates with R = sum(m*r)/sum(m)

    1. The density of part A (p)(kg/m^3) and in part B (p/2)(kg/m^3). Find the coordinate (x,y) of the center of mass of this object. (consider a piece of the object that is one meter in the z direction) 2. R = sum(m*r) / sum(m)...maybe? I am completely stumped on how to start this problem my...
  12. B

    Understanding Center of Mass: Common Misconceptions Explained

    so I am not too sure how to answer a question... ill word it the best i can: While doing an experiment where we use a Plumb Line and paper object and find the center of mass by letting the string (with mass attached to other end) hand and draw lines that eventually intersect at on specific...
  13. G

    Finding Center of Mass and Velocity of Three Moving Balls

    Three Balls are moving along a straight line having the instantaneous positions shown in figure 8.15. At that instant, find the location and velocity of the center of mass of this system. Figure 8.15 Mass A---------------------Mass B-----------------------------------Mass C...
  14. T

    Translational Motion and Center of Mass

    Homework Statement A 280-kg flatcar 25m long is moving with a speed of 6.0m/s along horizontal frictionless rails. A 95-kg worker starts walking from one end of the car to the other in the direction of motion, with speed 2.0m/s with respect to the car. In the time it takes for him to reach the...
  15. S

    Center of mass for uniform solid

    Homework Statement A uniform solid consist of a hemisphere of radius r and a right circular cone of base radius r fixed together so that their planes are coincident. If the solid can rest in equilibrium with any point of the curved surface of the hemisphere in contact with a horizontal plane...
  16. D

    When Can You Pretend All the Mass is at the Center of Mass?

    Homework Statement Discuss when you can and can’t “pretend all the mass of an object is concentrated at the center of mass”. Homework Equations The Attempt at a Solution So I don't really understand what this is supposed to mean. I know that you utilize center of mass when trying...
  17. T

    Rigid body solid: center of mass and tendency to topple

    Homework Statement A whimsical piece of furniture has the base shaped like a star. Formulate the condition in terms of the location of the center of mass of the object that the piece would not topple over. A sketch would be helpful. The Attempt at a Solution Can someone help me...
  18. J

    Center of Mass of draining Soda can

    Homework Statement A uniform soda can of mass 0.140kg is 12.0cm tall and filled with 1.31kg of soda. Then small holes are drilled in the top and the bottom (with negligible loss of metal) to drain the soda. The can is placed upright such that the soda only drains from the bottom hole. What is...
  19. H

    Center of mass and distance

    A 500-kg cannon and a supply of 59 cannon balls, each with a mass of 19.0 kg, are inside a sealed railroad car with a mass of 50000 kg and a length of 41 m. The cannon fires to the right; the car recoils to the left. The cannon balls remain in the car after hitting the wall. After all the cannon...
  20. L

    Center of mass for a cylinder with water

    Hi! :) This is my first post here, hoping this will help me understand a few things. I'm really stuck at this problem and it is annoying me because I think I understand the theory, but just can't seem to be able to do it. So any help, hints and tips to solve this kind of problem is highly...
  21. P

    Calculating y-Coord of Center of Mass in Unique Shape

    Homework Statement ____ l _ l l_l l_l Find the y-coordinate of the center of mass. Each side l / _ is length a. I got an answer of 1.25a, but that isn't right. Homework Equations The Attempt at a Solution Area of each section * x-coordinate of center of mass. (the area...
  22. C

    How Far Does the Boat Move When Juliet Joins Romeo at the Rear?

    Homework Statement Romeo (80.3 kg) entertains Juliet (58.0 kg) by playing his guitar from the rear of their boat at rest in still water, 2.73 m away from Juliet who is in the front of the boat. After the serenade, Juliet carefully moves to the rear of the boat (away from shore) to plant a kiss...
  23. F

    Man on a plank- Center of Mass pb.

    Homework Statement A man is standing at one end of a plank of length L = 10 m. The man has mass Mman = 100 kg and the plank has mass Mplank = 40 kg and the plank is atop a frictionless sheet of ice. At the other end of the plank sits a large rock of mass Mrock = 200 kg. The center of mass of...
  24. F

    Find center of mass using rotational equilibrium

    Homework Statement A nonuniform bar is suspended at rest in a horizontal position by two massless cords as shown in Fig. 12-45. One cord makes the angle θ = 15.5° with the vertical; the other makes the angle φ = 58.5° with the vertical. If the length L of the bar is 6.08 m, compute the...
  25. H

    How Do You Calculate the Center of Mass for a Plate with a Circular Hole?

