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. 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...
  2. 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...
  3. Y

    Finding the Center of Mass of D

    Homework Statement Let D be the region in the (x, y) plane bounded by the lines y=x, y=4x and the hyperboals xy=1 and xy=9. find the center of mass of D. Homework Equations The Attempt at a Solution My thought: to use the center of mass formula? and use the change of variables?
  4. G

    Center of mass and center of energy

    center of mass and "center of energy" Hi everyone, I recently came across a little thought experiment in a book that seemed to imply that the center of mass of an isolated system is not a strictly conserved quantity. Rather, it is the "center of energy" that remains fixed. The idea of a...
  5. redtree

    Center of mass of a ring of variable density

    How would I find the center of mass (COM) for a ring of variable density, where the variation in density can be described by a continuous mathematical equation. The density at each location along the ring is described by a function (d) whose independent variables are radius (r) and angle...
  6. W

    Center of Mass of a Triangle: Struggling to Find the Bounds

    Homework Statement Find the center of mass of the triangle with vertices at (0,0), (6,6) and (-6,6) if the density at (x,y) is equal to y. Homework Equations The Attempt at a Solution I am having trouble with these centroid/center of mass problems, and I can't even figure out the...
  7. L

    Center of Mass Movement in a perfectly inelastic collision

    1. There are two 50 kg skaters in a friction-less horizontal surface. Both have velocities of |V|=10 m/s, in opossite directions (that is, skater 1's velocity is 10 m/s i, while skater 2's velocity is -10 m/s i). They both skate parallel to each other with a distance of 1,5m. Skater 1 has a...
  8. K

    Finding Center of Mass for a Lamina in First Quadrant

    Homework Statement A lamina has the shape of the region in the first quadrant that is bounded by the graphs of y = sinx and y= cosx, between x = 0 and x = π/4. Find the centre of mass if the density is δ(x,y) = y. Homework Equations I know all the equations for moments and center of...
  9. D

    How can I derive the equation X(cm) = int(X dm)/M?

    OMG second time I'm opening a thread in the wrong forum FFS! Damn bookmarks! MODS move it please. Homework Statement Not a home work question, just something i cam across and need a clarification. Could One show me how to derive to the equation that X(cm)=int(X dm)/M int=the deformed S of the...
  10. J

    Mass of solid and center of mass

    Homework Statement a solid in the first octant is bounded by the planes y=0 and z=0 and by the surfaces z=4-x^2 and x=y^2. its density function =kxy, k is a constant. find the mass of the solid and the center of mass Homework Equations My= double integral of D[x*kxy]dA Mx=double...
  11. S

    Velocity and Acceleration of the Center of Mass?

    Homework Statement Given: g=9.8 m/ss Given: A uniform flexible chain whose mass is 7 kg and length is 5 m. Given: A small frictionless pulley whose circumference is negligible compared to the length of the chain. Problem: Initially the chain is hung over the pulley with nearly equal...
  12. S

    How far behind the front wheels is the center of mass located?

    To determine the location of the center of mass of a car, the car is driven over a scale. When the front wheels are over the scale, the weight recorded by the scale is 5.70 x 103 N. When the rear wheels are over the scale, the scale reads 6.30 x 103 N. The distance between the front and rear...
  13. L

    Center of Mass and Energy in an Explosive Collision: A Homework Question

    Homework Statement A block of mass M = 15 kg moves with velocity V and explodes at the origin into two pieces of masses m1 = 10 kg and m2 = 5 kg. The piece m1 moves in the positive direction of y-axis with speed v1 = 4.5 m/s, whereas the other piece m2 moves in the positive direction of...
  14. D

    Center of Mass of a Hollow 2D Triangle

    Homework Statement What is the center of mass a hollow 2D right triangle having legs of length A and B. The sides of the triangle are also extremely thin. Assume uniform density. Homework Equations \overline{x}=∫xmdx/∫mdx The Attempt at a Solution...
  15. A

    Finding the coordinates of the center of mass

    A solid hemisphere of radius r given by: x2+y2+z2= r2, z ≥ 0 Suppose the density of the hemisphere is the constant δ. The coordinates of the center of mass (a,b,c) are given by: a=(∭ xδdV) / (∭ δdV) b=(∭ yδdV) / (∭ δdV) c=(∭ zδdV) / (∭ δdV) Find a, b ,c ** So far what i...
  16. D

    Center of Mass question (help)?

