# What is Centre of mass: Definition and 219 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. ### I Optimizing Drumstick Bounce: Finding the Magic Fulcrum Point

When a drummer holds a drumstick, the fingers form what's called a fulcrum. The stick pivots about this fulcrum. This question is not about striking from the hand with the drumstick. Instead, the drumstick is held horizontal and then allowed to fall due to gravity. A drumstick has two ends...
2. ### Conservation of momentum for a robot on a space platform

The momentum of the robot is 95.0 x 1.4 m/s towards the platform. This must be equal and opposite to the momentum imparted to the beam. Dividing 133 kg m/s by 330.0 Kg gives a velocity of 0.403 m/s for the beam. So the relative velocity of the robot relative to the platform is 1.40 - 0.403 =...
3. ### Moment of force, rod with mass

OA*P_A = OH*P + OB*P_B ⇔ OA*P_A = ¾*OA*P + 2OA*P_B ⇔ 5*10 = ¾*0.4*10 + 2*P_B →P_B=23.5 →m2=2.35 (kg) My result doesn't match the answer hint, and when I change the rod's centre of mass to OH=1, it fits the answer hint. How do I determine the centre of mass in this situation?
4. ### Finding the Centre of Mass of a Hemisphere

Place hemisphere in xyz coordinates so that the centre of the corresponding sphere is at the origin. Then notice that the centre of mass must be at some point on the z axis ( because the 4 sphere segments when cutting along the the xz and xy planes are of equal volume) y2 + x2 = r2 We want two...
5. ### System of particles, impulse and conservation of angular momentum

So i was able to solve the angular velocity part but i don't know how to find the velocity of centre of mass . For the first part i simply conserved momentum about COM because if i consider the particles as a part of the same system as rod the collision are internal forces . I am mainly...
6. ### Centre of mass (composite rod) - length and densities provided

I feel like I'm missing something fundamental here. Given only the lengths and the densities, how am I supposed to find a numerical centre of mass?Thought process so far: Are we supposed to use the ratio of the densities to find this answer? like ##\frac{8g/cm^3}{2.7g/cm^3}##? and then use that...
7. ### Shift in position of plank when a man moves on it (Centre of mass)

Lets take the original position of the man to be our origin The plank is uniform so we can assume its mass to be concentrated at its center i.e. 4m from the origin Xcom= m1x1+m2x2/m1+m2 =50(0) +150(4) /50+150 =3m There is no external force on the system so the centre of mass does not move...
8. ### Moment of Inertia with varying distance from Centre of Mass

h = d1 + 0.08 d1 = h - 0.08 d2 = h + 0.08 I of the vertical portion = 1/12 m (l^2 + b^2) + md1^2 = 1/12 m (0.28^2 + 0.04^2) + m(h - 0.08)^2 I of the horizontal portion = 1/12 m (l^2 + b^2) + md2^2 = 1/12 m (0.28^2 + 0.04^2) + m(h + 0.08)^2 The moment of inertia for the whole T-shape about...
9. ### I Does the Normal contact force act through the centre of mass?

Consider the following situation: You have 1 rectangular block lying on a table, and an identical block is placed above the block on the table. Now, this new block is constantly pushed to the right, right before it topples off. Consider the torque about an axis passing through the rightmost...
10. ### Changes in Work done with changes in the centre of mass of a tool

Work done if centre of mass is away from us will be less. But how? Or should we consider torque?
11. ### 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...
12. ### Centre of mass of two boaters in a small rowboat

If I calculate the cm pos vector using the two boys' masses, I get 2m, which is the distance between them, but I don't know how to find the boat's displacement when they switch places. How do I represent the fact that they switch places? In what form?
13. ### T-Shaped Pendulum: Find the Total Mechanical Energy

My answer to part A is correct but for Part B I got an incorrect answer of 0.204J. My working out is sent as an attachment. Part A: Part B: Part C:
14. ### Why do objects always rotate about their centre of mass?

Why do unconstrained objects always rotate about the lines passing through their CMs when tangential forces are applied to them? I understand that if an object does not rotate about its CM, then its rotation will decay to the rotation about the axis passing through its CM. Also, when a roller...
15. ### How forces act on a shape and its centre of Mass

Hello all After a lot of support from people like Dr.D, mfig and collinsmark I have finally understood the concept of Centriods and Centre of Mass. I am now trying to understand how a uniformally distributed force and a non uniformally distributed force acts on a shape. If I had an oddly...
16. ### Defining the Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes

Homework Statement: Defining Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes Homework Equations: Defining Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes Hello all; I am trying to understand the terms:- - Centroid for a 2D shape and 3D shape - Centre of Mass for a 2D...
17. ### How to calculate the centre of mass

1) I'm not sure I quite understand this statement, is there an example that can be given to show this statement mathematically? 2) Is there any derivation for this equation to calculate position centre of mass below?
18. ### Does Newton's Second Law for Torque Work for the Centre of Mass of an Object?