    A thin rectangular plate of uniform areal density σ = 3.13 kg/m2 has length of 44.0 cm and width of 26.0 cm. The lower left hand corner is located at the origin, (x,y)= (0,0) and the length is along the x-axis. There is a circular hole of radius 7.00 cm with center at (x,y) = (16.00,11.00) cm...
  26. D

    How Is the Center of Mass Calculated for a Plate with a Circular Hole?

    Homework Statement A thin rectangular plate of uniform areal density σ = 3.06 kg/m2 has length of 37.0 cm and width of 25.0 cm. The lower left hand corner is located at the origin, (x,y)= (0,0) and the length is along the x-axis. There is a circular hole of radius 6.50 cm with center at...
  27. L

    Center of mass of a non-uniform rod

    1. A straight rod has one end at the origin and the other end at the point (L,0) and a linear density given by \lambda=ax^2, where a is a known constant and x is the x coordinate. Since this wire is not uniform, you will have to use integrtation to solve this part. Use M=\int_0^L dm to find the...
  28. E

    What happens to the center of mass after a collision?

    Hi! I'm trying to understand the concept of the center of mass. I know the center of mass is the point within the object where it could be considered that the entire mass of the object lies, thus forces are going through this center, but how is it possible that the center of mass could move...
  29. B

    Finding Center of Mass - Plate with hole

    Homework Statement https://wug-s.physics.uiuc.edu/cgi/courses/shell/common/showme.pl?cc/DuPage/phys2111/fall/homework/Ch-09-Momentum/plate_with_hole/3.gif A uniform circular plate of radius 11 cm has a circular hole of radius 3 cm cut out of it. The center of the plate is at the origin...
  30. C

    Center of mass involving changing positions

    Homework Statement Richard, mass 80 kg, and Judy, who is less massive, are enjoying Lake George at dusk in a 30 kg canoe. When the canoe is at rest in the placid water, they change seats, which are 3.0 m apart and symmetrically located with respect to the canoe's center. Richard notices that...
  31. C

    Center of mass of two metal solids.

    Homework Statement Figure 7-29 shows a composite slab with dimensions 22.0 cm multiplied by 13.0 cm multiplied by 2.8 cm. Half of the slab is made of aluminum (density = 2.70 g/cm3) and half of iron (density = 7.85 g/cm3), as shown. How far from the line joining the two metals is the center...
  32. K

    Finding Center of Mass Between Earth and Satellite

    The center of mass between two equally sized planets would be right in the center. When you find the center of mass between the Earth and a satellite how would I find the distance offset from the hypothetical midpoint of the earth? The Earth contributes more to the center of mass when paired...
  33. Z

    Solve Easy Center of Mass Homework for Club-Axe

    Homework Statement A club-axe consists of a symmetrical 16.0 kg stone attached to the end of a uniform 1.6 kg stick, as shown in the figure. The length of the handle is L1 = 71.0 cm and the length of the stone is L2 = 16.0 cm. How far is the center of mass from the handle end of the club-axe...
  34. E

    Center of mass of metal square with a hole

    Homework Statement http://img35.imageshack.us/img35/6177/uthw404.jpg Homework Equations The Attempt at a Solution I tried finding the center of mass of the square as if there were no hole in it, and then I added the radius of the circle to the x-coordinate of that. It's wrong...
  35. T

    Finding a centriod (center of mass)

    Studying for a test and I can't really grasp a full example problem. I would also like help on a couple further example problems that are not done for me. Homework Statement Find the exact coordinates of the centriod: a. y=4-x^2, y=0 b. 3x+2y=6, y=0, x=0 c. y=e^x, y=0, x=0, x=1 d...
  36. B

    What is the Center of Mass of a Rod with Varying Linear Density?

    Homework Statement A rod of length 30.0 cm has linear density (mass-per-length) given by λ = 50.0 g/m + 20.0x g/m2 where x is the distance from one end, measured in meters. (a) What is the mass of the rod? (b) How far from the x = 0 end is the center of mass? Homework...
  37. J

    Calculate the center of mass on the force plate

    Homework Statement The attached jpeg shows a force plate with four measured masses each corresponding to a corner node. Calculate the center of mass on the force plate. Homework Equations I think they would be: xcg = (x1M1 + x2M2 + x2M3 + x4M4) / (M1 + M2 + M3 + M4) The ycg...
  38. mg0stisha

    Is There a Center of Mass to the Universe?

