What is Center of mass: Definition and 915 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. Sam Jelly

    Center of mass acceleration for an inclined plane and mass m

    Why is the center of mass acceleration zero? Did I do anything wrong?
  2. Z

    Kinetic energy in center of mass reference frame

    Here is the relevant chapter. Suppose we have two masses ##m_1## and ##m_2## interacting via some force, and two reference frames, ##S## and ##CM##. The ##CM## frame is the center of mass reference frame. The origin of this reference frame is at the location of the center of mass of the system...
  3. J

    Finding the center of mass of a piecewise function

    I understand that I can divide this shape into a few parallelograms and a triangle and calculate the center of mass of each, but am confused as to what I should do after that. My physics teacher also wants us to use integrals, but I'm assuming I can calculate the COM of each parallelogram and...
  4. M

    Two stars orbit their common center of mass

    Why do we add the two masses (3M+M=4M) and use that for M in the equation of kepler's 3rd law? Namely why is it T^2=4pi^2R^3/G(3M+M)
  5. P

    Misunderstanding Work-energy theorem and center of mass properties

    To do this apparently, you need to use the work-energy theorem. You can calculate work done by gravity easily. However it was said that work done by the reaction forces from the hinge is zero, I don't get why. Reaction Force from the hinge is an external force on the rod, and all external...
  6. U

    I Do objects always rotate around center of mass?

    For example if airplane or boat move rudder, do they always rotate around center of mass? Or exist specific conditions when object rotate around center of mass?
  7. G

    I Question about the velocity of the center of mass reference frame

    I'm looking into center of mass and I saw the derivation of: ## V = \frac{\sum\limits_{i = 1}^{n} m_iv_i}{\sum\limits_{i = 1}^{n} m_i} ## I understand how it's derived, so no need to explain this further. It's a velocity of the frame in which total momentum of our objects is zero. Forget...
  8. JohnnyLaws

    Finding the center of mass of a simple 2D shape

    Here it is the image of the statement: As I mentioned in the "relevant equations" section, my approach to solving this exercise involves calculating the difference between the centers of mass of the square and the triangle. Starting with calculation of center of mass for the square. Starting...
  9. makamo66

    I Board Breaking: Why Strike Center for Good Break?

    I have read that a practitioner of a martial art striking a target of wood or concrete needs to strike as much as possible to the center of a target in order to get good break. What is the physical reason or formula behind this? I'm thinking maybe you don't want to create a torque.
  10. S

    B Does the inverse square law equalize gravity at different spots?

    Inverse square law would reduce the gravity from the parts of Earth that are farthest from our feet. It'll also reduce the gravity from Earth's center by a lesser amount, but would that be lesser enough so the gravity 20 kilometers under our feet is stronger than the core's gravity or even the...
  11. P

    I Rolling Motion direction of Velocity center of mass

    Hi I have come across something confusing in rolling motion. If an object moves with a positive V_cm meaning to the right its angular velocity will be clockwise or negative. The formula is V_cm=wR but for a positive V_cm you get a negative w as it moves clockwise if V_cm is to the right...
  12. Lotto

    B Can we say that a charged balloon has a center of charge?

    Doing so, we can consider the balloon to be a point charge (approximately). Can we do it in this case, when there are only electrons on its surface? Or is it stupid and we can't do it under any circumstances?
  13. String theory guy

    Why Measure the Center of Mass from the Same Position After a Collision?

    For example, in the problem below, if the center of mass is chosen to be measured initially at the center of the left mass, then it must be measured from the same position after the collision. This gives an initial COM of, and finial COM of, Which gives their change in center of mass of...
  14. paulimerci

    Determine the net torque acting about the center of mass

    Net torque = 2gx20xcos30 - 2gx35xcos20 - 2gx15xcos60 + 2gx30xcos80 = -357Nm Have I done it right?
  15. sachin

    Motion of center of mass under gravity

    While solving this question I could not figure out the concept of two blocks sticking together. the question is, Two particles A and B of masses 1 kg and 2 kg respectively are projected in the directions shown in figure with speed uA =200m/s and uB =50m/s. Initially they were 90m apart. They...
  16. M

    B Does a body behave as a point mass even at rest?

