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. V

    Centre of mass energy

    1. I have to work out the centre of mass energy of a beam collider of e-e+ each beam having an energy of 45.6Gev, That was no problem, i used the formula below and got The centre of mass = (2E)^2 = 91.2Gev. My problem arises when i apply this to a beam hitting a fixed target. The question...
  2. E

    Change in the centre of mass of a leg

    Homework Statement To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 92.0m long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a...
  3. L

    Calculate the centre of mass of the object shown below

    I have a few questions that are driving me nuts. First: Calculate the centre of mass of the object shown below. Assume that the mass density is uniform. Place the origin, O, at the lower left. If the positive x-axis points to the right along the page, what is the x coordinate of the centre of...
  4. P

    Centre of Mass and translational motion

    1. A 280kg flatcar 21 m long is moving with a speed of 7.0 m/s along horizontal frictionless rails. A 85kg worker starts walking from one end of the car to the other in the direction of motion, with speed 1.5 m/s with respect to the car. In the time it takes for him to reach the other end, how...
  5. D

    How to Calculate the Angle of Shifting for a Rotating Disc with Removed Portion?

    Homework Statement From a uniform circular disc of radius 2 centimeter( its centre of mass is at 'O') , a circular portion of radius 1 cm is removed such that shift in centre of mass is maximum.The disc is now rotated about about 'O' perpendicular to the plane through an angle then the...
  6. mr_sparxx

    Rolling motion on inclined plane: centre of mass acceleration, friction, torque

    Hi, When a circular section object (cylinder, sphere of radius r) rolls down an inclined plane (at angle theta) it experiences both linear and angular acceleration due to gravity. I have read multiple times that: mg sin(theta) - friction = m a where friction = Torque/r From there on...
  7. P

    Shifting Centre of Mass Puzzle?

    Homework Statement A child crouches on a swing. She is held AT REST, at 30 degrees to the vertical. The distance from her centre of mass (CM) while crouching to the suspension point is 3 m. The swing is released. When she reaches the lowest point, she SUDDENLY stands up. Her CM is now 2.6m...
  8. A

    Centre of mass of truncated sphere

    Please help me in derivation of the cetre of mass of truncated sphere
  9. K

    How Do You Calculate the Center of Mass for a Club-Ax?

    Homework Statement "Alley club-ax consists of a symmetrical 8 kg stone that is 18cm long, and is attached to the end of a uniform 2.5 kg stick that is 98 cm long." Homework Equations cm = centre of mass x(cm) = x1m1 + m2x2 / (m1+m2) The Attempt at a Solution m1=8 m2=2.5 x1=9...
  10. W

    Centre of Mass of carpenter's L square -

    Homework Statement A carpenter's square has the shape of an "L," as shown at the right. Find the coordinates of its center of mass, assuming it to be made of uniform, material. (Hint: divide the L-shape into two rectangles.) Homework Equations MaXa+MbXb/(Ma + Mb) The...
  11. P

    Centre of Mass of two Gravitating bodies from rest

    Hi How would I show that the centre of mass of two point particles, one of mass m1 and the other m2, when released from rest from infinite simultaneously, stays in the same place? I feel this is a very simple question, I just can't seem to get it right. Everything I've tried still involves...
  12. A

    Centre of mass of truncated sphere

    Homework Statement derivation of the centre of mass of the truncated sphere Homework Equations The Attempt at a Solution i tried to solve it with triple integrals but i failed to figure ut the integral limits
  13. U

    What Is the Maximum Inclination Angle for Equilibrium of a Cut Square Plate?

    Homework Statement A uniform square metal plate of side 10 cm has a square of side 4cm cut away from one corner. (a) Find the position of the centre of mass of the remaining plate. It is now placed on a rough inclined plane as shown in the diagram. The place is inclined to the horizontal at...
  14. J

    Centre of Mass of Isosceles Triangle?

    Homework Statement an isosceles with base=a, height=h. It has uniform density and negligible thickness. Homework Equations determine it's centre of mass by integration,using more than one orientation The Attempt at a Solution It is a two-dimensional continuous structure. Let the...
  15. G

    Find CM of Round Bottom Cone | Centre of Mass Problem

    Is there anyway of breaking up the parts of an a round spherical bottomed cone into a regular cone and a hemisphere? I can find the CM of a cone with its tip on the origin, and a hemisphere with its flat bottom in the x-y plane. Can I break the round bottomed cone up into these 2 parts? How...
  16. T

    Mechanics problem - centre of mass

    Homework Statement Three uniform rods, each of length L = 22cm, form an inverted U. The vertical rods each have a mass of 14g; the horizontal rod has a mass of 42g. What are (a) the x coordinate and (b) the y coordinate of the system's centre of mass? Homework Equations x(com) = 1/M...
  17. G

    Centre of mass of a solid hemisphere.

