Moments

What centre of gravity means

• The centre of gravity of an object is the single point through which its entire weight can be treated as acting
• When drawing a free-body diagram, the weight arrow is always drawn from this point pointing straight down
• The centre of gravity is a useful fiction; the object's weight is really shared across every particle in it, but treating it as a single force at one point gives the same moment about any pivot

Centre of gravity of symmetrical objects

• For an object with uniform density and a clear axis of symmetry, the centre of gravity sits on every axis of symmetry; for a fully symmetric object it sits at the geometric centre:
  • a uniform sphere → at the very centre of the sphere
  • a uniform cube → at the centre of the cube
  • a uniform thin rectangular sheet → at the point where the two diagonals cross
  • a uniform thin disc → at the centre of the disc

Finding the centre of gravity of an irregular shape

• For an awkward, irregular flat shape the centre of gravity is found experimentally using a plumb line:
  1. Punch or drill a small hole near one edge of the shape and hang it from a horizontal pin or nail so it can swing freely
  2. Wait until the shape stops swinging; the centre of gravity must now lie somewhere directly below the pin (otherwise there would still be an unbalanced moment about the pin)
  3. Hang a plumb line (a string with a small weight tied to its lower end) from the same pin so it falls straight down across the shape
  4. With a pencil, trace the line of the string onto the shape
  5. Take the shape off the pin, choose a second hole well away from the first, and repeat steps 1–4 from this new pivot
  6. The two pencil lines cross at a single point, and that intersection is the centre of gravity
• Repeating with a third hole and a third line gives a sanity check; if the third line does not pass through the same intersection, one of the readings was off

Centre of gravity can lie outside the object

• The centre of gravity does not have to be inside the solid material of the object:
  • a doughnut (torus) has its centre of gravity in the empty space at the middle of the ring
  • an L-shaped bracket has its centre of gravity inside the concave corner, outside the solid metal
  • a C-shaped magnet has its centre of gravity in the air gap between the two pole faces
• This is why such objects can balance in surprising ways; the relevant pivot only has to lie under the centre of gravity, even if that point is in empty air

Centre of gravity, posture and stability

• The position of an object's centre of gravity shifts when its shape changes:
  • a person standing upright has a centre of gravity in the lower abdomen, about one metre off the ground
  • the same person leaning forward to touch their toes has a centre of gravity that has moved forward and lower, far enough that if they overbalance, their feet leave the ground
• A wide base and a low centre of gravity make an object harder to tip over:
  • a sports car with a low centre of gravity corners more sharply without rolling than a tall delivery van
  • racing bicycles drop the rider's centre of gravity by tilting the body forward, improving stability into a turn