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. TutaTeacherAllan

    Calculating the center of mass

    Figure 4 shows three point masses glued to the vertices of a massless triangle of side L=2.0 m. Two of the massess are m=0.75kg and the other mass is twice that value, as shown in the figure. The device is allowed to rotate about point A in a verticle plane about the centre of the larger mass...
  2. Amik

    Velocity of the center of mass of the system

    I believe they are the same?I think i am wrong.
  3. Luke Tan

    Conservation of Energy of the Center of Mass

    In classical mechanics, the energy of a system of particles (say with 2 particles) in an external field is given by $$E=\frac{1}{2}m_1|\vec{v}_1|^2+\frac{1}{2}m_2|\vec{v}_2|^2+V(\vec{r}_1)+V(\vec{r}_2)+V'(|\vec{r}_2-\vec{r}_1|)$$ Where V is the potential energy of the external field, and V' is...
  4. F

    Inverted pendulum -- What force is applied at the center of mass?

    Summary: I know that in order to keep inverted pendulum balanced when it leans forward, I should accelerate the bottom of it, but what is the force which is applied to center of mass of the pendulum with balances torque due to gravity? Is it inertial force? We assume that our inverted pendulum...
  5. lLehner95

    Find the photon energy in the center of mass frame and vice versa

    I tried to use the Lorentz transformation: ##E^{*}_{\gamma }=\gamma _{cm}E_{\gamma }-\beta _{cm}\gamma _{cm}p_{\gamma }## We have a photon, so it becomes: ##E^{*}_{\gamma }=\gamma _{cm}E_{\gamma }(\beta _{cm}-1)## Unfortunately, the solutions say that the correct way is to use the inverse...
  6. brotherbobby

    Is the Calculation for the Center of Gravity of Two Spheres and a Rod Correct?

    I begin by drawing the problem. Let the center of the bigger sphere on the left by the origin for the system. Calculating ##x_C = \frac{10 \times 1.1 + 20 \times 2.15}{50+20+10} = \frac{54}{80} = 0.675\; \text{m} \;= \boxed{67.5 \; \text{cm}}\;##. Problem is, doesn't match the answer in the...
  7. Nimarjeet Bajwa

    How can we find the center of mass of a solid cone?

    is this method even possible? anyways here is my attempt Step1) y= 2H/3 ( H is the height of the cone) step 2) we take the density (ρ)= 3M/π R2 H. The problem i am facing is to Find "dm"
  8. A

    Center of mass and equilibrium

    ?
  9. brotherbobby

    Several parts of a system and its CM

    I have known and used this theorem for a long time solving problems ("Calculate the CM of the some given shape"). I took the theorem to be "obvious" and didn't know it could be proved (and that indeed it was a theorem at all). I can make no attempt at the proof. Any help would be welcome.
  10. jisbon

    Help Me Solve My Diagram Problem: Calculating Center of Mass

    Not sure what I went wrong here, anyone can help me out on this? Thanks. EDIT: Reformatted my request. Diagram: So as far as I know to calculate the center of mass for x, I have to use the following equation: COM(x): ##\frac{1}{M}\int x dm## And I also figured that to find center of mass, I...
  11. Prabs3257

    Velocity of the center of mass in constrained motion

    I tried getting the velocity of a which by using constraint came out to be 3 and by using the eqn for vcom i got the ans as 3.5 i just want to confirm is this the correct answer
  12. E

    Calculating the Center of Mass

    The center of mass of someone's thigh, shank, and foot are located at the following coordinates: Thigh: (2.7, 3.2) Shank: (2.1, 2.5) Foot: (1.5, 1.9) If the masses of them are respectively 9.6, 2.9, and 0.9 kg, where is the location of the entire leg? ANSWER: CM = (2.5, 3)I'm having trouble...
  13. brotherbobby