    Hi there, I can't get my head round how to do the math for this problem. I'm sure it's not as hard as I think... Homework Statement Show that the CoM of a uniform solid hemisphere of radius r lies at a distance (3/8)r from the centre of the flat face. You may find it convenient to regard the...
  18. A

    Question about centre of mass and load centre of gravity

    Can anyone tell me that if centre of mass and load centre of gravity of a heavy vehicle doesn't concide,what vl be its effect on vehicles acceleration?
  19. T

    Moments-calculate centre of mass

    Homework Statement A woman and a man support a non-uniform plank, AB of mass 10kg and length 4m. The woman holds the plank 1m from A and the man holds the plank at B. The vertical reaction provided by the woman is 56N find; A) the vertical reaction force provided my the man b) The position at...
  20. A

    The lowest centre of mass while filling a cup of water?

    Homework Statement Ok, there's a cylindrical mug of mass m. When full of water it has a mass 4m. Assuming the base of the mug has no mass, how full is the water in the mug when the centre of mass is the closest to the base? (as a proportion of the height of the mug I'm guessing) The...
  21. G

    Centre of mass and translational motion

    Homework Statement A 270 kg flatcar 20 m long is moving with a speed of 5.0 m/s along horizontal frictionless rails. A 90 kg worker starts walking from one end of the car to the other in the direction of motion, with speed 3.0 m/s with respect to the car. Homework Equations MVcm = m1v1...
  22. Q

    Rotational motion about a centre of mass

    Consider a free floating spacecraft in space sufficiently far from any gravitational or other external forces. The spacecraft is equipped with attitude control thrusters. The thrusters fail and fire randomly and intermittently but symmetrically about the craft's centre of mass i.e. the craft...
  23. D

    How can I find the centre of mass of a plate with constant density?

    Hi. I'm not sure how to picture it on the x-y-axis. I found this in the textbook: If i have a plate bounce by 0<= x <=2 and 0<= y <= 3. and the density is constant. I'm looking for two integrals that corresponding to stirps in the x-direction and y-direction. let say the density is...
  24. R

    How Does a Climbing Man Affect the Center of Mass in a Pulley System?

    Homework Statement A pulley fixed to a rigid support carries a rope whose one end is tied to a ladder and a man and the other end to the counterweight of mass M. The man of mass m climbs up a distance of h with respect to the ladder and then stops. If the mass of the rope and the friction in...
  25. R

    Solving the Centre of Mass Shift When Ice Melts

    SOLVED Homework Statement A plate of of Mass m contains an ice cube of mass M and side L. Find the shift in centre of mass when th ice melts. (Gravity absent) The Attempt at a Solution I think that the centre of mass will come down by ML/2(M+m) But my book gives it zero. why? What has gravity...
  26. F

    Attatchment depicts a truck which has a centre of mass C

    Hello. Could someone possibly help me with this question? 1. The problem: The attatchment depicts a truck which has a centre of mass C. • The truck is front wheel drive. • The geometrical position of C is defined. • The coefficient of friction between the tyres and the road is μ. •...
  27. W

    Calculating Centre of Mass for 2D Car with Added Load: Homework Help

    Homework Statement the centre of mass for a 2D car is 1.125m towards the rear wheels from the front wheels, there is then 1.675m to the rear wheels. If a load with a centre of mass 0.4m from the rear wheels towards the back of the car is added how heavy does this load need to be to move the...
  28. F

    Rotational Dynamics, centre of mass.

    Homework Statement A uniform plate of mass m is suspended in the way shown. determine immediately after the connection at B has been released ;: A) angular aceleration of plate B) acceleration of its centre of mass Homework Equations before b is detached t1 + t2 = mg (Linear equillibrium) t1...
  29. A

    Momentum and centre of mass

    Two identical balls, each with velocity v, strike each other at right angles to each other. They stick together. Ignoring friction, (a) describe the motion of the combined ball after the collision and (b) describe the motion of the centre of mass of the two balls before and after the collision...
  30. J

    Centre of Mass frame of colliding molecules

    I'm not sure this is the right place to post this question but here goes: In feynman vol.1 39-9 there is a situation where two molecules are about to collide in a CM frame, the frame has velocity Vcm and the two molecules have respective velocities v1 and v2, then there is no correlation...
  31. K

    Centre of mass vs. centre of gravity

    consider a symmetrical solid body. we say its centre of mass is at its geometric centre. that is, for uniform density we've taken mass to vary exactly as volume. now when the same body is taken from a place of gravitational field A to gravitational field B. say from the Earth to jupiter, this...
  32. S

    Where can I find the centre of mass in a 3D co-ordinate system?

    hi, Can anyone please help me to find the centre of mass of an object in a 3D co-ordinate system. Thanks in advance
  33. C

    Centre of Mass and Cosmic Expansion in 4D Universe

    Considering a infinite unbound universe in 4 spatial dimensions, could the cosmic expansion be explained if we consider all matter in the universe to be accelerating towards its centre of mass due to its grav pull
  34. W