    I tried this problem but only managed to somewhat understand Part A. But that answer is wrong as well, so can someone guide me through steps of finding the answer: Thank you. To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 0.92 m long (measured...
  17. LarryS

    Can Relativistic Speeds Affect the Calculation of a System's Center of Mass?

    Consider a system of several point-masses separated by large distances and moving with relativistic speeds in relation to one another. GR says that gravitational waves travel at the speed of light. SR says that simultaneity often does not exist for such point-masses. Question: Is it possible...
  18. T

    Elastic collision in center of mass frame

    In this problem we will examine the collision between a baseball (the cowhide) and a bat (the ash). We assume a one-dimensional problem. That is, the bat hits the ball squarely, so that the ball reverses its direction after the collision. We also assume that the ball hits the bat at the center...
  19. M

    Building a hovercraft, having an issue with finding center of mass

    Homework Statement I'm building a hovercraft, and I'm familiar with how to calculate center of mass--however, there's something that will be sitting on the surface of the hovercraft that's throwing me off. I just can't seem to get it to work. I'm just trying to find what the mass would be at...
  20. M

    Center of mass and cross product

    Let vectors vec{A}=(0,0,-3), vec{B}=(2,0,0), and vec{C}=(0,1,1). Calculate the following, expressing your answers as ordered triples (three comma-separated numbers). Calculate A x C. For this part, remember that only perpendicular components contribute to the cross-product. The vector C has...
  21. G

    Formally finding the center of mass

    Question: Determine the location of the center of mass of a "L" whose thin vertical and horizontal members have the same length L and the same mass M. Use the formal definition to find the x and y coordinates, and check your result by doing the calculation with respect to two different...
  22. C

    Center of Mass & Displacement Question

    Homework Statement A big olive (m = 0.051 kg) lies at the origin of an xy coordinate system, and a big Brazil nut (M = 0.56 kg) lies at the point (0.81, 3.0) m. At t = 0, a force Fo = (2.6 + 1.7) N begins to act on the olive, and a force Fn = (-4.4 -3.4) N begins to act on the nut...
  23. S

    Torque and center of mass question

    Homework Statement A baseball player holds a 36-ounce bat (weight = 10.0 Newton) with one hand at the point O. The bat is in equilibrium. The weight of the bat acts along a line 60 centimeters to the right of O. What is the torque exerted on the center of mass of the bat by the player...
  24. T

    How Does Force Affect the Center of Mass and Momentum?

    1. Ball 1, ball 2, and ball 3 have masses 1 kg, 2 kg, and 3 kg respectively, and are initially arranged at x = 5 m, x = 10 m, and x = 15 m respectively (see picture). y = 5 m for all three balls. Now suppose that a net force of 1 N is applied to ball 1 in the +y direction and a net force of 2 N...
  25. T

    Momentum Conservation and Center of Mass

    1. Ball 1, ball 2, and ball 3 have masses 1 kg, 2 kg, and 3 kg respectively, and are initially arranged at x = 5 m, x = 10 m, and x = 15 m respectively (see picture). y = 5 m for all three balls. Now suppose that a net force of 1 N is applied to ball 1 in the +y direction and a net force of 2 N...
  26. D

    Sun-Earth System given center of mass

    Homework Statement Consider the Sun and the Earth as a 'system' of two 'particles'. The center of mass of this system lies a distance L from the center of the Sun. Find L and compare it to the radius of the Sun. The center of mass of the Earth-Sun system lies? A. inside the radius of the...
  27. M

    What is the new distance between the man and the shore after he switched seats?