    Center of gravity and over-hanging blocks

    The red dots show the CM of each block. ##x## is the amount by which the upper block overhangs the lower block. The blue (dashed) line shows the CM of the combination. For maximum overhanging, this line lies on the edge of the table below. By symmetry, the CM lies exactly midway between the two...
  14. e2m2a

    Center of mass momentum of a photon and electron system

    Suppose there is a photon with momtum p=h/lambda moving in the positive x-direction. Suppose it collides with an electron at rest and is completely absorbed by the electron, and that after the collision, the electron moves to the right with the same momentum of the photon. This seems...
  15. S

    Classical Book about center of mass frame of reference and fictitious force

    Hello I have learned about conservation of momentum, Newton's law (1st, 2nd, 3rd law + free body diagram), conservation of energy and finding center of mass of several 2 D and 3 D shapes (non - calculus method). I watched youtube video about two objects connected by horizontal spring and in...
  16. R

    Center of Mass of acceleration and velocity

    I don't understand what I am supposed to do. I know how to find the acceleration of the system, but I am not sure how to find the com acceleration. My guess would be something along the lines of xcom = (m1a1*x + m2a2*x)/(m1a1 +m2a2). Then do the same for the y axis. Then to find the velocity all...
  17. P

    What are the steps for calculating the center of mass of this object?

    What are the steps to calculating the center of mass for this object? I don't want a numerical answer just the theory. I understand I need to calculate COM for all axis, we know in the Y axis it's dead centre because of symmetry but I don't know how to do it for the others. A hint would be...
  18. R

    What is the solution to the Center of Mass problem?

    I was able to find the y and z axis. To find the x-axis I was assuming they would be the same for both of the slab parts and since the center of mass is the middle of the cube it should be halfway between 14.0 cm, which is 7.0cm. I can't think of any other reason why it would not be.
  19. Lone Wolf

    Finding the center of mass of a homogeneous object

    The object is: My attempt at a solution: I divided the object into 3 different rectangles and found the coordinates for the center of mass of each one, considering the origin at point "O". Then I found the mass of each rectangle, assuming the object has an area density of σ. m1 = 15σ; m2= 6σ...
  20. P

    I Center of Mass of a jetpack-Earth System

    It is known that the center of mass of an isolated system will assume the same location no matter what internal forces there are (as long as no net external force occurs). My first question is if there exist two bodies of masses m1 and m2 in some space that has no friction, gravity, or...
  21. A

    Rigid body mechanics, solution seems incorrect?

    I've attached the problem and solution as picture. To my understanding, the gear E and the rod OB are taken together as the rotating rigid body. However, the equations of motion and (##∑F = macm##) are applied to the center of mass of the rod, G, rather than the center of mass of the rigid body...
  22. D

    How to find the center of mass of a cube?

    Homework Statement Find the center of mass of a homogeneous solid cube with side ##L## analytically. Homework Equations None. The Attempt at a Solution I don't understand how to find the center of mass on three dimensions. I know that since it is homogeneous, if I center the cube on the...
  23. D

    Find the center of mass of a plate

    Homework Statement Find the center of mass of the next plate if: A) Is homogeneous B) Its density per unit mass is ##\sigma=Axy##, where ##A## is a constant. Homework Equations ##X_{cm}=\frac{\int\sigma x dA}{\int\sigma dA}## ##Y_{cm}=\frac{\int\sigma y dA}{\int\sigma dA}## The Attempt at a...
  24. J

    Finding the POSITION of the center of mass

    Homework Statement Three spheres are placed around a coordinate grid: one of mass m at the bottom-left, one of mass 3m a distance of 3L above the first, and one of mass 2m a distance of 4L to the right of the first. 4 points between these three spheres are labeled: A near (1.3L, 1.5L), B...
  25. H

    Effect of the Location of the Center of Mass on the Falling Time....

