Mass and weight, Balances, Centre of gravity

School Science Lessons
(UNPh08)
2024-07-31a

Mass and weight
Contents
8.1.0 Mass and weight
8.4.0 Balance, balances
8.2.0 Centre of gravity, centre of mass
8.3.0 Gravity

8.1.0 Mass and weight
Masses, iron, hexagonal, 1 kg, Masses, slotted, 1000 g total, Masses, slotted, 100 g total
See diagram 2.7: Weight

Mass, kilogram
The mass of a body is the quantity of matter it contains, as measured by its acceleration produced by a force, or by the force exerted on it by a gravitational field.
Every body or particle within a body has mass and is attracted towards the centre of the Earth by a force = mg, where m = mass, and g = the attractive force by which objects are attracted towards the centre of the Earth.
The intensity of gravity is measured by the acceleration caused by the gravitational force.
Inertial mass
The standard of mass is the kilogram, based the existence of a particular cylinder of platinum-iridium alloy.
This standard can be referred to as the inertial mass, m_i..
So mass is defined by its inertia.
Gravitational mass
However, mass is conveniently measured by using the weight, W, of the body, i.e. the force of gravity attracting it to the Earth.
W = mg, where g is the acceleration of fall that varies slightly in different places on the surface of the Earth.
To define mass in terms of the gravitational force it can produce, i.e. gravitational mass, m_g,
Use the formula: m_g = Fd² / MG where M is a standard body distance d from another body of mass m_g,
F is the gravitational force between the bodies and G is the universal gravitational constant.
However, m_i = m_g.
The kilogram is the base unit of mass.

Experiments
4.0 Balances (Primary)
11.1.0 Conservation of mass

8.2.0 Centre of gravity, centre of mass
Centre of mass, centre of gravity, balancing
The "Mini Balancing Eagle", (toy product) has its centre of mass below its beak.
See diagram 8.2.0a: Centre of mass of uniform rod.
Centre of mass is a point representing the summary of all gravitational forces and there is no torque about this point whatever the position of the body.
Centre of mass is a point representing the mean position of mass in an object.
In a uniform gravitational field, centre of mass and centre of gravity are in the same place.
For any object we deal with in our daily life, the centre of mass and centre of gravity are in the same position relative to the object.
For a regularly shaped object made of homogeneous material, the centre of mass is at its geometrical centre.
A symmetrical homogeneous object such as a sphere or cube has its centre of mass at its physical centre.
A hollow shape, e.g. a cup, may have its centre of mass in space inside the hollow, so the line of action of weight may pass through a point outside the body.
A tightrope walker in a circus may carry a long pole to ensure equilibrium, so that the line of action of his or her weight falls inside the rope.
The centre of mass of the tightrope walker must be vertically above the rope so that the weight and normal reaction are in the same line, otherwise a turning couple will occur.
Movement of arms and legs can adjust position on the rope, but this position is more easily adjusted by carrying a long pole.
So if falling to the right a movement of the pole to the left of the rope brings the centre of mass into a position vertically above the rope.
If the centre of mass of a tennis racquet is too far from the hand, the player has trouble holding the "head heavy" racquet horizontally.
Experiments
8.2.2 Centre of mass and stability of an object, balancing pins
8.2.3 Centre of mass of a double cone, uphill roller, circular cone with two heads on a ramp
8.2.4 Centre of mass of a shape, wobbling circles
8.2.5 Centre of mass of irregular-shaped object
8.2.6 Centre of mass of man and woman
8.2.7 Centre of mass of maps
8.2.8 Centre of mass of pine trees
8.2.9 Centre of mass of playing dice
8.2.10 Centre of mass with a plumb line, plumb bob
8.2.11 Centre of mass outside an object, U-shape cardboard
8.2.12 Centre of mass of biased bowls
8.2.13 Centre of mass glider
8.2.14 Centre of mass of thrown rod
8.2.15 Centre of mass of domino castle
8.2.16 Centre of mass of pencils
8.2.17 Centre of mass of hanging belt, "sky hook"
8.2.18 Centre of mass of suitcase with false bottom
8.2.19 High jumping
8.2.20 Overhanging pile, leaning tower of Lire
8.2.21 Point of balance of circular shape with circular cut-out
8.2.22 Rolling uphill
8.2.23 Stable, unstable and neutral equilibrium
8.2.24 Shared centre of mass, stability brace
8.2.25 Throw a disc
8.2.27 Tipping glass of water

8.3.0 Gravity
8.3.01 Gravity, gravitational force, law of gravitation, gravitational field
Experiments
8.3.1 Acceleration of marbles down an incline
8.3.2 Ball bearings fall together
8.3.3 Ball projected upwards from a moving cart
8.3.4 Bullet projected upward from a stationary gun
36.54: Discover weightlessness
8.3.5 Falling ball and paper
8.3.6 Gravity-powered lamp to enter field tests
8.3.7 Measure "g" with Kater's reversible pendulum
8.3.8 Satellite launcher
8.3.9 Shape of a hanging cable or flexible chain, catenary curve
8.3.10 Velocity of an arrow
8.3.11 Weight of a falling chain
8.3.12 Weightlessness

8.4.0 Balance, balances
4.0 Balances (Primary)
Balance, digital, 5000, MTA
Balance, digital, 200, MTA
Experiments
8.4.2 Balance with a metre stick
8.4.3 Balance with a see-saw
8.4.4 Balanced forks on glass rim
8.4.5 Balanced H-shape
8.4.6 Balanced mobile

8.1.0.1 Mass
Mass is measured with a beam balance.
Mass is the quantity of matter in an object as measured by its inertia, i.e. its resistance to acceleration.
Mass determines the acceleration produced in an object by a given force acting on it, the acceleration being inversely proportional to the mass of the object.
Mass determined in this way may be called the inertial mass.
In the SI system, the base unit of mass is the kilogram.
A platinum iridium cylinder of one kilogram is the standard unit of mass to which you can compare all other masses.
The mass also determines the force exerted on an object by gravity on Earth.
Mass determined in this way may be called the gravitational mass, although this attraction varies slightly from place to place.
inertial mass = gravitational mass.