    Homework Statement A man decides to boat out on the lake for a relaxing day of fishing. At one point during the day, he finds himself facing the shore. He judges that he is a distance Dinit = 35 ft from the shore. The man has a mass of Mman = 190 lbs, and the boat a mass of Mboat = 270 lbs. The...
  28. B

    Conservaion of Angular Momentum and Center of Mass

    Homework Statement A puck with a mass of 0.08 kg and a radius of 0.04 m slides along an air table at a speed of 1.5 m/s. It makes a glancing collision with a second puck at rest having a radius of 0.06 m and a mass of 0.12 kg such that their rims just touch. Because their rims are coated...
  29. R

    Y coordinate of the system's center of mass?

    Homework Statement In Fig. 9-39, three uniform thin rods, each of length L = 28 cm, form an inverted U. The vertical rods each have a mass of 12 g; the horizontal rod has a mass of 31 g. What are (a) the x coordinate and (b) the y coordinate of the system's center of mass? (Give your answer in...
  30. T

    Center of Mass for 8 kg Stone & 2.5 kg Stick

    Homework Statement alley club-ax consists of a symmetrical 8 kg stone attached to the end of a uniform 2.5 kg stick that is 98 cm long. How far is the center of mass from the angle end of the club-ax? Homework Equations Xcm = (sum) x*m / m The Attempt at a Solution I don't know...
  31. Simfish

    Why are objects with higher center of mass less stable?

    Is it because less force (from, say, potential objects that could tip the object) is required to exert the same amount of torque on objects with higher center of mass?
  32. M

    Center of Mass and Linear Momentum of pellet gun

    1. A pellet gun fires ten 2.0 g pellets per second with a speed of 500 m/s. The pellets are stopped by a rigid wall. What are (a) the magnitude of the momentum of each pellet, (b) the kinetic energy of each pellet, and (c) the magnitude of the average force on the wall from the stream of...
  33. A

    Center of gravity vs center of mass

    Is it different from the center of mass? My book says yes but doesn't answer the question.
  34. L

    How Is the Lowest Center of Mass Calculated for a Draining Soda Can?

    Homework Statement If x is the height of the remaining soda at any time in a can of soda, find x (in terems of M, H, and m) when the center of mass is the lowest. {{The soda is draining out of the bottom of the can}} Also, the Can and soda have uniform density. So the can is set up so...
  35. H

    Center of mass of a rod question.

    Homework Statement A rod of length 24.5 cm has linear density (mass-per-length) given by the following equation, where x is the distance from one end. λ = 50.0 g/m + 20.5x g/m2 (a) What is its mass? (b) How far from the x = 0 end is its center of mass? Homework Equations...
  36. N

    How Does a Spring's Compression Affect Scale Readings?

    Urgent ! Center of mass and relevant stuffs Homework Statement A vertical spring of constant k is attached at the bottom to a platform of mass mp, and at the top to a massless cup. The platform rests on a scale. A ball of mass mb is placed in the cup. What is the reading on the sacle when...
  37. N

    Center of mass of baseball bat problem

    Homework Statement A baseball bat of length L has a peculiar linear density given by \lambda = \lambda0 * (1+X2/L2) Find the x coordinate of the center of mass in terms of L Homework Equations Mxcm= mr The Attempt at a Solution So I use integration The integrand I have is x*\lambda dx...
  38. E

    Center of mass on inclined plane

    Homework Statement A box with its center of mass off-center as indicated by the dot is placed on an inclined plane. In which orientation does the box tip over? Homework Equations Teacher didn't even mention anything related to this The Attempt at a Solution Since the CM was at...
  39. C