    The set-up of the system is like the following: The thing is, a weight was attached to the meter stick and it was placed in different locations. Then the system composed of the meter-stick and the added weight was then released and the time it took for the meter stick to hit the floor was...
  26. Q

    Boat with Friction: CoM / Minimum Time Problem

    Homework Statement A person of mass ##m## stands at the left end of a boat of mass ##M## and length ##L##. Both the person and the boat are initially at rest. The coefficient of kinetic friction between the boat and the water is ##\mu##. If the person starts moving to the right, what is the...
  27. E

    Where is my mistake? (Pure rolling and center of mass)

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  28. K

    Problem when calculating the center of mass of a triangle

    Homework Statement if the black dot is assumed by (0,0).find the center of mass coordinate of this triangle [/B] i'm sorry but since the pic won't show ill attach the link here https://ibb.co/4Ptw5T7 <Moderator's note: picture added> Homework Equations centroid is 2/3 of median [/B] using...
  29. Hiero

    Torque about center of mass due to Coriolis force, is it 0?

    It’s essentially a question about cross product identities. I want to analyze a problem in a frame of reference which is rotating with angular velocity ##\vec \Omega## relative to an inertial frame. In this non-inertial frame, we have a rigid body rotating with constant angular velocity ##\vec...
  30. E

    Angular momentum conservation and center of mass

    Homework Statement Two bodies with an equal mass of M are attached by a pole with no mass with a length of L. The system is placed on a horizontal table and at first it is at rest. At t=0 a bullet with a mass of m hits the pole, as described in the picture. The collision is completely elastic...
  31. F

    Body rotating about its center of mass

    Hello, As many know, when an external force acts on a rigid body and the force's line of action does not pass through the body's center of mass ##c.m.##, the force will cause the body to both translate and rotate exactly about the ##c.m.##. Otherwise, the body will solely translate without...
  32. M

    B Moving Center of Mass Collisions

    If I fire an electron and positron bunch both in the same direction away from my experimental accelerator rest frame such that the electron bunch is fired with higher energy towards a "read end" collision with an already fired positron bunch and they collide a short distance away from the...
  33. L

    Torque and Angular Momentum - Origin Misconception

    Homework Statement (Problems/diagrams referenced are attached as images.) Homework Equations Net torque about an origin = time derivative of the angular momentum vector about the same origin. The Attempt at a Solution I've solved these problems before, but I'm now looking back at them and...
  34. S

    Center of mass and rotation question

    Homework Statement A 100 g ball and a 250 g ball are connected by a 34-cm-long, massless, rigid rod. The balls rotate about their center of mass at 150 rpm . Homework EquationsThe Attempt at a Solution I solved by first getting the center of mass, then converting rpm into m/s I treated rigid...
  35. E

    Finding z component of center of mass of a complex shape

    Homework Statement The rigidly connected unit consists of a 2.5-kg circular disk, a 2.8-kg round shaft, and a 4.2-kg square plate. Determine the z-coordinate of the mass center of the unit.Homework Equations ∑zm/∑m The Attempt at a Solution Circular disk: mass = 2.5 kg z = 0 zm = 0 Round...
  36. J

    What is the magnitude of the acceleration of cylinder's com?

    Homework Statement In the figure below, a constant horizontal force app of magnitude 18 N is applied to a uniform solid cylinder by fishing line wrapped around the cylinder. The mass of the cylinder is 19 kg, its radius is 0.11 m, and the cylinder rolls smoothly on the horizontal surface. (a)...
  37. K

    How do masses affect the acceleration of the center of mass?