I am reading Tipler & Mosca 5th edition. On pages 289-290 there is an example problem which seems to assume that Newton's second law for torque works just as well when we apply it to the centre of mass of an object. However, before this example problem was introduced, the authors did not state...
19. ### I Centre of mass of a semicircle using polar coordinates

I am labelling this as undergraduate because I got it from an undergraduate physics book (Tipler and Mosca). The uniform semicircle has radius R and mass M. I am getting the wrong answer but I can't see where I am going wrong. Any help would be appreciated. My solution: The centre of mass...
20. ### Intuition behind centre of mass equation for a rigid body

So my basic understanding of an integral is that it finds the area underneath a graph. I understand the idea behind an integral being the summation of f(x) * delta x, where delta x approaches zero. If I look at the integral it's telling me that there's a change in mass that is being...
21. ### I Force required to tip an object

So I need to find equations for the force required to tip over a cylindrical object. This is fairly straight forward in the case when the force is applied above the centre of mass, by taking moments about the pivot corner. However, in the case where the force is applied below the centre of mass...
22. ### I Technique to find CG seems like should be for C.M. instead

When thinking over the method of finding the centre of gravity that Julius Sumner Miller shows in this classic video, I wondered about if it would work in some other extreme situations. Imagine a uniform, continuous plank of length equal to 1 Earth radii positioned at the surface of the Earth...
23. ### Need help in finding COM of a hollow hemisphere

Homework Statement Hey everyone, I'm studying for my physics and came across a question for the COM of a hemisphere.I made my attempt to calculate the COM. Homework Equations The Attempt at a Solution I tried to calculate the y coordinate of COM this way,please go through it- But,I am...
24. ### B Help me understand the meaning of these lines

What does the author want to say through these lines? What is the *harm* he is talking about? I will be thankful for any help!
25. ### Centre of mass of a hollow pyramid

Question: Calculate the centre of mass of a uniform, square-based pyramid of height H and base length B (taking the centre of the base as the origin). Hence, or otherwise, derive an expression for the centre of mass of such a pyramid with a pyramid-shaped hollow cavity (of height h and base...
26. ### Volume of smaller pyramid is less than the volume of the whole.

I have a test coming up next week and while doing some practice questions I found one I can't wrap my head around. The question is: A pyramid (assume uniform density) is divided in two parts by a horizontal plane through its center of mass. How do the masses of the two parts compare ? There are...
27. ### Centre of mass of binary system calculation

Homework Statement Given a graph (see below) containing the velocities of two stars with respect to the sun, I am asked to calculate the velocity of the centre of mass of a binary system. I am not given the mass of either star, nor the shape of the orbit nor the velocity of the centre of mass...
28. ### I Displacement of the centre of mass of the Earth

What will happen if the center of mass of Earth was displaced for some distance away from its origin? Will that affect the Earth's rotation around itself or even around the Sun?
29. ### Unstable equilibrium, centre of mass problem

Homework Statement Two masses, A and B, are placed on the edges of a mass of 20 kg which is homogeneously distributed figure such that an unstable equilibrium is achieved. If the mass of A is 12 kg and the mass of B 16 kg, how large is the distance x from the left end of the board to the...
30. C

### Centre of mass of non-uniform rod

Homework Statement The mass of non uniform rod increases linearly with distance from lighter end. If m is mass of the rod and l it's total length a the linear density at lighter end, then found the distance of centre of mass from lighter end Homework Equations I put λ= kx+a where lands is...
31. ### Can the centre of mass do work?

Hello, Consider two equal masses moving away from each other at the same speed. The total momentum of the system is zero, so the total momentum of the system is zero. Therefore, the centre of mass has no speed. The total energy in the system is ##mv^2##, due to the kinetic energy of the two...
32. ### Work done on the centre of mass and kinetic energy of system

If we have two objects forming an isolated system and their centre of mass is ##X_{com}##, we know by work energy theorem that work done on centre of mass will be ##\int F_{ext}.X_{com}= 0## as no external force is acting on the system. However, if there is internal forces between the objects...
33. R

### B Understanding Centre of Mass in Stationary Objects

are 2 objects in the same frame if their centre of mass is stationary wrt each other?
34. M

### Find the centre of mass for this sheet of paper with a cutout

Homework Statement A uniform, rectangular sheet with sides of lengths a and b has a hole of dimensions a/4 by b/4 punched in it as shown below. Find the centre of mass of the sheet after the hole is made. Homework Equations CM= ∫xdm/∫dm The Attempt at a Solution I'm confused on what to...
35. ### I Centre of mass of the observable universe

Ok so I have read several of the threads regarding the impossibility of determining the centre of the universe based on observations of expansion of the universe. That one seems to have been beaten to death. So this is a slightly different question. When we look at all the matter that we can...
36. ### Find Centre of Mass: Angle AC & Horizontal- 9.46°

Homework Statement The lamina is free to rotate about a fixed smooth horizontal axis, perpendicular to the plane of the lamina, passing through the point A. and hangs in equilibrium. Find the angle between AC and the horizontal. I have attached a the solution to the question, it seems like...
37. ### How far has the boat moved and in which direction?