    Hi, A body with center of mass behaves as a point mass when a force is applied. So when ##F_{ext}=0## then does it also behave as a point mass with ##a_{com}=0##, at rest. If yes, How can we prove this? (And can somebody please answer my other question I posted a week ago...
  17. C

    I What is the Order of Integration in a Double Integral?

    Hello everyone! I've been reading Mr. McMullen's book and took some curiosity in an equation on the cover art, it is as follows:$$y_{cm} = \frac \rho m \int_{r=0}^R\int_{\theta=0}^\pi (r\sin \theta)rdrd\theta$$I'm trying to understand what it means, firstly the limits of integration for the...
  18. N

    I Solving for the trajectory of the center of mass

    I'm working on the physics engine component of a game engine I'm building, and I need some guidance with this particular situation. Consider a square with mass M that is free to translate in the xy plane and free to rotate about any axis perpendicular to the page (Fig. 1) If a linear impulse J...
  19. V

    Angular momentum of a disk about an axis parallel to center of mass axis

    I am using the following formula to solve this problem. $$ L_a= L_c + \text { (angular momentum of a particle at C of mass M)}$$ Because the point C is at rest relative to point A, so the second term in RHS of above equation is zero. Hence, the angular momentum about A is same as angular...
  20. nomadreid

    I A gravity conundrum involving a solid cylinder

    Given a cylinder of height 2k with constant density and total mass M, and another object (for simplicity, a point mass) with mass m on the top of the cylinder; the force of gravitation is calculated between the centers of mass, which for the cylinder is at a distance k from the point mass...
  21. A

    Find the center of mass of a uniform semicircular plate of radius R

    $$rcm = \frac{1}{M}\int_0^\pi(rdm)$$ $$dm = \sigma{dA}$$ $$dA = (\pi*R^2)*\frac{d\theta}{2\pi}$$ $$\sigma = \frac{M}{\frac{\pi*R^2}{2}}$$ $$dm = M*\frac{d\theta}{\pi}$$ $$r = R(cos(\theta)\vec i + sin(\theta)\vec j)$$ $$rcm = \int_0^\pi{\frac{R}{\pi}(cos(\theta)\vec i + sin(\theta)\vec j)} =...
  22. I

    B Calculate needed dimensions of table - Center of mass, levers etc

    My high school physics days are long ago ;) This is not homework, well, other than it is work, at home. For a real application: Very space constrained "workshop", got a bench drill press, and want to build a table on wheels for it, to be able to move it into a corner when not needed. Those...
  23. A

    Calculating the center of mass as an astronaut moves on a shuttle

    I had solved this question but it didn't seem to be appropriate to post in the classical physics problem as my question is still homework-based. Originally I had thought this might be a conservation of momentum problem. But since we don't have any initial conditions it leaves too much to guess...
  24. SpectraPhy09

    Gravitational Forces between two masses

    Density of the Sphere = 3M/4πR³ Mass of carved out sphere = density × 4π/3 × R³/8 = M/8 The position of center of mass of The Sphere {M(0) - M/8(R/2)}/M-M/8 -R/14 So total distance between centers of the two bodies is R/14 + 3R = 43R/14 So now I found force between the Mass 7M/8 (left out...
  25. lela

    Is the force exerted by a pivot always towards the center of mass?