    Find the Center of Mass: Tutorials & Problem Solutions

    Hey guys, does anyone know a good website teaching center of mass? I need to know how to find the center of mass of objects like spheres, cubes etc. and also do problems such as "a man is standing on a boat and moves from one side to the other, how far does the boat move." Stuff like that...
  40. K

    Calculating Center of Mass of a Car: Step-by-Step Guide

    Homework Statement To determine the location of the center of mass of a car, the car is driven over a scale. When the front wheels are over the scale, the weight recorded by the scale is 5.05 x 103 N. When the rear wheels are over the scale, the scale reads 6.20 x 103 N. The distance between...
  41. N

    What is the center of mass of a cone?

    1. A right circulat cone of constant density (5kg per m^3) is 4 meters from the base to the tip. the diameter of the base is 6 m. find each of the following using inergrals A) Find the volume and mass of the cone B) find the center of mass of the cone C) find the moment of inertia of the...
  42. C

    Center of Mass of a System Problem

    Homework Statement From an inertial reference frame S, the vector position of a particle of mass m1 = 1kg is given by r1(t)=(txhat - t^2yhat)m. The vector position of a particle m2=2m1 is given by r2=(t)=(txhat +t^3yhat)m Assume t=1second Find the position, the velocity and the...
  43. C

    What is the acceleration of the center of mass of the cylinder?

    If you guys would be kind enough to show me if my attempt to problems posted are correct or incorrect. Homework Statement A uniform 3.0 kg cylinder of radius 0.15 m is suspended by two strings wrapped around it, as shown in the figure below. As the cylinder descends, the strings unwind from...
  44. M

    Calculating center of mass and moment of inertia using integrals.

    Homework Statement I am very fuzzy about how to find the center of mass of an object and, similarly, the moment of inertia for an object using integrals. Here are the formulas as I remember them. Center of mass: (1/M) \int \vec{r} dm Moment of Inertia: \int R^{2} dm What I was...
  45. T

    How Does Friction Torque Affect Angular Speed in a Spinning Spherical System?

    A uniform spherical shell of mass 9.00 with diameter 54.0 has four small masses of mass 2.00 attached to its outer surface and equally spaced around it. This combination is spinning about an axis running through the center of the sphere and two of the small masses What friction torque is...
  46. S

    Did I Find the Correct Mass and Center of Mass for the Solid?

    Homework Statement Find the mass and center of mass of the solid bounded by the planes x=0, y=0, z=0, x+y+z=1; density\delta(x,y,z)=y Homework Equations M=\int\int_D\int\delta dV M_{yz}\int\int_D\int x \delta dV;M_{xz}\int\int_D\int y \delta dV;M_{xy}\int\int_D\int z \delta dV...
  47. U

    Period of uniform plank with spring not at Center of Mass

    Before begining, I would like to point out I did look at the following thread previous to this one, and thus have a few questions about that thread and it's application to this one: https://www.physicsforums.com/showthread.php?t=143536 Homework Statement The problem...
  48. B

    Center of mass of physical pendulum

    Homework Statement A pendulum consists of a uniform disk with radius r=0.100m and mass 0.500 kg attached to the end of a uniform rod with length L=0.500 m and m 0.250 kg. It pivots at the other end of the rod. a) Calculate the rotational inertia of the pendulum about the pivot point. b) What...
  49. B

    Conceptual Center of Mass acceleration problem

    Homework Statement 3 identical rectangular blocks on frictionless surface have equal magnitudes of force and in the same direction are exerted on them. The CoM is the same for all the blocks and in the center of the rectangle. For block 1, the force is exerted to the left of the CoM, and...
  50. T

    How to Calculate FM in a Walking Person Using a Cane?

    Homework Statement http://img201.imageshack.us/img201/2135/21592808vx2.th.jpg http://g.imageshack.us/thpix.php In class you learned that when a person walks, their centre of mass must be positioned directly above the foot their weight is resting on. The figure shows the anatomy of the...
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