    Homework Statement The vector position of a 4.00 g particle moving in the xy plane varies in time according to 1 = (3i+3j)t +2jt^2 where t is in seconds and is in centimeters. At the same time, the vector position of a 5.95 g particle varies as 2 = 3î − 2ît^2 − 6ĵt. Determine the acceleration...
  38. G

    Cylinder with Displaced Center of Mass Rolling Down Incline

    Homework Statement A rigid cylinder of radius ##R## and mass ##\mu## has a moment of inertia ##I## around an axis going through the center of mass and parallel to the central axis of the cylinder. The cylinder is homogeneous along its central axis, but not in the radial and angular directions...
  39. J

    Calculating Center of Mass in NH3 Molecule

    Homework Statement In the ammonia (NH3) molecule of the figure, the three hydrogen (H) atoms form an equilateral triangle, with the center of the triangle at distance d = 9.40 ✕ 10^−11 m from each hydrogen atom. The nitrogen (N) atom is at the apex of a pyramid, with the three hydrogen atoms...
  40. J

    When the spring is maximally extended, find v_1f

    Homework Statement A mass m_1, initially moving at a speed v_0, collides with and sticks to a spring attached to a second, initially stationary mass m_2. The two masses continue to move to the right on a frictionless surface as the length of the spring oscillates. At the instant that the...
  41. S

    Finding the acceleration of the center of mass

    Homework Statement Two forces F1 →= 2i ^ + 3j^N acting on a mass m1 =1.5 kg, andF2→= i^+ j^N acting on a mass m2= 1.0 kg.The magnitude of the acceleration of the center of mass of the system is?
  42. L

    Center of Mass Balancing Question

    The question is: The orange and blue cat are currently balanced on the see saw. The orange cat then eats a bird, which of the following could make the see saw balanced again? - I attached the picture that the question is referencing. And the possible answers are: A. Orange cat stays at same...
  43. B

    Use triple integral to find center of mass

    Homework Statement Find the centre of mass of a uniform hemispherical shell of inner radius a and outer radius b. Homework Equations ##r_{CoM} = \sum \frac{m\vec{r}}{m}## The Attempt at a Solution Using ##x(r,\theta,\phi)## for coordinates...
  44. Igor Oliveira

    Describing the translation and rotation of a square frame

    Homework Statement Four equal discs of mass ocuppy the vertices of a square frame made by four rigid bars of length and negligible mass. The frame is at rest on a horizontal table, and it can move with negligible friction. An instantaneous impulse is transmitted to one of the masses, in the...
  45. Sune Irl

    Calculus 2 - Center of Mass and Pappus Centroid Theorem

    Homework Statement determine the center of mass of a thin plate of density 12 and whose shape is the triangle of vertices (1,0), (0,0), (1,1). Then, using the appropriate pappus theorem, calculate the volume of the solid obtained by rotating this region around the line x = -2. Homework...
  46. E

    What are the forces on the center of mass?

    Homework Statement [/B] A light rope passes over a light frictionless pulley attached to the ceiling. An object with a large mass is tied to one end and an object with a smaller mass is tied to the other end. Starting from rest the heavier object moves downward and the lighter object moves...
  47. F

    Center of Mass to lab-frame

    Homework Statement a) [Solved] What is the threshold energy in the center of mass for production of an antiproton in a proton-proton collision? Make sure quantum numbers are conserved. (This one was easy and I manage to solve it) b) In a fixed target experiment calculate the proton beam energy...
  48. P

    Period of a physical pendulum with a pivot at its center

    Homework Statement The picture illustrates a simple pendulum and and two physical pendulums ,all having the same length ,L. Class their period in ascending order. Homework Equations T = 2π / ( I/mgh) I = Icm + mh2 Icm=(ML2/12) The Attempt at a Solution I have found the period for first...
  49. G

    Pure rolling of sphere having non uniform mass density ?

    in case of rolling without slipping of a solid sphere having uniform mass density the condition is Vcm (velocity of center of mass ) = Rω or [a][/cm] = Rα ,which comes from the fact that if an object that rolls without slipping the geometric center of the body travels 1 circumference along...
  50. lfdahl

    MHB A center of mass determined triangle: Find the angles of the triangle P_1P_2P_3

    Masses $m, 2m$ and $\sqrt{3}m$ are located at points $P_1, P_2$ and $P_3$ on a circle $C$ so that their center of mass coincides with the center of $C$. Find the angles of the triangle $P_1P_2P_3$.
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