Center Definition and 1000 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.

View More On Wikipedia.org
Recent contents Top users
  1. jdawg

    Center of Mass for a cubical box with a missing top

    Homework Statement A cubical box has been constructed from uniform metal plate of negligible thickness. The box is open at the top and has edge length L=40cm. Find (a) the x coordinate (b) the y coordinate, and (c) the z coordinate of the center of mass of the box. Ok, so my professor...
  2. A

    Why is the center of mass assumed to be on the vertical in an arc?

    Homework Statement image attachment Homework Equations The Attempt at a Solution i have the solution , but in it , he assumes the center of mass is on the vertical and thus the distance of the vertical from the center is rsin(1/4∏)/(1/4∏) where r is the radius , so why did he...
  3. G

    Does the expanding Universe have a knowable center?

    Since the speed of light is a constant, no matter what location from which a search of the universe is made, matter would appear to be accelerating in the expansion in the same way to the observer. It is understood that about 14 billion years ago the expansion started. From any vantage point...
  4. Demon117

    How Does a Particle's Guiding Center Move Near a Current-Carrying Wire?

    Homework Statement What is the motion of the guiding center of a particle in the field of a straight current carrying wire? What happens to the particle energy? Homework Equations The field is tangential to the Amperian loop, so the magnetic field is simply: \oint B\cdot dl =...
  5. A

    Finding Center of Mass of Disc with Hole

    Homework Statement Find the center of mass of a uniform sheet in the form of a circular disc with a hole bounded by the equations x^2 + y^2 ≤ 1 and (x - 1/2)^2 + y^2 ≥ (1/16). Homework Equations The Attempt at a Solution
  6. T

    Magnetic Field at Center of Loop

    Homework Statement What is the strength of the magnetic field at the center of the loop in the figure? See attachment Homework Equations B = Mu_0*I/(2*R) The Attempt at a Solution I plug in the information to this equation and get .00038 Tesla which is incorrect. Does the...
  7. D

    MHB Center of Gravity Calc: $\bar{y}$

    The y center of gravity is found by \[ \bar{y} = \frac{M_x}{M}. \] In my calculus book, they are always finding the center of gravity of a flat object or objects spaced apart (like planets). Suppose we have a block with length and width \(1\) and height \(a\). On top of this block is another...
  8. T

    How Does Throwing a Medicine Ball Affect Beam Displacement?

    1. Homework Statement A person with mass m1 = 61 kg stands at the left end of a uniform beam with mass m2 = 104 kg and a length L = 2.7 m. Another person with mass m3 = 68 kg stands on the far right end of the beam and holds a medicine ball with mass m4 = 8 kg (assume that the medicine ball is...
  9. B

    Solving a Charge at the Center of a Sphere Problem

    Homework Statement A positive charge q is placed at the center of a hollow electrically neutral conducting sphere (inner radius R1 9cm, outer radius R2 10 cm. Using Gauss' law determine the electric field of every point in space, as a function of r (the distance from the center of the...
  10. L

    Question about the E-field along center axis of charged ring

    http://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html I'm having trouble understanding the derivative notation. I know that "dE" and dQ" have to do with the derivative, but what exactly does that mean in the context of this problem? For some reason I find it difficult to...
  11. S

    Help with center of mass question

    Homework Statement The problem is attached in this post. Homework Equations x=My/M=(∫xf(x)dx)/(∫f(x)dx) from a to b The Attempt at a Solution My=4∫xtan(x^2)dx from 0 to √π/2 = 2ln(2/√2) M=4∫tan(x^2)dx from 0 to √π/2 =2xln(2/√2) My/M= 2ln(2√2)/2xln(2/√2) = 1/x or 1/A The...
  12. L

    Find the net electric field at the center of the foil

    Homework Statement A large, thin plastic disk with radius R = 1.1 meter carries a uniformly distributed charge of Q = -7e–5 C. A circular piece of aluminum foil is placed d = 3 mm from the disk, parallel to the disk. The foil has a radius of r = 6 cm and a thickness t = 1 millimeter. a...
  13. B

    What is the potential at point P at the center of this arc?

    Homework Statement What is the potential at point P due to a point charge Q at a distance R from P? Set V=0 at infinity. The same charge has been spread uniformly over a circular arc of radius R and central angle of 40 degrees. What is the potential at point P at the center of this arc...
  14. A

    Center of mass of non-uniform plate

    Hello Bruce .. I have problem regarding finding the center of mass of a rectangular plate practically. Also it is a geometric property i.e it mid point is the COM . but the plate is not homogeneous and high dense regions occur at the plate at some portions of the plate ... SO my COM is not at...
  15. P

    Center of mass vs. center of gravity

    Are there any realistic scenarios for which center of mass is not almost exactly equivalent to the c.o.g., i.e. they must be treated separately?
  16. H

    Moment of Inertia of Hollow Sphere about Center Axis x-y-z method

    Homework Statement Find the moment of inertia of a hollow sphere about a vertical axis through its center in terms of its mass M and radius R. Homework Equations I=\int r^{2} dm The Attempt at a Solution I've been curious about different methods for finding moments of inertia...
  17. S

    MHB How do I calculate the center of a square given its position and orientation?

