Grade by Grade
D. Structures and Mechanisms · Form and Function

Centre of Gravity

D2.2 A1.2 A1.4About 40 minutes
Learning goals
  • I can explain what the centre of gravity of an object or structure is.
  • I can find the centre of gravity of a flat shape by balancing and by the plumb line method.
  • I can predict how adding or moving a load changes where the centre of gravity is.

Words to know

centre of gravity (CG)
The point where all the weight of an object seems to act. An object can balance on this point.
balance point
A point where an object can be held up without tipping one way or the other.
plumb line
A string with a small weight hanging on it. Gravity makes it hang straight down.
vertical
Straight up and down, the way a plumb line hangs.
uniform
The same all the way through, for example a ruler that is equally thick and heavy along its length.

Hook: the balancing fork trick

Have you ever seen two forks wedged together balanced on the edge of a glass by a single toothpick? It looks like magic. It isn't. The trick works because of where the centre of gravity of the forks and toothpick ends up: right under the spot where the toothpick touches the glass.

Today you will learn what the centre of gravity is and how to find it.

Remember from Grade 3

In Grade 3 you learned that the centre of gravity is the balance point of an object, and that structures with heavy parts low down are hard to tip.

In Grade 7 you will find the centre of gravity of shapes, see how it moves when loads move, and (next lesson) use it to explain stability.

What is the centre of gravity?

Gravity pulls on every bit of an object. But for many problems, we can pretend that all of the object's weight pulls at one single point. That point is the centre of gravity (CG).

  • If you hold an object up directly under its CG, it balances.
  • For a uniform object with a simple shape, like a ruler, a brick or a ball, the CG is at the centre.
  • If one part is heavier, the CG moves toward the heavy part. Tape a coin to one end of a ruler, and the balance point shifts toward the coin.
  • The CG can even be in empty space outside the material! The CG of a ring or a doughnut is in the hole. The CG of a boomerang or an L-shaped bracket is in the space between its arms.
balances at the middle (15 cm)coin taped on the end:balance point moves toward the coinCG(in empty space!)L-shaped bracket
Left: adding weight to one end moves the centre of gravity toward the heavy part. Right: the CG of an L-shape is in the empty space between its arms.

Finding the CG of a flat shape: the plumb line method

  1. Poke a small hole near the edge of the shape. Hang it loosely from a pushpin so it can swing freely.
  2. Hang a plumb line from the same pin. When everything stops swinging, the CG is somewhere on the plumb line, because gravity makes the shape settle with its CG directly below the pin.
  3. Trace the line of the string on the shape.
  4. Make a second hole somewhere else near the edge, hang it again, and trace a second line.
  5. The CG is where the two lines cross. (A third line from a third hole should cross at the same point, which is a good check.)
CGhole 1hole 2a ruler balances on its CG(the middle, if it is evenall along)
Left: a cardboard triangle was hung from hole 1, then from hole 2. The two traced plumb lines cross at the centre of gravity (CG). Right: a ruler balances on a narrow support (like a pencil) when the support is right under its CG.
Worked example

Let's predict the CG

A 30 cm ruler balances at the 15 cm mark. You tape a heavy coin at the 30 cm end. Predict where it will balance now.

  1. Without the coin, the ruler is uniform, so its CG is at the centre, 15 cm.
  2. The coin adds weight at the 30 cm end.
  3. The CG moves toward the heavier side, so it will be somewhere between 15 cm and 30 cm.
  4. Test by sliding the ruler across your finger until it balances. (With a light ruler and a heavy coin, it may balance around 18 to 22 cm.)

The new balance point is past 15 cm, toward the coin end. The exact spot depends on how heavy the coin is compared with the ruler.

Worked example

Let's find the CG of a structure in a real situation

A canoe holds two paddlers sitting on the seats near each end. One paddler stands up in the middle. What happens to the canoe's centre of gravity?

  1. The CG of the whole system (canoe + people + gear) depends on where the mass is.
  2. When a paddler stands up, their body mass moves higher.
  3. So the CG of the whole system rises.
  4. A higher CG makes the canoe easier to tip (you'll see why next lesson). That is why paddlers are told to stay low and stay seated.

