Center of Gravity

The center of gravity of a body, noted G, is the point of application of its weight: all the forces of gravity can be replaced by a single force, the total weight, applied at G. This makes it possible to treat an extended object as a point mass.

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How to measure it in class

The center of gravity is not measured with a sensor: it is located either by balancing or by the trajectory. FizziQ allows the second approach, the richer one, thanks to the video analysis module.

Steps:

Film an object or a person performing a jump, a dive or a free rotation, with a ruler or a known length in the field of view
Open the video in FizziQ’s kinematic analysis module and set the scale
Mark, frame by frame, the point you estimate to be the center of gravity (for a bent human body, it is near the pelvis, sometimes outside the body)
Plot the graph y = f(x) of the marked positions
Check the criterion: if the marking is correct, the trajectory is a regular parabola; otherwise, it undulates
Repeat, correcting the marking until you obtain the parabola. Film from far enough away, phone fixed on a support and without zooming during the shot: a parallax error distorts the trajectory regardless of the marking

Scientific activities on this topic

Center of gravity of a diver - mark the center of gravity of a diver frame by frame and validate the marking by the parabolic character of the trajectory
Pole vault: energy balance

Learn more

Locating G, and distinguishing it from the center of mass

For a homogeneous and symmetric object, the center of gravity is located at the geometric center; otherwise, it is determined by calculation or by experiment. It must not be confused with the center of mass, or center of inertia, the purely geometric barycenter of the masses. The two points coincide as long as the gravitational field is uniform over the extent of the object, which is the case for any object in a laboratory, a classroom or a sports field. They only separate for very extended bodies, such as a satellite in orbit or the Earth itself: in secondary school, the distinction is conceptual, never numerical.

One of the most notable contributors to the understanding of the center of gravity was the Greek mathematician, physicist and engineer Archimedes (287-212 BC). Archimedes formulated fundamental principles concerning moments, equilibria and the concept of center of gravity. The saying attributed to him, “Give me a place to stand and a lever long enough, and I will lift the Earth”, is reported by much later authors, but it aptly sums up his work on levers and moments, which founds our understanding of equilibria.

A point that can lie outside the matter

Nothing requires G to be inside the object. The center of gravity of a ring is at the center of the hole; that of a boomerang or a horseshoe is in the air, next to the matter. This is also true of the human body: standing, arms at the sides, G is located roughly at the height of the navel, inside the pelvis; but as soon as you bend, it leaves the body. This is the principle of the high jump in the Fosbury flop: by strongly arching the back above the bar, the jumper makes their center of gravity pass under the bar while their body passes over it. The energy to be supplied is that needed to raise G, not to raise the bar: the technique gains several centimeters for the same effort.

Stability and base of support

A resting object remains in equilibrium as long as the vertical line through G falls inside its base of support, that is, the surface delimited by its points of contact. As soon as this vertical line leaves the base, the moment of the weight tips the object over. Two levers to gain stability: lower G (hence the ballast at the bottom of sailboat hulls, the floor-mounted battery of electric cars) and widen the base (spread the feet, the stabilizers of a crane). The Tower of Pisa does not fall because, despite its tilt, the vertical line through its center of gravity remains within its base.

The center of gravity in motion

The most useful result is the following: the center of gravity of a system obeys Newton’s second law applied to the total mass and to the external forces alone. A gymnast in free flight can rotate, tuck, stretch out: whatever she does, her center of gravity follows a parabola imposed by the weight alone, because the internal forces of her muscles cannot deflect it. This is exactly the criterion used in the FizziQ activity. The same goes for a firework: after the explosion, the center of gravity of the fragments continues the parabola of the whole rocket.

Distinguishing center of gravity, center of mass and center of buoyancy

Three points often confused. The center of mass is geometric. The center of gravity is the point of application of the weight. The center of buoyancy is the point of application of the buoyant force, located at the center of gravity of the volume of displaced fluid. The stability of a boat depends on the relative position of its center of gravity and its center of buoyancy: it is because the hull heels that the center of buoyancy shifts and creates a righting couple.

Where this appears in the curriculum

The notion is introduced in lower secondary school for equilibrium and the representation of weight, reused in the following years to model an object as a point mass, then in the final years whenever a force balance or a theorem of the center of inertia is written. It is also the basis of video analysis: when you mark “the object” frame by frame, it is indeed its center of gravity that you are tracking.

Formula

Position of the center of mass of a set of point masses:

OG⃗ = (Σ mᵢ · OMᵢ⃗) / (Σ mᵢ)

Projected onto an axis, for two masses:

x_G = (m₁x₁ + m₂x₂) / (m₁ + m₂)

For a continuous body:

OG⃗ = (1/M) ∫ OM⃗ dm

Equilibrium condition for a solid suspended from a point A: G settles on the vertical line through A, below A.

where:

G: center of gravity (coinciding with the center of mass in a uniform gravitational field)
mᵢ: mass of point mass i (kg)
Mᵢ: position of point mass i
M: total mass of the system (kg)
x_G: x-coordinate of the center of gravity (m)

Application examples

A dumbbell formed of two masses of 5 kg and 15 kg separated by 60 cm, with a bar assumed light: x_G = (5×0 + 15×0.60)/20 = 0.45 m, that is, 45 cm from the small mass - G is always closer to the larger mass.
The center of gravity of the Earth-Moon system is located about 4,700 km from the center of the Earth, that is, inside the Earth whose radius is 6,370 km: the Earth therefore does not revolve around an external point, it oscillates around this barycenter.
A Fosbury jumper clearing 2.20 m only raises their center of gravity to about 2.10 m: the arching makes G pass under the bar.
A sports car has a low center of gravity (about 45 cm from the ground) while an SUV places it around 70 cm: for the same track width, the SUV tips over at a lower lateral acceleration.
The loading of an airliner is calculated so that the center of gravity stays within a narrow range around the aerodynamic center; outside this range, the aircraft becomes uncontrollable at takeoff.
A tightrope walker holds a long flexible pole: it lowers the center of gravity of the whole and greatly increases the moment of inertia, which slows the sideways fall and leaves time to correct.

FAQ

Q: Center of gravity and center of mass, are they the same or not? A: They are two different definitions that give the same point in almost all practical cases. The center of mass is the barycenter of the masses, a purely geometric notion. The center of gravity is the point of application of the weight, which depends on the gravitational field. They coincide as soon as this field is uniform over the object, which is true at our scale. They differ for a very extended object, such as an elongated satellite in orbit.

Q: Can the center of gravity be outside the object? A: Yes, and it is frequent. Ring, boomerang, banana, horseshoe, bent human body: in all these cases G is in empty space. This is logical, since G is not a piece of matter but a calculated point.

Q: How do you find the center of gravity of a plate of arbitrary shape? A: Suspend it from one point, draw the vertical line through that point (plumb line), then repeat by suspending it from another point. G is at the intersection of the two vertical lines. A third trial serves as verification.

Q: Why does an object tip over? A: Because the vertical line from its center of gravity has left the base of support, that is, the zone delimited by its contact points. The weight then creates a moment that makes the object rotate instead of holding it in place.

Q: In video analysis, which point should be marked on an athlete? A: The center of gravity, otherwise the resulting trajectory is not a parabola. Marking the head or a foot gives a curve that undulates, because these points rotate around G. It is precisely this criterion that tells you whether the marking is good.

Chronophotography - Archimedes’ Principle