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.

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  1. Mcmenhweilleisi

    Equilibrium with the string in the center exactly horizontal

  2. R

    How to interpret the center of aero forces on a sports car?

    Hello. I am working on a physics project for a simulation title and have stumbled upon on an interesting challenge. Below is the example from wind tunnel data of a Dodge Viper GTS sports car. Wheelbase: 2,44m Lift front axle: 54kg Negative lift rear axle: 26kg Can somebody please explain to...
  3. G

    Find the center of gravity of a combined cone and cylinder

    Could I please ask for help with the following: Given: The centre of gravity of a uniform solid right circular cone of vertical height h and base radius a is at a distance 3h/4 from the vertex of the cone. Such a cone is joined to a uniform solid right circular cylinder of the same material...
  4. O

    MHB Finding the Center of a Circle

    Hello! I have an application where I need to find the center of a circle where I am having trouble coming up with a simple way to do this. The diameter of the circle is known and i want to be able to determine the location of it where only a portion of the circle is known. (see the image...
  5. A

    A What are the probabilities of a die with an offset center of gravity?

    With a pure die, all odds are equal. With a pure die, the center of gravity is exactly in the middle of the die. But what if the center of gravity is not in the center? How are the odds then. For example, how do the odds become if the center of gravity is exactly on the line that runs through...
  6. B

    Induced EMF in a small square loop of wire at center of an AC Circuit

    I used the voltage of the power supply and resistance to solve for the current in the larger circuit (20V/5ohms=4 amps). I am not sure if the equation listed above is the correct one I should be using, but I tried it using the following numbers. For omega, I used 2*pi*frequency. N should...
  7. G

    To find the center of gravity of a particular quadrilateral

    Could I please ask for help with the following question. Part 2 is my problem. I have no idea how to begin, any hints would be much appreciated: 1) Prove that the center of gravity of a uniform triangular lamina is the same as that of three equal particles placed at the vertices of the lamina...
  8. 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...
  9. kyphysics

    Would You Get Vaccinated at Recreation Center If Offered?

    Assume: i.) You are obese (I'm about 35-40 lbs overweight). ii.) You live with someone in a very high risk category who has not been fully vaccinated (they've only received 1 shot and need the second). iii.) You are going at a time after the center has opened (several hours afterwards) with...
  10. 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? $$ \int_0^\infty (x^2-\frac6{x^4})dx=0 $$
  11. H

    The magnitude of the magnetic field at the center of the loops is....

    I calculate like this and I think the answer is E not D. N( µoI/2a + µo2I/2*2a) = 2µoNI/2a = µoNI/a
  12. H

    Potential difference between the center of a sphere and a point 4.0 cm away?

    I can use integral as attached to obtain the answer. However, I wonder why I can not use " V = KQ / r " the get the correct answer. Thanks.
  13. 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!
  14. A

    Engineering Finding the center of instantanous rotation

    [Mentor Note -- Improved versions of the two pictures are posted in a reply a few posts down] Good day and here is the solution I have a problem in finding the value of AC and BC, I couldn't figure it out? many thanks in advance!
  15. J

    Yaw stability -- center of gravity must be ahead of center of pressure

    If we just prove (https://www.physicsforums.com/threads/does-every-object-rotate-around-its-center-of-gravity.998359/) that object don't rotate about CG ,why then center of gravity must be ahead of center of pressure ,for yaw stability? Can you explain physics behind this phenomen?
  16. J

    Does every object rotate around its center of gravity?

    It is said that rocket,plane rotate about center of gravity ,why this is is not case for boats? Boat pivot point is not in center of gravity.
  17. 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 :
  18. 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...
  19. F

    I Exploring the Galactic Center: Adaptive Optics & GR

    https://indico.cern.ch/event/977179/ Description The proximity of our Galaxy's center presents a unique opportunity to study a galactic nucleus with orders of magnitude higher spatial resolution than can be brought to bear on any other galaxy. After more than a decade of diffraction-limited...
  20. 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
  21. patrykh18

    Centre of Mass of a Uniform Cuboid -- Show that it is at the Center

    So, I volunteered to run a seminar to first year students in my college. They got a question like this for homework recently and a lot of them made a mistake in the calculation. I am not asking for help with the question itself because I know how to do it. However, a lot of students made a...
  22. 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?
  23. 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...
  24. PhysicsTest

    The net magnetic field at the center

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  25. 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...
  26. agnimusayoti

    Electric field a distance z from the center of a spherical surface

    Well, I really don't understand what is the use of the hint. I try to solve this problem with Coulomb's Law and try to do in spherical coordinates and got very messy infinitesimal field due to the charge of infinitesimal surface element of the sphere. Here what I got: $$\vec{r}=\vec{r_P} +...
  27. 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...
  28. G

    The velocity of the center of the base of a rolling cone

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  29. R

    B Why do we need a separate center of gravity idea when we have CoM?