    Hi, I want to find the world coordinates of the center of a square given: -The top left corner is the reference point -Its position relative to the world frame -Its orientation relative to the world frame The example that I have been using is that the square is at (0,2), has 0 orientation...
  18. C

    Temperature is inversely proportional to position away from center

    A metal sphere with center at origin has its temperature that is inversely proportional to the position from center. At point P(1,2,2) temperature is found to be 120 degrees What is the rate of temperature change in direction of Q(2,1,3)?
  19. J

    Problems understanding how universe has no center.

    I've already googled this question, and its not making sense to me. They keep trying to describe the universe as 4d, but the universe has 3 spatial dimensions that we live in, so how can it not be like a 3d expanding ball?? They use the surface of a 3d balloon as an analogy, but universe is...
  20. noffya

    Center of Mass in a system of cylinders

    Homework Statement A collar brass 50 mm length is mounted on an aluminum bar 80 mm in length (density of brass 8470 kg / m 3 density of aluminum 2800 kg / m 3 ). Find the height to which is the center of mass of the composite body. Homework Equations COM= m1*x1+m2x2/m1+m2 The...
  21. S

    How Is the Center of Mass Calculated for an Astroid?

    Homework Statement Find the center of mass of Astroid ##x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}} ## for ##x,y\geq 0## and ##a>0##.Homework Equations ##x_T=\frac{\int x\left | \dot{\vec{r}}(t) \right |}{\int \left | \dot{\vec{r}}(t) \right |}## The Attempt at a Solution...
  22. N

    Easy center of mass/toppling question?

    [Mentor's note: this was originally posted in a non-homework forum and then moved here, so it doesn't use the standard homework forums template.] I am studying the edexcel A level M2 mathematics module and came across this question that i don't know how to do.Even the book doesn't have any...
  23. B

    MHB How to calculate center coordinates of two reverse arcs in 3D space

    Hi, Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? Points C1 and C2 are centers of two reverse arcs which are tangent to each other at point...
  24. B

    How to calculate center coordinates of two reverse arcs in 3D space

    Hi, Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? See this image. Points C1 and C2 are centers of two reverse arcs which are tangent to...
  25. B

    Center of mass and moment of inertia of catenary

    Homework Statement A homogeneous catenary ##z=acosh(x/a)##, ##y=0## and ##x\in \left [ -a,a \right ]## is given. Calculate the center of mass and moment of inertia Homework Equations The Attempt at a Solution I started with ##x=at##, for##t\in \left [ -1,1 \right ]##, therefore...
  26. B

    Center of mass of a curved surface

    Homework Statement Let ##\Sigma## be surface given as ##f(x,y)=\frac{1}{2a}(x^2+y^2)## above circle ##K((0,0),a)## for ##a>0##. Calculate the center of mass. Homework Equations If you are not familiar with ##K((0,0),a)## , that is a circle in point ##(0,0)## with radius a. The...
  27. omarMihilmy

    Understanding Center of Mass Distribution in Rotational Equilibrium

    Isn't the point of center of mass is where the masses on both sides are equally distributed? Why is it not the case here?
  28. R

    Falling Through Earth's Center

    Homework Statement With what speed would mail pass through the center of Earth if falling in a tunnel through the center? Homework Equations W = -K K = 0.5mv^2 (vf)^2 = (vi)^2 + 2aΔx The Attempt at a Solution My main question is why can't I use the last equation I listed...
  29. B

    Center of mass with gymnastics as an example

    Hi there, I was recently watching a gymnastics video, and the three gymnasts balance on each other in a crazy tower. Here is the part: http://youtu.be/Q5L4MT8LdOQ?t=2m10s Does the person on the bottom feel a shift in the center of mass due to the top person leaning so far back, making it...
  30. Y

    Electric potential at the center of a sphere

    Homework Statement The electric field at the surface of a charged, solid, copper sphere with radius 0.19m is 2800 N/C , directed toward the center of the sphere. What is the potential at the center of the sphere, if we take the potential to be zero infinitely far from the sphere...
  31. P

    Center of gravity of an orbitting body not at center of gravity

    It just occurred to me that in the right gravitational field, say, in a low orbit 100 km from the core of a neutron star with a mass equal to that of the sun, an orbiting object's center of gravity is somewhat noticeably displaced from its center of mass. Specifically, 1 meter closer to the...
  32. S

    Finding center of mass of surface of sphere contained within cone.