    I thought that the force by the pivot A on the pole AB would be the reaction force to the x-component of the gravitational force on AB. This would mean that the force by the pivot would be parallel to the pole, but in my notes from class the force vector seems to be more along the bisector of...
  26. C

    B E.P.E. -> ? when 2 masses are attached to a spring

    Two masses m and M are attached to a compressed spring. When the spring decompresses, the masses won't be pushed off the spring. What will happen to the masses and the entire system? By conservation of energy, the elastic potential energy of the spring will convert into kinetic energy, but which...
  27. L

    A slanted cylinder full of liquid about to fall

    The cylinder will cease to be in equilibrium when the sum of the torques on the cylinder calculated with respect to the rightmost point of contact of the cylinder with the plane will be unbalanced. Now, the liquid is homogeneous and the cylinder has negiglible mass so the forces (normal force of...
  28. T

    Understanding Acceleration and Center of Mass in Shock Absorption

    I don't attempt solving a problem until I fully understand it, conceptually. After the hit (when maximum velocity is reached) the person starts losing momentum, having a constant upwards acceleration. The forces acting on the person are gravity and the normal to the ground. $$N - mg = ma$$...
  29. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    I recently learned how to calculate the centroid of a semi-circular ring of radius ##r## using Pappus's centroid theorem as ##\begin{align} &4 \pi r^2=(2 \pi d)(\pi r)\nonumber\\ &d=\frac {2r}{\pi}\nonumber \end{align}## Where ##d## is the distance of center of mass of the ring from its base...
  30. Istiak

    Find the center of mass of an arc

    In my mind, I had cut half of B and, thought it's semi-circle. Then, I was trying to find Center of Mass by taking it as semi-circle. But, I get an answer which is approximately, close to main answer. Someone else had solved it another way This way I can get the accurate answer. But, the...
  31. PiEpsilon

    Freely hinged rods on a table - Linear velocities of CM after the impulse

    We know that impulse is $$\vec J = \vec F \Delta t = \Delta \vec p$$ Let ##l, m## be the length of single rod and its mass respectively. Analyzing torques and forces on each rod separately we have: Rod ##AC##: $$F\Delta t +N_x\Delta t = mV_{ac,x} \space\space\text{ eq. }(1)$$ $$F\Delta t\cdot...
  32. P

    Does Raising My Arm Change the Center of Mass?

    Let me imagine myself standing on the Earth with my arm in the resting position perpendicular to the ground. Now if I decide to raise my right arm by 90 degrees, now that it is parallel to the ground. I have shifted my center of mass in this process. But the center of mass will not accelerate...
  33. A

    A Regarding center of mass of an infinite area

    Regarding finding centers of mass of infinite figures, how one can show that $$ \int_{-\infty}^\infty \left(\frac1{x^2}-\cos \frac1x\right)dx=\pi $$ for instance, and other similar integrals, like the following? $$ \int_0^\infty (x^2-\frac6{x^4})dx=0 $$
  34. Norashii

    Calculate the center of mass of a deformed hollow cone trunk

    I couldn't make progress in this problem, I would appreciate some suggestion on how could I attack this problem. Thanks in advance!
  35. R

    Find the Center of Mass for each of these figures

    hi I am struggling with simple question i need to find the mass center for each one :
  36. bieon

    Pendulum, Rotational Inertia and Center of mass

    This is the figure given. My Attempt ##T=2\pi \sqrt\frac {I}{0.5gd}## ##\frac {m_r} {l} ## ##dm= \frac {m} {l}dx## ##dI = dm_r x^2## ##dI=(\frac {m_r} {l}dx)x^2## ##I= \int_l^0 (\frac {m_r} {l}dx)x^2 \, dx ## ##I_c.m=\frac {m_r l^2}{3}## ##I_,, = \frac {m_r l^2}{3}+m_p x^2## Given...
  37. patric44

    Proof related to the center of mass

    hi guys in the proof of the parallel axis theorem this equation is just put as it is as a definition of the center of mass : $$\int[2(\vec{r_{o}}.\vec{r'})I-(\vec{r_{o}}⊗\vec{r'}+\vec{r'}⊗\vec{r_{0}})]dm = 0$$ is there is any proof for this definition ? and what is the approach for it
  38. Adams2020