8.1.0.2 Weight
Weight is measured with a spring balance (spring scale, newton meter).
Mass is commonly measured by using gravitation.
The weight of an object is the force of gravity on it.
Weight, W, = mg (mass × the acceleration of free fall, g).
Weight is a force, measured in newton, N, the SI unit of force.
A spring balance supporting an object measures this invisible force.
At a given place, equal masses experience equal gravitational forces.
You can compare the mass of objects by comparing the weight of objects at the same place.
However, g is not the same on all parts of the surface of the earth.
So the same body, mass m, could have different weights in different places.

8.1.0.3 Weight and gravitational force
The weight of an object also depends on its location.
Measured in a reference system fixed to the earth, the weight of an object is the same as the earth's gravitational force acting on it.
This force decreases as the object moves away from the earth and will eventually become negligible.
The weight of the object is changing under the above circumstances.
Content of matter of the object, measured in kg does not change, unless you are dealing with relativistic physics, where objects experience speeds approaching that of light.
An astronaut whose mass on the surface of the earth is 90 kg still has the same mass of 90 kg on the surface of the moon.
However, the astronaut's weight, which is 90 kg weight on the earth's surface, would only be about 15 kg weight on the moon's surface.
Using SI units, the mass is m kg, but the weight is mg Newton.
Since g at the moon is about one sixth of g at the earth, the weight of a man on the moon will be one sixth of his weight on the earth.

8.1.0.4 Weight in space
A space-ship in orbit is still within the earth's gravitational field.
Its weight is exactly the force required to keep the ship in orbit.
In a reference system attached to the ship, everything inside the ship is weightless.
With a slight push against one wall of the cabin a man can propel himself towards the opposite wall.
Further away from the earth, the gravitational force becomes negligible and the space-ship will move in a straight line unless acted upon by forces from its own engine or from other objects like the moon.
Outside the space-ship a man could, if he were completely free to move, push himself off in any direction never to return.
To avoid such a possibility, safety lines are attached to the space suits of astronauts, who work in space.

8.1.2 Beam balances
Beam balances (two pan balances), measure mass when the beam is perfectly horizontal.
Also, lever balances (substitution balances), using a single lever arm measure mass.

8.1.3 Electronic balances
Electronic balances use electric force to measure mass.

8.1.4 Spring balances
Spring balances, force meter, top loading balance, compression balance, kitchen scale, bathroom scale, measure weigh
1. Use a spring balance marked in newton, attach an object, e.g. an orange, to the hook at the bottom, and allow the pull of gravity to stretch the spring in the balance.
2. Check the scale by hanging a 1 kilogram mass from the hook.
In most places, the weight of a 1 kilogram mass = 9.8 newton.
The mass of an object is always the same, but the weight of the same object can vary slightly in different places.
3. Add 100 g masses to a spring scale and record the extension on the scale in a table showing load and extension of the spring.
On a graph plot the total loads along the y-axis and the total extensions along the x-axis.
Take off the 100 g masses one at a time and check the total extensions.
The straight line graph show that the extension is proportional to the load.
4. However, the spring balance may be marked to indicate the total load that can be applied.
If too much load is applied to a spring, it may stretch too much and not be able to return to its original shape.
The load that causes this one way stretch is called the elastic limit.
So we can say that within the elastic limit the extension is proportional to the load.
The formal statement of this property of materials is called Hooke's law.

8.2.2 Centre of mass and stability of an object, balancing pins
See diagram 8.2.2: Balancing pins.
Put a large cork stopper on the table with the larger area down.
Hold a long plastic knitting needle vertically over the centre of the smaller area of the cork.
Push the pointed end down through the centre of the cork.
Cut off the head of the knitting needle and insert the cut end into a Styrofoam ball.
Push two other plastic knitting needles through the holes of one-hole rubber stoppers, so the stoppers reach the heads of the knitting needles.
Insert the pointed ends of these knitting needles into each side of the cork.
Place the end of the knitting needle pushed through the cork on any convex point and the system remains balanced about the convex point pivot.
If you move the rubber stoppers up the knitting needles the system is less stable, because the centre of mass of the system approaches the level of the pivot.
By slanting the knitting needles with the rubber stoppers pushed down, the centre of mass of the system is lowered below the pivot, and the stability of the system is increased.
The bottom of a racing car is as low as possible to obtain the lowest possible centre of mass and prevent the racing car rolling over during a turn.

8.2.3.1 Centre of mass and falling over, toppling
See diagram 8.2.3.1: Toppling body.
An object taller than broader falls over, topples, when a vertical line drawn through its centre of mass does not pass through it base.