The centre of gravity moves up, so the canoe becomes less stable.

The two-finger trick

Hold a broom or a metre stick level across two pointing fingers, far apart. Slowly slide your fingers toward each other. They will always meet under the centre of gravity, even if the broom's heavy head is on one side! Try it.

Just for fun

A doughnut's centre of gravity is right in the middle of the hole. So technically, the most important point of a doughnut is the one part you can't eat!

Hands-on activity

Find the centre of gravity of cardboard shapes

You need: a piece of thin cardboard (a cereal box works), scissors, a pushpin, a long piece of string with a small weight tied to it (a washer, a key or a paper clip), a pencil, a ruler, and a corkboard or a piece of foam to pin into. A grown-up should help with the scissors and the pushpin.

  1. Cut out three shapes: a rectangle, a triangle and an odd shape of your own (like a map of Ontario or a maple leaf).
  2. Predict where each CG is and mark it lightly with an X.
  3. Use the plumb line method: make 2 holes near the edge, hang the shape and the plumb line from the pin, and trace the string line each time. Mark where the lines cross.
  4. Check: stand a pencil upright and try to balance the shape on its eraser end at that point.
  5. Cut an L-shape and find its CG. Is the CG on the cardboard, or in the empty space?
  6. Record: predicted CG, measured CG, and whether it balanced.

What to look for: the rectangle's CG is in the exact middle (where its diagonals cross), the triangle's CG balances about one-third of the way up from the middle of a side, and an L-shape's CG may be in the empty corner area or very close to the inside corner, depending on the arm sizes.

Your turn

1

What is the centre of gravity?

2

Where is the CG of a uniform metre stick?

3

Which way does the CG move if you add a heavy object to one end of a board?

4

Why does a plumb line hang straight down?

5

In the plumb line method, why do you need two lines?

6

Can the CG be outside the material of an object? Give an example.

More practice: warm-up

1

What does CG stand for?

2

Where is the CG of a uniform rubber ball?

3

What tool is a string with a small weight hanging from it?

4

True or false: an object balances when you hold it up directly under its CG.

5

When a paddler stands up in a canoe, does the CG go up or down?

More practice: on your own

1

A hammer has a heavy steel head and a wooden handle. Is its CG nearer the head or the end of the handle? Explain.

2

You load a wagon with heavy rocks at the front only. Where is the wagon's CG now compared with when it was empty?

3

Why would the CG of a bookshelf be lower if you put the heavy books on the bottom shelf?

4

You trace two plumb lines on a cardboard shape but they cross outside the cardboard. Did you make a mistake?

5

Explain the fork-and-toothpick trick using the term centre of gravity.

Challenge

1

A 1 m plank has a 2 kg paint can at one end and an identical 2 kg paint can at the other end. Where is the CG of the plank and cans together if the plank is uniform? What if you add a third 2 kg can on top of one of them?

2

A crane lifts a heavy steel beam with its long arm. Explain what happens to the CG of the crane-plus-beam, and why cranes carry a heavy counterweight on the opposite side.

3

Design a fair test to find out how the position of a coin taped to a ruler changes the ruler's balance point. What do you change, keep the same and measure?

Check your understanding

Quiz

1.

The centre of gravity is...

2.

Where is the CG of a uniform 30 cm ruler?

3.

A heavy coin is taped to one end of a ruler. The balance point...

4.

In the plumb line method, the CG is where...

5.

Where is the CG of a ring?

6.

Why should paddlers stay seated in a canoe?

For parents and teachers

Big idea (D2.2): The centre of gravity is the point where an object's weight seems to act. This lesson finds it and shows how it moves; the next lesson links it to stability.

How to help: Try the two-finger broom trick and the coin-on-a-ruler test together. When loading a car or a canoe, ask "Where is the centre of gravity now?"

Common difficulties:

  • Thinking the CG must be on the object. Rings, L-shapes and boomerangs have CGs in empty space.
  • Thinking the CG is always the geometric middle. That's only true for uniform objects.
  • Safety: supervise the pushpin and scissors. In real canoes, everyone wears a lifejacket (PFD), whether or not they stand up.
  • Fair tests: the challenge question practises planning a fair test (A1.2).