    I guess this is a relatively simple question, but depending on the answer can have complex follow-up questions. So here goes with the first: is there a center of mass to the universe? Or is this not possible? P.S. Is this site being weird for anyone? It only works on my phone's web browser...
  39. R

    Calculating the Center of Mass of an Arc

    Homework Statement There is an arc with an angle of 2\alpha and radius R What is its center of mass. (It is implied that it is to be measured from the center of the circle the arc is a part of.Homework Equations Integration, trigonometry.The Attempt at a Solution Well, the formula for a center...
  40. L

    Center of mass of the earth and the moon

    The center of mass of the Earth is approximately at the center. The center of mass is also where the force vector of the gravitational force is pointing. (down that is). Now the total gravitational force excerted on me should be the sum of the force excerted by the Earth plus anything else...
  41. P

    Understanding Center of Mass: Exploring Its Location and Impact on Objects

    can some one explain me why,center of mass of an object can lie outside the surface of the object also?
  42. T

    Why Is My Calculation of the Center of Mass for a Uniform Wire Incorrect?

    Homework Statement Find x coordinant of the the center of mass point of a uniform wire that subtends an arc of 22 degrees if the radius of the circular arc is 1m. Homework Equations I begin with: length = r*theta Therefore M = density*r*theta dm = density*ds = denisty*r8*d(alpha)...
  43. G

    Bowling Bowl - Center of mass - Friction - Angular Acceleration/Velocity

    Homework Statement A bowling ball with Mass M and Radius R is thrown down an alley so that the balls center of mass has an initial horizontal velocity of Vo and the ball is not rotating. Kinetic friction force is f^k. Find: a) an expression for the acceleration and velocity of the...
  44. P

    Conservation of Energy in a Sliding Box on an Incline

    Homework Statement A 1.90 kg box slides down a rough incline plane from a height h of 1.51 m. The box had a speed of 2.99 m/s at the top and a speed of 2.56 m/s at the bottom. Calculate the mechanical energy lost due to friction (as heat, etc.).
  45. S

    Rotation about the center of mass and spin angular momentum

    I needed to refresh my classical physics knowledge and I was going through Prof. Walter Lewin's physics videos at ocw.mit.edu and at some point he proposed the following theorem without proof: "The angular momentum due to rotation about the center of mass is universal and does not depend on...
  46. T

    Center of Mass: Plate with hole

    Homework Statement A uniform circular plate of radius 14 cm has a circular hole of radius 2 cm cut out of it. The center of the plate is at the origin of the coordinate system and the center of the hole is located along the x-axis a distance 4 cm from the origin. What is the position of the...
  47. K

    Earth & moon rotate round center of mass

    [b]1. Homework Statement Calculate the time the moon rotates once around the earth, and show that you have to add 4 hours to that since the moon & the Earth rotate around their center of mass. hint: use the reduced mass: \overline{m}=\frac{m_{1}m_{2}}{m_{1}+m_{2}} mass of moon...
  48. S

    How to Balance a Meter Stick with Unequal Masses at Different Positions?

    1. A uniform meter stick is pivoted at its center. The meter stick has a M1 = 103 g mass suspended at the D1 = 28 cm position. Which is measured from the left end. At what position should a M2 = 74 g mass be suspended to put the system in equilibrium? Give your answer D2 in cm from the left end...
  49. S

    Calculating Center of Mass in Binary Systems

    What is the correct formula for center of mass of a binary system? I had seen that it is R=\frac{m_{1}r_{1}+m_{2}r_{2}}{m_{1}+m_{2}} If that is the right equations what exactly does r one and two stand for? I had also seen a much more complicated formula. Side note: This is not...
  50. P

    Walking in a Boat (Center of Mass?)

    Homework Statement A(n) 64 kg boat that is 8 m in length is initially 8.5 m from the pier. A 32 kg child stands at the end of the boat closest to the pier. The child then notices a turtle on a rock at the far end of the boat and proceeds to walk to the far end of the boat to observe the...
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