8.2.4 Centre of mass of a double cone, uphill roller, circular cone with two heads on a ramp
See diagram 8.2.4: Uphill roller, double cone paradox.
1. Two cones joined at their bases are place on a rail system in a V-shape with the widest part of the V is lower than the apex of the V. The centre of mass of the cones at the apex of the rail is higher than at the widest point of the rail.
If placed at the apex, the cone rolls up the incline to the widest point.
Even though the cones appear to be rolling upward, their centre of mass is moving downhill, because the two rails are spread apart, allowing the centre of mass to sink lower.
2. As a double cone moves up an set of inclined rails its centre of mass lowers.
The double cone appears to roll uphill.
A double cone rolls up an inclined track.
As the double cone moves "up", the increasing width lowers the double cone, so that its centre of mass actually moves "down".
2.1 Make a circular cone with two heads as follows:
Tape together two plastic funnels at their mouths with a smooth connection.
Cut out two identical equilateral triangles from cardboard with one side in the shape of an arc.
Roll the triangles, beginning at a straight side, then tape together the straight sides to form two circular cones.
Rub smooth the bottom of each cone and tape them together.
2.2 Make a ramp as follows:
Cut out a long narrow strip of cardboard and fold in half to make a V-shape.
Cut cardboard in rectangle shape and tape it to the V-shape to make a ramp.
2.3 Use two rulers leaning on a book or use two drinking straws.
Put the cones on a lower end of the cardboard ramp.
If the surfaces of the cone and ramp are smooth the cone rolls up along the ramp.
Adjust the upper distance between two rulers or straws to make the cone roll up or down the ramp.
Hold the upper ends of the rulers or straws to first make the distance between them small, then move them apart until the cone begins to roll upward.
Before the cone arrives at the top of the ramp decrease the distance between the two ends of the rulers or straws to make the cone roll down.
Measure the height of the top of the cone before and after the rolling.
The top of the cone after rolling is lower than the height before the rolling.
The cone is symmetrical, so its centre of mass is on the line the connecting of the two tops, so the height of the tops is the height of the centre of mass of the cone.
When the bottom of the ramp is narrower, the cone at the lower place of the ramp has a higher centre of mass.
While the cone rolls up to the top of the ramp due to the wider width of the top, the centre of mass of the cone is lower.
So the centre of mass of the cone is higher at bottom, the centre of mass of the cone is lower at the top, so the cone does not roll upward, but downward.
3. An object in the shape of a cylinder cannot roll up itself on such a ramp, because its centre of mass will rise.
Repeat the experiment with a ball.
Put a ball with a suitable size on the ramp.
It can roll up itself and the speed of rolling is faster than the speed of a cone.
4. A loaded disc is put on an inclined plane so it rolls uphill.
A large wood disc weighted on one side will roll uphill or to the edge of a table and back.

8.2.4 Centre of mass of a shape, wobbling circles
See diagram 8.2.5: Spinning card.
All flying objects spin around their centre of mass.
Cut cardboard into a kidney shape.
Support it on your finger to let it balance and find its centre of mass then mark the centre of mass of it by a pencil.
Draw concentric circles around the centre of mass on one surface of the cardboard, called A surface, by a compass and thicken the with the marker.
Draw the same concentric circles on other side of the surface, B surfaces, but around a spot about 3 cm away from the centre of mass.
All you have done above is in the condition without having the students' observation.
Let the A surface of the card face to the students, hold the edge of it by your fingers, throw it by using wrist action to make it spin vertically in the air.
Catch it when it comes down.
Then reverse it to make the B surface face to the students, throw it again.
What are the circles on the card like as the card flies in the air?
The situations of the spinning on two sides of the card are different.
The circles are steady in the first spin and wobbling in the second.
All objects spin around their centre of mass.
On the A side of the cardboard the concentric circles are around the centre of mass, so they stay steady as they spin.
When you see the other side B, the circles wobble, because it is off the centre of mass and the circles spin around a point outside the of mass, making a wobbling motion.

8.2.5 Centre of mass of irregular-shaped object
See diagram 8.2.1: Centre of mass of irregular-shaped object, potato.
Draw an irregular picture on cardboard, e.g. a potato.
Cut out the picture with scissors.
Punch 3 small holes at the edge of the cardboard shape such that the distances between holes are about the same.
Pass a thread through one hole and tie it.
Suspend the cardboard shape by lifting the thread.
The centre of mass must be directly below the point of suspension.
When the cardboard shape stops moving, put the edge of a ruler on the thread and slide the ruler over the cardboard shape vertically down along
the direction of the thread.
Hold the ruler and cardboard shape together tightly with your fingers, lay them on the table and draw a straight line on the cardboard shape with the ruler.
Untie the thread, tie it in other holes and repeat as above until you have drawn 3 lines on the card.
The 3 lines intersect at the centre of mass of the cardboard shape.
Test that the point is the centre of mass, by supporting the cardboard shape with the end of sharpened pencil below the point.
The cardboard shape will remain in equilibrium.

8.2.10 Centre of mass with a plumb line, plumb bob
See diagram 8.169: Find centre of mass.
A plumb line is a ball of lead attached to a string used to define a vertical line.
1. Cut out a map and punch holes where cities or town are shown on the map.
Hang the map from a peg through a hole and use a plumb line to draw a vertical line on the map down from the hole.
Repeat with other holes.
Where the lines intersect is the centre of mass.
2. Find centre of mass with a plumb bob.
Use a chalk line on the plumb bob and snap it to make a quick vertical line.
Suspend various regular shapes and an irregular board from several points and use a plumb bob to find the centre of mass.
3. Hang a potato from several positions and stick a pin in at the bottom in each case so that all pins point to the centre of mass.
4. Place a block on an incline and raise the incline until the block tips.

8.2.19 High jumping
See diagram 8.169: Find centre of mass.
1. The simplest method used in schools is to jump so that the body and centre of mass passes over the bar.
The body is like a pair of scissors where first one blade (leg), then the cross-blades handle (upper body, then the other blade (leg), passes over the bar.
2. The Western Roll used by athletes was to kick the leg nearest the bar strongly so that body passed over the bar parallel to it.
3. Later the Straddle technique used by athletes was to kick the leg away from the bar strongly, so that the body passed over the bar face down.
4. The 'Fosbury Flop' technique, first exhibited by Stefan Holm, Sweden, Gold Medal at the 1968 Olympics, in Mexico City, which allowed a backwards flip over the bar, allowed the jumper to curve the body around the bar after jumping backwards and then landing on the shoulders.
In this technique, the centre of mass of the body remains below the bar, just like the centre of mass of an L-shaped object.
This techniques has allows jumps up to 6 metres, but requires the expert placing of excellent air bags for the jumper to land on and so this technique
is too dangerous for use in schools.
5. If a high jumper, mass M, launches vertically upward at speed U, for height H that can be reached, U² = 2gH, where g = acceleration due to gravity and H = height of the jumper's centre of gravity.
Energy of motion of jumper's take-off, 1/2 MU² = potential energy MgH of the jumper at height H.
So U² = 2gH.