    Centre of mass - board on ice

    Homework Statement A person of mass M is standing at one end of a board of mass m and length l. The board rests upon frictionless ice surface, and its mass is uniformly distributed along its length. The person now walks to the center of the board and stops. In terms of the given quantities...
  35. O

    Calculating Centre of Mass on a Board on Ice

    Board on Ice 1. A person of mass M is standing at one end of a board of mass m and length l. The board rests upon a frictionless ice surface and it's mass is uniformly distributed along its length. Calculate how far from the person, d, the centre of mass is of the system. The person then walks...
  36. J

    Centre of mass of velocity (q61)

    Hi, for the following problem http://tinyurl.com/249gz8 the answer is d What i have done is Centre of mass of velocity =(m1*v1+m2*v2)/(m1+m2) but I am not sure how does it leads to the result d... can anybody help on this?
  37. M

    Exploring the Center of Mass When Cutting a Circle in a Square

    If I cut a perfect circle into a square piece of tin, how will the centre of mass be affected? I am making the assumption that I know the dimensions of the square as well as the diameter of the circle being cut. Let's say that I also know the position of the centre of the circle relative to the...
  38. C

    Force applied at centre of mass

    If a force is applied at centre of mass the object will only translate and not rotate. Can anyone proove this mathematically?
  39. MathematicalPhysicist

    Centre of mass (in relativity).

    consider a head on elastic collision of a bullet of rest mass M with a stationary target of rest mass m. prove that the post-collision gamma factor of the bullet cannot exceed (m^2+M^2)/(2mM) im given as a hint to prove that if P,P' are the pre and post 4 momentum of the bullet and Q,Q' are...
  40. N

    Centre of mass is a unique point

    Homework Statement Prove that centre of mass is a unique point Homework Equations definition of CM The Attempt at a Solution I started with O as origin.G as CM and G' as assumed 2nd CM in the same sysyem of particles. OG=R=(1/M) sum(i) [m_i*r_i] OG'=R'=(1/M) sum(k)...
  41. S

    Centre of mass of a solid cone

    im actually bugged of finding a solution for d topic mentioned can any 1 pleasezzz help me
  42. A

    Relationship between Lift and Centre Of Mass (aerodynamics)

    Hey here is a quick question. I have a question on an assignment that is asking me to define lift in an aircraft (done) and its relationship to the aircraft's centre of mass. I'm unsure how to do that second bit as lift acts through the centre of pressure...not the centre of mass (also known as...
  43. Reshma

    Proving Magnitude of Centre of Mass Vector Equation

    Prove that the magnitude of R of the position vector for the centre of mass from an arbitrary origin is given by the equation: M^2R^2 = M\sum m_ir_i^2 - {1\over 2}\sum m_i m_j r_{ij}^2 Well the centre of mass is given by: \vec R = \frac{\sum m_i r_i}{M} But squaring this doesn't seem to...
  44. J

    Calculating Centre of Mass with Added Weight on a Station Wagon

    A station wagon of mass 2000kg has a wheelbase (distance between front and rear axles) of 3.0m. Ordinarily 1100kg rests on the front wheels and 900kg on the rear wheels. A box of mass 200kg is now placed on the tailgate, 1.0m behind the rear axle. How much total weight now rests on the front...
  45. J

    Spherical Coordinates and Centre of Mass

    Wondering if someone could help me get this answer. I don't get spherical coordinates at all. The volume of the region given in spherical coordinates by the inequalities 3 less than or equal to rho less than or equal to 5 0 less than or equal to phi less than or equal to pi/6 -pi/6 less...
  46. G

    HELP : Centre of mass and Centre of gravity

    hello , could anyone explain to me the difference between a.) the centre of mass b.) the centre of gravity much aprreciated =) Cheers
  47. D

    Help with centre of mass please

    http://i3.tinypic.com/wmjdqh.jpg its question number 10 hi all i was wondering if you could help me with my centre off mass question sthanks the diagram shows a thing uniform sqaure of metal which has a corner folded over. Calculate the distance the centre of mass of ABCDEF is from A...
  48. Q

    Why a Wise Cyclist Carries His Load on the Bicycle: Understanding Centre of Mass

    why is it that a wise cyclist carries his load in packs on his bicycle intead of on her back? this is a poor question because of an assumption it makes. i understand that by carrying the load on the bicycle instead of on his back the cyclist lowers the ecntre of mass so that it is harder for...
  49. J

    Centre of Mass Theorem false for a rotating body?

    For a distributed mass, F = M dv/dt where F is the total external force, M the total mass, and v the velocity of the centre of mass. But take the case of a gyroscope, where the force acting on the centre of mass is gravity, assuming it is uniform over the body. The gyroscope doesn't topple...
  50. T

    How can we calculate the centre of mass using integrals?

    Hi, I appreciate that we can calculate the x-coordinate of the centre of mass of a system of bodies like this: x = (m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3} + ... + m_{n}x_{n}) / M where M is the total mass. This could be rewritten as x = \frac {1} {M} \sum_{i=1}^n m_{i}x_{i} My next step...
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