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. G

    How Far Does Dave Johnson's Center of Mass Move Up During the High Jump?

    Dave Johnson, the bronze medallist at the 1992 Olympic decathalon in Barcelona, leaves the ground at the high jump with vertical velocity component 6 m/s. How far does his center of mass move up as he makes the jump?
  2. TheSodesa

    Problem related to the center of mass of a binary star.

    So here's my problem: The distance between two stars is constant(d = 4,3 * 1010m), and they have a common center of mass. Ms = mass of our star, ma = 0,82 * Ms and mb = 2,2 * Ms. What I'm supposed to do is calculate the period of orbit of both stars, which is the same for them both, since the...
  3. moriheru

    Quantum scattering theory (lab frame, center of mass frame)

    Greetings, Here is my question: What is the difference between the labframe and the center of mass frame in QM scattering theory? Is it that in the labframe both particles are moving towards each other and in the center of mass frame one particle is static and a second one is moving towards the...
  4. M

    Center of Mass Calculation for a Club-Axe: How Far from the Handle is the COM?

    Homework Statement A club-axe consists of a symmetrical 23.0 kg stone attached to the end of a uniform 2.8 kg stick. The length of the handle is L1=91.0m and the length of the stone is L2=13.0cm. How far is the center of mass from the handle end of the club? Homework Equations Center of mass...
  5. C

    Finding center of mass of three disks

    Homework Statement All three disks are made of sheet metal of the same material, and the diameters are 1.2m ,2.4m , and3.6m . Assume that the x-axis has its origin at the left-most point of the left-most object and it points to the right. A) Determine the location of the center of mass of the...
  6. S

    Center of Mass of Three Beads Equation

    Homework Statement Three beads are placed on the vertices of an equilateral triangle of side d = 1.8 cm. The first bead of mass m1 = 150 g is placed on the top vertex. The second bead of mass m2 = 55 g is placed on the left vertex. The third bead of mass m3 = 85 g is placed on the right...
  7. X

    Recalculation of center of mass on a grid

    Hi, I checked the forum rules and homework / coursework is not allowed so I am not sure if this is appropriate or not. Feel free to guide me through appropriate forum and/or lock this thread if such questions are not allowed. The question I have is mostly related with my research and since I am...
  8. C

    Does the Center of Mass Have Horizontal Velocity in a Frictionless System?

    Hi. There are two masses connected by a massless bar, and from the unstable equilibrium position shown in the figure is slightly inclined so it falls down, being the final state of the system that both masses are in contact with the surface. There is no friction between the floor and m2. The...
  9. Satvik Pandey

    Find Velocity of Rod Center of Mass When Making 30° Angle with Horizontal

    Homework Statement A thin rigid rod of length ##l## is placed perfectly vertical on a smooth ground. A slight disturbance on the upper end of rod causes lower end of rod to slip along the ground and rod starts falling down. Find velocity of center of mass of rod when the rod makes ##30## with...
  10. M

    Center of mass for two small particles

    1. Two small particles of mass m1 and mass m2 attract each other with a force that varies with the inverse cube of their separation. At time t0 , m1 has velocity v directed towards m2 , which is at rest a distance d away. At time t1 , the particles collide. How far does m1 travel in the time...
  11. H

    The role of center of mass in motions

    Consider the motion of a system composed of a rigid bar and a rigid ring attached to it as is shown in the attached figure.The rigid system is rotating around the point O. Why we can not treat this motion as rotational motion of "the center of mass"? Namely why we can not get the correct result...
  12. V

    How Are Distances from the Center of Mass Determined in Classical Mechanics?

    I have attached the image as an attachment Homework Statement I am reading a classical mechanics textbook and I don't understand how they found that. r1' = -\frac{m_{2}}{m_{1}+m_{2}}r and r2' = \frac{m_{1}}{m_{1}+m_{2}}r r1' is the vector from the center of mass R to m1 and...
  13. E

    Force accelerates center of mass same regardless of where applied?