8.2.6 Centre of mass of man and woman
See diagram 8.2.6: Woman and man lifting.
The centre of mass of a woman is usually about 2-3 cm lower and further back than the centre of mass of a man.
A woman has a greater proportion of weight below the waist, in the region of the hips, buttocks and thighs.
Men have comparatively broader chest and shoulders, so their centre of mass is usually above the waist.
When a man bends forward like a snow skier, his centre of mass is above his toes, but a woman's centre of mass is above her heels.
The lower centre of mass of women is supposed to help them to be better dancers than men.
1. Sit in a straight-backed armless chair with you back against the back of the chair.
Cross the arms about the chest to grab the top of the shoulders.
Keep the feet flat against the floor and stand up.
When sitting upright the centre of mass is at the base of the spine.
If you try to stand up while keeping the back straight you prevent the centre of mass from moving over the feet, the support base.
So when people sitting upright want to get out of the chair and stand up they first lean forwards.
2. Place a chair with its back to the wall.
Stand a pace back from the wall so that the chair is in front of you.
Lean forward over the chair until your forehead rests against the wall.
Reach down and pick up the chair by its arms.
Stand up straight while keeping hold of the chair.
Usually men cannot do it, but women can do it, because they have a lower centre of mass.
Men usually have bigger feet, so when a man leans forward his centre of mass is farther from his base than for a woman.
This experiment works best if the large man wears big boots and the small woman wears high heels.
3. Women can bend forward in the kneeling position, but if a man tries to do that he usually falls forward, because in this position their centre of mass is in front of their knees.
Kneel on the floor at the distance of your forearm from a box of absorbent tissues placed upright on the floor.
Hold you hands behind your back and lean forward to knock the box over with your nose.
Most women can do it, but most men fall forward onto the box.
4. Stand with heels against the wall and heels together.
Drop a coin on the floor to land in front of the feet.
Pick up the coin without moving the feet or bending the knees.
When standing against the wall the centre of mass is over the feet.
When bending over the centre of mass moves forward, so to keep balance the feet must also move forward to keep a stable base under the centre of mass.
The coin cannot be picked up without falling over.
5. Stand with heels, hips and shoulders against the wall.
You cannot jump without leaning away from the wall and falling over.
6. Stand with the right side, cheek and right foot pressed against the wall.
Lift the left foot up and away from the wall.
It cannot be done, but if the wall was a screen, when you raised the left foot, the screen would move slightly to the right.
7.1 Bend forward, bend the knees slightly, hold the toes with both hands, and jump forward.
To jump forward the centre of mass must move forward of the base, which cannot be done while holding the toes.
However, the jump could be done if the leg muscles were strong enough to lift the body off the ground and support the unbalanced position when jumping.
7.2 Bend forward, bend the knees slightly, hold the toes with both hands, and jump backwards.
This jump is possible, because the support base moves first and the centre of mass maintains a balanced state.
8. Put a small cushion on the floor and squat down so that it is below and in front of the nose.
Place a broomstick across the groove of the bent knees and crook the elbows around it.
Lean forward to touch the cushion with the nose.
As you lean forward, the centre of mass moves forwards from over the feet, so you may become unstable, fall forwards and your nose hits the cushion.
9. Half open the door and stand with your nose and stomach just touching the edge of the door.
Place the feet each side of the edge of the door and just touching the ankles.
Rise on your tiptoes.
The door prevents you leaning forward so you cannot transfer the centre of mass forward.
Stand away from the door, stand on tiptoes and feel the forward movement.

8.2.11 Centre of mass outside an object, U-shape cardboard
See diagram 8.4.9: U-shape.
1. To verify the existence of a centre of mass not on the object, lay a U-shape magnet on a piece of cardboard, then draw along the boundary between magnet and cardboard to make an U-shape picture.
Cut off this U-shape cardboard.
Draw two vertical lines used to hang the cardboard.
Define two points in two ends of each thread, up and down AA' and BB', punch four holes at these four points.
Thread through A and make a knot.
The other end of the thread goes through A', around back under the cardboard, and meets again with thread in A'.
Tie them together.
Fix another thread at BB' with the same operation.
Make each thread as tight as possible.
Pick up the cross point of two threads with tweezers, the U-shape cardboard will maintain horizontally in the air.
This shows that the point is its centre of mass.
Punch another hole at the C point besides the A' (or B').
Untie A' (or B') and move to C, tie it.
Pick up the new cross point of two threads with tweezers.
You cannot balance the U-shape cardboard.
2. The centre of mass of an object that is empty in the centre may not be on the object.
You cannot see the centre of mass so to verify its existence place an U-shape magnet on a piece of cardboard and draw a line along the boundary between magnet and cardboard to make an U-shape picture.
Cut out this U-shape.
Draw two vertical lines parallel to the ends of the U-shape to cross its arms.
Punch four holes AA' and BB' on the two vertical lines.
Pass a thread through A, make a knot, then attach the other end of the thread tightly to A'.
Fix another thread through BB' in the same way.
Make each thread as tight as possible.
Pick up the cross point of two threads with tweezers and the U-shape cardboard will stay horizontal, showing that the cross point is its centre of mass.
Punch another hole at C besides A'.
Untie A' and move to C, tie it.
Pick up the new cross point of two threads with tweezers.

8.2.7 Centre of mass of maps
To find the centre of mass of an irregular shape, e.g. map of a country, drill holes through places indicating large cities.
Use a peg to suspend the map from one of the holes and hang a plumb bob form the peg.
Mark the plumb line with a pencil.
Repeat the suspension by using other holes in the map.
The centre of mass is where the plumb lines intersect.