    Homework Statement Problem (also attached as TheProblem.jpg): Find the center of mass of the surface of the sphere x^2 + y^2 + z^2 = a^2 contained within the cone z tanγ = sqrt(x^2 + y^2), 0 < γ < π/2 a constant, if the density is proportional to the distance from the z axis. Hint: R_cm =...
  33. S

    Center of Mass problem: 40kg child on a boat catching a turtle

    Homework Statement A 40 kg child stands at one end of a 70 kg boat that is 4 meters in length. The boat is initially 3 meters from the pier. The child notices a turtle on a rock near the far end of the boat and proceeds to walk to that end to catch the turtle. Neglect friction b/w the boat and...
  34. Q

    Center of Mass Velocity of rotating object

    Homework Statement As part of a carnival game, a 0.523-kg ball is thrown at a stack of 19.5-cm tall, 0.227-kg objects and hits with a perfectly horizontal velocity of 9.9 m/s. Suppose the ball strikes the very top of the topmost object as shown to the right. Immediately after the collision...
  35. P

    Does the center of the galaxy intersect the ecliptic?

    Does the center of the galaxy intersect the ecliptic? If so, is it a temporary coincidence?
  36. W

    Angular velocity when mass is added at center of rotation

    Homework Statement A guy is spinning on a chair with his hands at rest on his lap. As he is spinning, a large mass drops into his hands/lap. Does the guy continue spinning at the same rate, a slower rate, or a faster rate? This video demonstrates what happens when the guy drops mass...
  37. S

    What does an off center applied torque do?

    I know that if a force is applied off center (as in: not in the center of gravity), one can split them up in an equal force and a moment in/about the center of gravity. But what about a torque applied offcenter? Maybe this image clarifies my problem: So the force will be split op in a...
  38. S

    Tricky center of mass question

    Homework Statement Explain how you could measure both your mass and the position of your body’s centre of mass if all you had as experimental equipment was two spring balances (calibrated, of course), and a light, rigid plank, set up as shown in the figure. Figure: #1...
  39. H

    How to Calculate the Longitudinal Center of Buoyancy for a Blimp Model?

    I am looking to calculate the center of buoyancy of a model blimp envelope. I have calculated various cross sections of the envelope to get the total volume (and thus the lift) and surface area. By center of buoyancy I mean the point that one could tie a string and add a weight and the...
  40. G

    Center of gravity & moment of inertia

    Dear forum, while standing, spreading your legs helps your stability because you have a wider base. but doesn't spreading your legs lowers your center of gravity, thus shortening the distance (r) from your center of gravity to the ground, and therefore lowering your moment of inertia = making...
  41. K

    Calculating the velocity of the center of mass of a system

    Hello everyone. First post here. I'll go the question straightaway. Homework Statement A gun of mass "M" placed on a smooth horizontal surface fires a bullet of mass "m" with a velocity "u" at an angle "θ" with the horizontal (ground). The velocity of the center of mass of the gun+bullet...
  42. M

    Gravity, center of planets, stars

    Hello , it is said that at the center of planets and stars there is zero gravity because , assuming spherical geometry, the gravity from all sides comes together and cancels out in the middle and everything that is in close approximation to it. Now that said I also read that the pressure at sun...
  43. AakashPandita

    Angular acceleration x radius= Acceleration of center of mass ?

    For a rotating body, (angular acceleration x radius)= Acceleration of center of mass ? But a rotating body can have angular acc. even if it is not translating. what is wrong?
  44. S

    Find Center of Mass for Semicircular Wire

    Homework Statement Find the center of mass of a semicircular wire. The wire is a semicircle, but the important thing to realize is that the wire is a closed loop. Think of a solid half disk, and just take the outer edge. Where is the center of mass? This will be positioned with the center at...
  45. M

    Derivation for Moment of Inertia of Rectangle rotated through center

    Homework Statement I need to know how to derive the equation for the moment of inertia of a rectangle rotated about an axis through its center. The rectangle has sides a and b. I know the equation to be (1/12)M(a2+b2), but I am having trouble deriving it. I have searched all over the...
  46. U

    Where is the center of mass of this structure?

    Homework Statement So I'm taking the safe side by posting this question as a h/w problem, since even though its just to satisfy my curiosity, I don't want to take the risk of being penalized for putting a specific case like this one on another forum. Please look at...
  47. D

    Gravity distortion at the center of a massive object?

    I am not totally sure what I am trying to ask, but wondering about gravity at the center of a massive object such as the sun, earth, highly massive star. If you look at the time dilation at the center vs the surface or far in space time at the center time is running slowest. Now obviously for...
  48. O

    Find mass of black hole in center of galaxy given eccentricity+

    Homework Statement Astronomers believe that there is a massive black hole at the center of the Milky Way galaxy. What evidence is there for that? A group of astronomers have observed a star "S2" in a 15.2-year orbit around the center of the galaxy. They measured the period of revolution T =...
  49. F

    Center of mass and angular momentum

    Homework Statement On a frictionless table, a glob of clay of mass 0.240kg strikes a bar of mass 1.560kg perpendicularly at a point 0.390m from the center of the bar and sticks to it. a) If the bar is 1.260m long and the clay is moving at 9.300m/s before striking the bar, what is the final...
  50. S

    Center of Mass w/ variable density

    Homework Statement A box with one corner at the origin and the opposite corner at (a,b,c). The density is:##\rho=\rho_o\frac{z^2}{c^2}## Homework Equations ##r_{cm}=\frac{\int\vec{r}dm}{\int dm}## The Attempt at a Solution I calculated the mass...
Back
Top