    While deriving the formula for the position vector of the center of mass of a system of ##n## particles, we assume some external force ##\vec{F_i}## on each particle and internal forces ##\vec{f_{ij}}## between any two particles. In the derivation, we come to the conclusion that the position...
  30. Michael_L

    I How to find the distance from a star to the center of the Milky Way Galaxy?

    The distance of the sun to the Galactic Center is about 7.4–8.7 kiloparsecs. If I use the known data of stars in various stellar catalogues, how can I calculate (approximately) a distance of a given star from the Center? What kind of data I need to look for? can it be calculated from Right...
  31. Spinnor

    Center of Isaias passes near, pressure drop graph local weather

  32. K

    I Is space curved at the center of gravity?

    At the core of the earth, or sun, there is no net force of gravity...every direction is up...does this mean that space is not curved, or more generally, becomes less curved from the surface of an object towards its center?
  33. 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...
  34. 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...
  35. rxh140630

    Drilling a hole through the center of a solid sphere

    The volume of the sphere = \frac{32{\pi}r^{3}}{3} The answer given at the back of the book is (\frac {32}{3} - 4\sqrt{3}){\pi}r^3 To drill a hole completely through the sphere, the hole would have to have a length of 4r. To get the answer in the back of the book, it requires setting the...
  36. tanaygupta2000

    Resultant field at the center of two semicircular current arcs

    So the magnetic field induced at the center of a current-carrying loop is given by: B = μ0 i /2r where r is the radius of the loop In the case of a semi-circular loop, this becomes B = μ0 i /4r In the question, i = 2A, r1 = 1m and r2 = 2m So, field induced at the center of first semicircular...
  37. 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...
  38. H

    MHB Volume and surface area of "One World Trade Center"

    Hi to everyone, do you know the "One World Trade Center"? Well, I've to calculate two things about it: -The volume, according to its particular shape -The surface of the glass plates which cover the whole structure Searching on internet i found two dimensions: 1) Total height without...
  39. 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...
  40. S

    { help } Calculating the B-field at the center of a solenoid

    Problem 52: A solenoid is 40 cm long, has a diameter of 3.0 cm, and is wound with 500 turns. If the current through the windings is 4.0 A, what is the magnetic field at a point on the axis of the solenoid that is (a) at the center of the solenoid, (b) 10.0 cm from one end of the solenoid, and...
  41. P

    Electric field at the center of a sphere

    My first impression was the electric field is 0 at the center of the sphere, but it turned out not the case. My understanding when problems refer surface charge density, is that the charge exists only on the surface and it is hollow inside the sphere. Am i correct? Using the electric field...
  42. G

    Determing the center of gravity of a shaded section

    Determine the volume of the shaded area around the Y-axis by using the theorem of Pappus Guldinus, where value of R = 143,3 cm. a) Determine the area of the shaded section. b) Determine the center of gravity of the shaded section. c) Detrmine the volume by using the theorem of Pappus Guldinus...
  43. Leonardo Muzzi

    I Why the center of black holes should be different from neutron stars?

    Physically speaking, why the insides of the least massive black hole should be any different from the heaviest possible neutron star?
  44. 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##...
  45. 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...
  46. Z

    Center Selection Problem: Center for a Single Site

    I have underlined the word single center, is this single center or single site? What is coverng radius? Why it can have bad solution? Somebody please guide me. Zulfi.
  47. Z

    What Is the Center Selection Problem in Geometry?

    Hi I have some questions about the center selection problem: 1) K= 4, Are we finding the center of 4 circles? 2) r(C) = maxi dist(si, C) = smallest covering radius. In case of r(C), I can’t understand what is meant by ##max_i dist(s_i, C) ## How the max will work? Some body please guide...
  48. 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...
  49. E

    Calculating System Work and Displacement of Center of Mass

    <work done a system> a) w=5N*(0.4m)=2J I think this is right, but b) center of mass initial = ((0.5kg)(0.05m)+0.5kg(0.65m))/1kg = 0.35m center of mass final = ((0.5kg)(0.45m)+0.5kg(0.4+0.1+0.3+0.25))/1kg = 0.75m I'm not sure for this one... How can I calculate the displacement of the...
  50. TutaTeacherAllan

    Where Is the Center of Mass in a Triangle with Unequal Masses?

    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...
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