8.2.12 Centre of mass of biased bowls
See diagram 8.2.10: Path of lawn bowl After Steve Ritchie, The Australian Science Teachers Journal, Vol. 33 No. 1.
The bowl used in lawn bowls has a biased shape, so when bowled in a straight line it follows a curved path.
To find the bias of a bowl and how this affects its path of the bowl you need a lawn bowl, metre stick, chalk, and a carpet or evenly grassed surface.
Observe different size discs on the sides of the bowl.
Stand the bowl on its rolling surface and push it gently.
When it stops it leans to one side or falls onto its side with the small disc closer to the ground.
Stand the bowl with its rolling surface resting on a bench.
Place a metre rule horizontally across the top of the bowl.
Mark this spot with a piece of chalk.
Hold the piece of chalk at this spot while rotating the bowl, keeping it in its upright position during rotation.
After one complete rotation you have drawn a complete chalk circle on the bowl, called the running line.
The running line is not in the centre of the running surface.
Draw a cross-section of the bowl and include the running line.
Identify the sides of the bowl in your diagram with a large and small disc.
The running line of the bowl is closer to the end marked by the large disc.
Bowls are made of a plastic composition and there is more plastic mass on the small disc side of the running line.
The bowl is unbalanced when bowled, because there is more mass on one side of the running line.
The bowl is weighted or biased to one side of the bowl.
The bowl is biased on the small disc side of the running line.
When a bowl is delivered, i.e. bowled, it curves in the direction of the bias.
Refer to the diagram to predict the direction of aim, A or B, taken for the bowl to approach the target, T.
Change the bias, i.e. turn the bowl around.
Try to deliver the bowl in such a way that the bowl stops on the target.
The bowl curves towards the bias, because throughout the path of the bowl, the bowl gradually leans over towards its bias.
This leaning of the bowl shifts its centre of its mass, and running line, sideways, thus causing a curved path.

8.2.8 Centre of mass of pine trees
See diagram 8.2.11: Pine trees.
A pine tree growing by itself with no other trees near it receives sunlight from all sides.
Its largest branches are near the bases of the tree and the branches get smaller towards the top.
Consequently its centre of mass is in the trunk near the base.
However, when pine trees grow close together in a forest their sides are shaded by other trees so the lower branches do not develop much and later dies.
They receive light only from above.
Consequently forest trees have the centre of mass in the trunk near the top of the tree.
They are "top heavy" and compared to free standing trees of the same age, their trunks are longer and thinner and their roots do spread out so far.
During a very strong wind the free standing tree will just bend with little damage except to the tip of the tree.
However, around a clearing in a forest, many trees may be knocked down due to their high centre of mass.

8.2.18 Centre of mass of suitcase with false bottom
See diagram 8.2.12: Gold smuggler.
Make a double bottom, "false bottom", in a small cardboard box with a lead weight in the space below.
Shake the lead weight to one end of the box.
You can place the box on the table so that only the lead weight is over the table and the box does not fall.
A smuggler make carry gold in the false bottom of a suitcase.
When the smuggler carries the suit case through the customs office, it tilts to one side so that the centre of mass, now at the side, is vertically below the suspension point of the suitcase handle.

8.2.17 Centre of mass of hanging belt, "sky hook"
See diagram 8.2.13: Hanging belt
Dissemble a plastic two arm clothes peg and remove the clip.
Hang a heavy leather belt evenly over one arm of the clothes peg and use the clip to secure the belt to the arm.
You can raise the system by raising only your finger tip under one end of the clothes peg arm.
The system is stable when the clothes peg arm is inclined slightly downwards away from you.
At this angle, the belt hanging down is slightly inclined towards you so the centre of mass of the system lies vertically below your finger tip.

8.2.24 Shared centre of mass, stability brace
See diagram 8.2.14: Braces between verticals.
Ride your bicycle parallel to another cyclist and at the same speed.
Take hold of the handle bar nearest to you of the other cyclist.
The other cyclist takes hold of your nearest handle bar.
Straighten both your arms while still holding the other's handle bar.
You can both slow to a stop and not fall over, because the arm brace of the two cyclists has given the system a common centre of mass.
Within construction towers and scaffolding, braces between vertical elements form stable triangles with two brace sides and one vertical element side.

8.2.26 Tipping bottle
See diagram 8.2.15: Tipping bottle.
Put a plastic bottle with an oval base, e.g. a shampoo bottle on a sloping window sill as follows:
(1) Full bottle with longer axis of the oval base parallel to the edge of the window sill.
(2) Half full bottle with longer axis of the oval base parallel to the edge of the window sill.
(3) Full bottle with longer axis of the oval base at right angles to the edge of the window sill.
(4) Half full bottle with longer axis of the oval base at right angles to the edge of the window sill.
The bottle remains stable on the sloping window sill only if the centre of mass is supported by the surface of the window sill.
(1) and (3) have a higher centre of mass than (2) and (4).
(1) and (2) are more likely to topple over than (3). and (4).
So the most stable system is probably (4) then (3) then (2) then (1), depending on whether the centre of mass is supported.

8.2.27 Tipping glass of water
See diagram 8.2.15.1: Tipping glass of water.
Put a glass of water on a table with a table cloth.
When nobody else is looking put a rod on the table, but under the table cloth.
Slowly pull the table cloth across the rod so that the glass of water tips up.
The centre of mass is still in the same place in the glass of water.

8.2.21 Point of balance of circular shape with circular cutout
See diagram 2.0.2a: Balanced circle with circular cutout, golden ratio.
Find the point of balance of a circular piece of metal by balancing it on a point.
If the metal is cut in a perfect circle the point of balance is at the centre.
Cut a small perfect circle from the piece of metal so that they have the same point on their diameters.
The balancing point is on the edge of the cutout on the same diameter as the original centre of the circular piece of metal.
The ratio of the original diameter to the diameter of the circular cutout is the golden ratio, 1.61803.