    I The center of mass & relativistic collisions

    In special relativity (especially relativistic collisions), is the center of mass frame as useful as Newtonian mechanics?
  39. Hamiltonian

    Doubt related to the velocity of the center of mass

    In question 1. since there is no external force on the system of particles(and since it was initially at rest) shouldn't the ##V_{cm}## be zero? But the correct answer applies the above stated formula for ##V_{cm}## and gets ##V_{cm} = v/2## and in question 2 again as there is no external force...
  40. Hamiltonian

    Finding the center of mass of a solid hemisphere

    for this derivation, I decided to think of the solid hemisphere to be made up of smaller hemispherical shells each of mass ##dm## at their respective center of mass at a distance r/2 from the center of the base of the solid hemisphere. also, I have taken the center of the base of the solid...
  41. K

    B Earth Center of Mass: GR & Inertial Object Acceleration

    Super-basic question that I'm embarrassed to ask. It's just what the summary says: Taking Earth's center of mass as our reference frame, how does GR account for an inertial object near the surface approaching with an acceleration of G? I assume (perhaps incorrectly) that this is an inertial...
  42. Nexus99

    Moment of inertia, center of mass and vincular reaction of a rod

    I'm struggling doing point 5, i have no idea how to solve that question. In point 1 i obtained the following result: ## I=\frac{ML^2}{2}## calculating the integral of dI, the infinitesimal moment of inertia of a small section of the rod of length dr. 2) Through the conservation of angular...
  43. Hamza M khan

    The center of mass of a semicircular arc of non-negligible width

    My attempt: 1) I am going to start this with a goal of setting up a reimann sum. First I divide the "arc"(?) of angle pi into n sub-arcs of equal angle Δθ 2) The total center of mass can be found if centers of mass of parts of the system are known. In each circular arc interval, I choose a...
  44. J

    Trivial : Use of "negative sign" when calculating centroid

    I realize that this is to be solved by breaking up the object into simple objects and using their known center of mass to find the center of mass of the entire object. 1. In the solution the circular gap is also considered in the calculations with a negative center of mass, why is this done? 2...
  45. S

    Center of mass energy for two relativistic colliding particles

    Starting from the center of mass energy S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2}) knowing that E^2 = m_{0}c^4 + p^2*c^2 one has S = (E_{1} + E_{2})^2 - (\vec{p_1}+\vec{p_2}) = ( m_{0}c^4 + p_{1}^2*c^2) + m_{0}c^4 + p_{2}^2*c^2)^2 - p_{1}^2 - p_{2}^2 - 2p_{1}p_{2}cos \{theta} and then...
  46. person_random_normal

    I Visualizing the Fourier transform using the center of mass concept

    I found this video on youtube which is trying to explain Fourier transform using the center of mass concept At 15:20 the expression of the x coordinate is given in the video. I believe it is wrong, and it should be: ##\frac{{\int g(t)e^{(-2 \pi ift)}.g(t).2 \pi f.dt}} { \int g(t).2 \pi...
  47. Adesh

    Oscillation of a boat in still water (Metacenter and Center of Mass)

    Let’s say we have a boat whose longitudinal axis is the y-axis (which goes into the screen in the figure below) standing upright in a still water . ##S## is the Center of Mass of the boat and ##C## is the Center of Mass of the displaced water.On ##S## lies the force ##\mathbf W##...
  48. V

    Calculate the center of mass of a non-uniform metal rod

    I know that if they had the same density they would have the center of mass at 1,5 m. But now that they don't the center of mass will be shifted towards the part of the rod with higher density. they will have their center of mass where they have equal mass p1*v=p2*v now i don't know how to...
  49. Adesh

    An isolated object can rotate only about its center of mass

    I was talking to someone about the equilibrium of fluids and we reached at some stage where we had to prove that in an external field the translational forces add to zero along with moments (torques) should also add to zero. The first one was quite easy but during the discussion of second...