Homework Statement Jane and her little brother John are fishing, sitting at opposite ends of a boat (Jane on the left, John on the right). Jane is having incredible luck! John, not so much. Jane offers to switch places in the boat with John. Jane has mass of 50 kg, john has a mass 30kg, and the...
38. ### Find the centre of mass of the system?

Homework Statement All three disks are made of sheet metal of the same material, and the diameters are 1.0 m , 2.0 m , and3.0 m . Assume that the x-axis has its origin at the left-most point of the left-most object and it points to the right.Determine the location of the center of mass of the...
39. ### Two stars rotating about a centre of mass -- Find speed

Homework Statement The diagram shows a binary star system consisting of two stars each of mass 4 * 1030 kg separated by 2 * 1011 m. The stars rotate about the centre of mass of the system. (a) (i) Copy the diagram and, on your diagram, label with a letter L a point where the gravitational...
40. ### The Centre of mass of an exploding projectile

I know that the centre of mass moves in the path (the parabola) that the intact projectile would have followed but does the answer change if the new (broken) masses also have a vertical component of velocity?
41. ### B Differences between Centre of Gravity and Centre of Mass

Hello, I need help in understanding the concept of centre of gravity and centre of mass.I really get confused in the two of these.It is written in books that the centre of gravity is never changed in an object,its position is constant.But as far as I know the position of an object determines the...
42. ### Centre of Mass of an empty and filled goblet

1. A goblet consists of a uniform thin hemispherical cup of radius r, a circular base of the same material thickness and radius as the cup, and an intervening stem of length r and whose weight one quarter of that of the cup. (a) Show that the height of the centre of gravity above the base is...
43. ### I General formula for centre of mass of polygons

The position vector of the center of mass of a triangle is ##\frac{1}{3}(\mathbf{a}+\mathbf{b}+\mathbf{c})##. Is the position vector of the center of mass of a planar four-sided figure ABCD ##\frac{1}{4}(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d})##? Does this generalise to n-sided figure...
44. ### I Prove centre of mass of an arc is rotationally invariant

Suppose the coordinates ##(\bar{x}, \bar{y})## of the centroid (or the centre of mass) of an arc is defined as follows ##\bar{x}=\frac{1}{L}\int x\,ds## and ##\bar{y}=\frac{1}{L}\int y\,ds##, where ##L## is the arc length. Could you prove that the centroid is invariant under a rotation of...
45. ### Centre of mass velocity special relativity

Why is it that the ZMF velocity is given by the total momentum of the system divided by the total energy? Many thanks :)
46. ### Centre of mass of this boomerang?

Homework Statement I have an image with the question here: http://imgur.com/TGG9VTx.jpg The red text is the answer. I haven't been able to get it without contradicting what I've been told. Homework Equations I know m1x1 + m2x / m1+m2 is centre of mass in 2d plane, but I don't know how that...
47. ### Elastic gravitational collision

Hi all. Our lecturer gave us an exercise the other day regarding an elastic gravitational collision between a planet and a satellite where the satellite slingshots using the gravitational field of the planet. The question asks to show that ##v_{f} - v_{i} = 2v_{0}## where ##v_{f}## is the final...
48. ### Find Centre of Mass of a Metal Sheet

A thin sheet of metal of uniform thickness is cut into the shape bounded by the line x=a, y=kx^2 and y=-kx^2 . Find coordinates of center of mass. My attempt at the solution : To apply the formula r(c.m) = (Σm1a1)/Σa1 ; a is the area; we need to know the area but we have just been given the...
49. ### Momentum in centre of mass reference frame

my book says " the total momentum is zero in the centre of mass reference frame.This should not surprise you" but ITS NOT INTUITIVE FOR ME. I am considering a completely elastic collision. 1. I know the v_cm is const because there are no ext forces on the system of the two masses undergoing 1D...
50. ### How to find the x coordinates of the centre of mass?

Homework Statement A sphere of mass M and Radius R had two spheres of R/4 removed. the centres of cavities are R/4 and 3R/4 from the centre of the original sphere (at x=0). what is the x coordinate of the centre of mass of this object? there is a drawing next to the question literally showing...