8.2.9 Centre of mass of playing dice
The cube dice have the opposite side labelled 1 and 6, 5 and 2 4 and 3. Sides 4, 5 and 6 touch and side 1, 2 and 3 touch.
If the spots on the dice are made by drilling out a small holes, then the centre of mass is below the geometric centre of the cube and nearer the smaller number.
So when such dice are thrown, the faces with more than three spots are more likely to be uppermost.
However, if the spots on the dice are made by painting on the spots, the faces with less than four spots are more likely to be uppermost.

8.2.23 Stable, unstable and neutral equilibrium
1. A thin block on a cylinder is stable, a thick block is not.
2. Stick two forks and a match together and balance on a glass while pouring out the water.
3. Balance ten landscape spikes on the head of a single upright spike.
4. Hang a giant food service spoon with curved handle end on your nose.
5. Toy horse has an attached weight to lower the centre of mass as an example of a stable equilibrium of a centre of mass object.
6. Wires form a support at the centre of mass of a laboratory stool.
Construct a stool so that wires crossed diagonally will intersect at the centre of mass.
The stool can be oriented in any direction.
7. Hide heavy weights in the ends of a stool's legs so it will balance on a vertical rod placed under the seat.
8. Spread the bristles and a straw broom will stand upright.
9. Stick the neck of a bottle through a hole in a slanted board and it stands up.
10. Use adhesive tape to fix a weight in the corner of a cardboard box, e.g. a shoe box.
Push the box towards the edge of the table until it is about to fall down, and so locate the centre of mass of the box.

8.2.20 Overhanging pile, leaning tower of Lire
See diagram 8.2.18.1: Leaning tower of Lire.
1. A set of eight blocks, e.g. wooden slabs, blocks, books, dominoes, is stacked so the top block is completely over the edge of the table.
Step lengths go as L / 2n.
2. Find the maximum number of objects used to construct a pile over similar objects with at least part of any object overhanging a void beyond the edge of the table.
In theory, an infinite number of objects may be used, but in practice most people can stack no more than 10 overhanging objects.
3. Use diagram 8.2.18.1 to construct a pile of object with each object contributing to maximum overhang of the pile.
A regular rectangular slab in an original position on the edge of a table, such that its right side is vertically above the edge of the table, may be moved to the right for almost half its length before falling off the table.
Maximum overhang exists when the centre of mass of the object is almost above the edge of the table.
(The following calculation and diagram 8.2.18.1 are based on "The Leaning Tower of Lire" created by "Interactive Real Analysis" with the help of Jillian Gaglione.)
The combined centre of mass of two objects, X, of mass m1 and m2 at distances x1 and x2 from the left = X = x1m1 + x2m2 / (m1 + m2) 1a.
A set of slabs has each slab with width 2, mass 1 and centre of mass -1.
Place slab 1 at the original position with its right side above the edge of the table, i.e. 0 on the number line.
1b. Move slab 1 almost half its width to the right so that its centre of mass is over the edge of the table.
The overhang distance, d = 1.
2a. Raise slab 1 vertically up, place slab 2 in the original position (right edge at 0), lower slab 1 vertically down.
2b. Move the block of 2 slabs by to the right so its centre of mass is at 0.
The overhang distance, d = 1 + X = x1m1 + x2m2 / (m1 + m2), = (-1) ×1 + 0 × 1 / 2 = - 3a.
Raise the block of 2 slabs vertically, place slab 3 in the original position, lower the block of 2 slabs vertically down.
3b. Move the block of 3 slabs to the right until they almost tip over the edge of the table.
This position is when the slab 3 has moved 1/3 to the right.
The overhang distance, d = 1 + + 1/3.
X = x1m1 + x2m2 / (m1 + m2), = (-1) × 1 + 0 × 2 / 3 = -1/3
4a. Raise the block of 3 slabs vertically, place slab 4 in the original position, lower the block of 3 slabs vertically down.
4b. Move the block of 4 slabs to the right until they almost tip over the edge of the table.
This position is when the slab 4 has moved to the right.
The overhang distance, d = 1 + + 1/3 +.
X = x1m1 + x2m2 / (m1 + m2), = (-1) × 1 + 0 × 3 / 4 = -3/4.
and so on.
X = x1m1 + x2m2 / (m1 + m2), = (-1) × 1 + 0 × n / 1 + n = - 1 / (n +1)
The overhang distance, d = 1 + + 1/3 + 1 / (n -1)
This problem is also called the "book-stacking problem" or "brick-stacking problem".
An overhang of two brick lengths is possible, as in a corbel arch.
However, it is unstable, because of outward pressure on the up walls.
The Roman arch with a keystone supplanted the corbel arch, because it is more stable.

8.2.13 Centre of mass glider
The glider has two freely swinging pendulums.
As the pendulum swings back and forth the glider moves in such a way that the centre of mass remains at rest.
If the pendulums swing in opposite directions, the glider does not move.
The centre of mass is stationary.
If the pendulums swing together, are synchronous, the glider moves.
The centre of mass moves forwards and backwards.

8.2.14 Centre of mass of thrown rod
See diagram 8.2.20: Centre of mass rod
A rod has unequal masses attached at each end.
A light globe powered by a small battery can be attached anywhere along the rod by sliding it.
The rod is thrown in the dark with the light globe switched on.
If the light globe is at the centre of mass it describes a parabolic path when the rod is thrown.
If the light globe is at another position it describes a very complicated path when the rod is thrown.