    Is it true that the center of mass of an object will be accelerated with the same direction and magnitude from the same force no matter if the force is applied to the center of the object or to the edge of the object? I have heard some people say this is not true because some of the "force"...
  14. D

    Center of mass for grav. potential (please reply quickly)

    Homework Statement I have a conceptual question which is not HW, although the moderators of this forum think different. In the "hoop" example, and the example after that here: http://tonic.physics.sunysb.edu/~dteaney/F06_Ph2034/lectures/lexam3.pdf do you ALWAYS have the change in...
  15. A

    Finding the Center of Mass of a pendulum

    Homework Statement Assuming that the combined mass of the balls is greater than the combined mass of the cords, but that the cords do have some mass, where is the center of mass for the system shown below? Homework Equations None The Attempt at a Solution I tried approaching...
  16. A

    What is the concept of center of mass and how is it related to potential energy?

    Homework Statement Image 1 Shows both system 1 and system 2 at rest Images 2 and 3 show answer choices: Given M2 > M1, which of the following pendulum systems has the GREATEST change in potential energy from a state of rest? Passage says: The greatest Potential energy is always associated...
  17. N

    Internal and center of mass energies

    Homework Statement An object with kinetic energy K explodes into two pieces, each of which moves with twice the speed of the original object. Compare the internal and center-of-mass energies after the explosion. The Attempt at a Solution Let K1 be the kinetic energy of the main...
  18. M

    Question on finding moment of inertia about the center of mass

    So I have a question. I (moment of inertia) is basically mr^2 right? And r is supposed to be the distance from the axis of rotation. When the axis of rotation is directly through the center of mass, how is there Icm (moment of inertia about the center of mass). It's confusing to me, because so...
  19. M

    Torque Equilibrium to find center of mass.

    Homework Statement 1. A non-uniform beam with a mass of 85kg has a length of 6.2m and is attached to a wall and supported at an angle of 28 degrees from the wall by a horizontal rope which is attached to the wall above the beam . The tension in the rope is found to be 310N. How far from the...
  20. K

    Why Does a Car's Front Lift When Accelerating?

    I got one multiple choice on our exam incorrect, and I was wondering exactly why this answer is correct: When you accelerate your car, the front of the car lifts up slightly. Or when you brake, the front dips down. The primary reason is because the A) center of mass of the car and its...
  21. S

    Would The Position of the Center of Mass Change Position?

    Take a uniform density bar in space for example. Would the center of mass move if the bar was struck at the very tip perpendicularly by some weight? If so, please explain.
  22. chongkuan123

    Finding Center of Mass of a Nonuniform Bar

    Homework Statement A nonuniform, horizontal bar of mass m is supported by two massless wires against gravity. The left wire makes an angle ϕ1 with the horizontal, and the right wire makes an angle ϕ2. The bar has length L. Find the position of the center of mass of the bar, x, measured from...
  23. J

    Derive conservation of center of mass position

    Hello there, If a mass, m, is placed at one end of an boat of mass M, which is located on a frictionless ocean. If the mass, m, is moved from one end of the boat to the other end, conservation of center of mass position ensures that the boat shifts a small distance in the opposite direction...
  24. 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...
  25. 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...
  26. 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
  27. 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...
  28. 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...
  29. 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...
  30. 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?
  31. 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...
  32. 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...
  33. 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...
  34. 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...
  35. 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?
  36. 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...
  37. 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 =...
  38. 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...
  39. 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...
  40. 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...
  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. 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?
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. B

    Angular Velocity of Center of Mass

    Homework Statement On a frictionless table, a 0.50 kg glob of clay strikes a uniform 1.58 kg bar perpendicularly at a point 0.38 m from the center of the bar and sticks to it. If the bar is 1.30 m long and the clay is moving at 6.60 m/s before striking the bar, what is the final speed of the...
  48. B

    Rotation around center of mass

    I understand that the center of mass is a point which can be considered to contain all of an object's mass, for the purpose of calculations involving universal gravitation. I also understand that the center of mass of an object of uniform density is located at the centroid. In this case, I...
  49. JK423

    Coupling only to the center of mass degrees of freedom?

    I am interested in the following scenario. Suppose that you have a composite system, e.g. ion, atom or whatever you want. This system will have internal degrees of freedom (d.o.f.), i.e. relative positions and momenta, and also center of mass d.o.f..What i am looking for is to find an...
  50. ShayanJ

    Center of mass of a uniform sphere

    We know,by symmetry,that the center of mass of a uniform sphere is at its center.So we expect the formula r_{com}=\frac{\int r \rho d\tau}{\int \rho d\tau} to give us zero for this case.So let's see: r_{com}=\frac{\int_0^{R} \int_0^{\pi}\int_0^{2\pi} r^3 \sin{\theta} d\phi d\theta...
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