8.2.15 Centre of mass of domino castle
See diagram 8.2.21: Domino castle.
Place dominoes 1, 2 and 3 on the table, upright and equal distance apart.
Add domino 4 horizontally across them.
Continue adding dominoes until all the 28 dominoes in the set are in the structure.
VERY CAREFULLY remove dominoes 1 and 3 leaving domino 2 to support the whole structure.
VERY CAREFULLY add dominoes 1 and 3 to the top of the structure.
The centre of mass is the point in a body around which its weight is evenly balanced.
In this structure, if it is still standing, the centre of mass is a point at the bottom of domino 2 equidistant from its two sides.

8.2.16 Centre of mass of pencils
Use long (18 cm) and short (8 cm) identical pencils, sharpened to the same point.
Stand the two pencils upright and then release both of them at the same time.
The short pencil falls to the table before the long pencil.
The pencil moves in the direction of the fall, so that the bottom end of the pencil has moved a certain distance, the long pencil more than the short pencil.
By balancing the pencils over a horizontal pivot, e.g. your index finger, you can estimate the position of the centre of mass inside the pencil.

8.2.22 Rolling uphill
Remove the top and bottom from a cylindrical coffee can.
Attach a lump of wet clay or plasticine (modelling clay) inside the can equidistant from the two open ends.
Use a grease pencil to mark the outside of the coffee can to show the exact position of the clay lump.
Make a ramp by resting the edge of a book on a similar book.
Place the coffee can on the bottom edge of the book ramp, parallel to the edge of the book and with the grease pencil mark touching the edge of the book ramp.
Turn the coffee tin in different positions keeping the grease spot mark away from the edge of the book until the grease spot mark is once again above the edge of the book, but at the top.
Start turning the coffee tin so that the grease spot is over the book ramp.
The coffee tin will start to move uphill over the book ramp.

8.2.25 Throw a disc
See diagram 8.2.24: Motion and centre of mass.
Cut out a circular disc of PVC plastic foam board.
Cut out a central circle for inserting a light bulb and battery.
Cut an off-centre circle to remain as a plug.
Screw a light bulb into the centre of the disc.
Hold the disc near the centre and throw it to a student.
The light bulb shows the centre of mass.
Change the centre of mass of the disc bye exchanging the foam plug for a lead-weighted plug.
Move the bulb off-centre to the new centre of mass.
Hold the disc near the centre and throw it to a student.

8.3.01 Gravity, gravitational force, law of gravitation, gravitational field
Gravity is the attractive force between two masses or between celestial bodies or between a body and the Earth.
Standard gravity, g, is the standard value of gravitational acceleration at sea level on the Earth.
The intensity of gravity is measured by the acceleration produced by this attractive force, i.e. gravitational force.
Gravitational force is said to be propagated by gravitational waves through space, but nobody has yet detected them.
The law of gravitation states that the force between two bodies is directly proportional to the product of their masses and inversely proportional to the distance between them.
The gravitational constant is = 6.67 × 10^-11 N m²kg ^-2.
The force of gravity exerted by the earth pulls everything down with the same force no matter what is the mass with a constant acceleration of 9.8 ms^-1.
A gravitational field surrounds a massive object where another object with mass can experience a force of gravitational attraction.
An electric field occurs where an electric charge experiences a force and is usually caused by the location of other charges.
Electrical fields can be both attractive and repulsive and so can be shielded.
However, gravitational fields are only attractive and so cannot be shielded.

8.3.1 Acceleration of marbles down an incline
Use a 3 m plank of wood with a groove down the centre.
Incline the plank so that marbles can roll down the groove.
Arrange small tin flags hung from wires so that the marbles hit them and make "clinks" sounds.
Put the flags at regular intervals, e.g. 25, 50, 75, 100 cm, from the end of the plank.
Roll a marble down the groove and listen to the time intervals between "clink" sounds.
The time intervals between the "clinks" reduce as the ball rolls down the incline.
Arrange the flags so that the clinks occur at equal intervals of time.
Measure the distance between the flags.
The distance between the flags increases down the incline in the ratio 1:3:5:7:9.

8.3.2 Ball bearings fall together
See diagram 8.234: Simultaneous fall.
See diagram 14.2.4: Spring-loaded device.
1. A spring loaded device drops one ball and projects the other horizontally.
2. Two balls simultaneously dropped and projected horizontally hit the floor together.
Drop one billiard ball and shoot another out simultaneous.
One ball is projected horizontally as another is dropped simultaneously.
Instructor rolls a ball off the hand while walking at a constant velocity.
3. Use two clothes-pegs, a pair of ball bearings and a wide rubber band.
Fix the band lengthways around one peg.
Then open the peg and force a ball against the tension of the rubber band between the prongs of the peg.
Grip the other ball with the second peg.
Hold the pegs side-by-side, pointing away horizontally above the floor.
Squeeze both pegs at once.
At the same moment, one ball begins to fall vertically, and the other is shot forwards.
Note what happens by looking and listening very carefully.
Repeat the experiment from different heights and with a tighter rubber band.
If the experiment is done correctly, while the ball bearings land in different places they strike the ground simultaneously.

8.3.9 Shape of a hanging cable or flexible chain, catenary curve
See diagram 8.2.18: Catenary curve.
See diagram 2.0.5: Conic sections.
See diagram 2.0.6: Parabola.
The shape of a hanging chain, called the catenary, has the minimum potential energy of any possible shape and the lowest possible centre of mass.
It is the shape of high voltage cables and overhead cables for electric railways.
The Gateway Arch, in St. Louis, Missouri, is an upside-down catenary, the optimal shape for a self-supporting arch, because it minimizes the shear stresses, being directed along the line of the arch towards the ground.
Hanging cables or wires are never horizontal, because no horizontal force can have a vertical component to overcome the weight of the hanging cable.
If symmetrical about the y axis, y = c cos h, where c = the point where it intersects the y axis.
A suspension bridge has to support the weight of its cables + the horizontal deck of the bridge.
The cable of a suspension bridge hangs in a parabola, not in a catenary, if the droppers allow the load to be suspended horizontally, e.g. Clifton
Suspension Bridge, Brooklyn Bridge.
A parabola is the intersection of a cone with a plane parallel to the side of the cone.
Y² = 4 ax, where a = distance from the focus to the origin.

8.3.7 Measure "g" with Kater's reversible pendulum
See diagram: Kater's reversible pendulum.
From 1906 until 1968, this pendulum, invented by Captain Henry Kater, 1818, was adopted as the standard to measure the value of the acceleration due to gravity, g, for the world gravity network.
A simple pendulum can be used to measure the value of g by measuring the length L and the period T and substituting in the equation: T = 2π L /g.
The value of T is found by timing many swings of the pendulum, but measurement of the length of the pendulum is exact enough for most purposes.
Also, the exact position of the centre of the mass is not known.
However, with Kater's reversible pendulum, two knife edge pivot points and two adjustable masses can have adjustable positions on the rod, so that the period of swing is the same from either edge.
The period for this pendulum is the same as with a simple pendulum with length L is the distance between the two knife edges.
It consists of a rod provided with two knife edges pivots facing inward on opposite sides of the centre of mass.
The bobs are adjustable in position.
These positions are adjusted until the periods are the same when the pendulum is suspended from either knife edge.
So g = 4πi /T (h1 + h2), where (h1+ h2) is the distance between the knife edges.
Experiment
Measure L with a precision of 0.05 cm, and use a digital wristwatch or analog timer to measure the period.
The pendulum knife edge support is a U-shape piece of aluminium clamped onto a rigid bench rod.
Adjust the distance between the m to a predetermined distance, e.g. 50 cm on a 0.7 m iron rod.
Adjust of the positions of the knife edges and masses until the two periods are equal.
Measure the time for of 100 swings of the pendulum to give a value of about T = 1.4 seconds.
Calculate the local value of g using the formula: T = 2π L / g.

8.3.6 Gravity-powered lamp to enter field tests
A report in 22/07/13 of trials in Africa and Asia that will test the performance of the lamp that converts energy from a descending weight into light.
It could provide an alternative to kerosene and solar lamps in rural areas.
The device, a gravity-powered LED lamp called "Gravity Light", works by attaching a weighted bag below it from a cord.
As the bag slowly descends, gears convert the weight into energy to provide up to 30 minutes of light, depending on the weight of the bag.
Each device comes with an empty bag, which can be filled with up to 12 kilograms of material such as earth, rocks and sand.
It needs no battery.
It overcomes the limitations of solar lamps that need sunlight and have a limited battery life, and kerosene lamps that cause indoor pollution and are expensive to refuel.
Each device is expected to cost between US $5 and US $10.
It has no pollution disadvantages and the LED should provide better lighting.

8.3.5 Falling ball and paper
Select a solid ball, e.g. a marble, golf ball and a square sheet of paper.
Squeeze the paper tightly in you fist to make a paper ball the same diameter as the solid ball.
If the paper ball is too small use a larger square of paper.
If the paper ball is too large cut the paper to make a smaller square.
Flatten the paper ball with your hand and spread the paper to make it a flattened square sheet.
Select a second sheet of paper and cut it to be the same size as the flattened square sheet.
1. Hold the solid ball and the second sheet of paper above your head at the same height and let them drop at the same time.
The ball fall straight down, but the sheet of paper flutters from side to side.
The solid ball hits the ground before the sheet of paper hits the ground.
2. Squeeze the second sheet of paper tightly in your fist to make a paper ball the same diameter as the solid ball.
Hold the solid ball and paper ball above your head at the same height and let them drop at the same time.
The solid ball and the paper ball both fall straight down and reach the ground at the same time.
3. Feel the weight of the solid ball and paper ball.
The solid ball is probably heavier than the paper ball.
The time taken by an object to fall and reach the ground does not depend on the weight of the object.
However, objects with greater surface area fall slower, because there is more resistance from the air.

8.3.12 Weightlessness
See diagram 36.105: Weightless toy soldier.
To study the motion of an object you need a reference system, e.g. something relative to which it is possible to describe the location of the object at any time.
For many experiments you choose a reference system that is fixed to the earth, as for instance when you study a falling object.
In such a reference system the earth is at rest.
However, if you want to study the seasonal changes, you prefer a reference system where the sun is at rest and where the earth will be moving in an orbit.
You see from this that the answer to the question whether an object is moving or not depends on what reference system you choose.
Not only the position, but the weight of an object depends on the reference system.
The following experiment will show weightlessness.
1. Tie a string with a toy soldier or other object suspended from it loosely across the top of the three pieces of wood joined as shown in the diagram.
Lift the entire apparatus and when it is hanging motionless release the string.
While the soldier is falling, he can be seen to remain in the same position inside the frame.
Since he is not supported by either the string or the frame, he is in a weightless condition with regard to his surroundings, e.g. the reference system being used.

8.3.8 Satellite launcher
See diagram 36.106: Simulated satellite launcher.
Materials required are a bucket, a football, a coat hanger, sinker or weight, a piece of string and a test-tube or a cap of some sort.
Place the ball securely in the bucket.
Bend the wire so that about 30 cm of it is straight and the rest is curved into a circular base as shown in the sketch.
Using masking tape, secure the circular portion on the ball, allowing the straight, 30 cm portion to stand upright in the centre of the top of the ball.
Attach the sinker or weight to the string.
Fasten the other end of the string to the test-tube or cap with tape.
Invert the cap on top of the upright wire, see diagram.
Explain that the ball represents the earth, and the sinker represents the satellite.
All that it takes to set the sinker into motion in any direction is the tap of a finger.
Let the students find out what happens when the satellite is launched in the following different ways:
1. With a slight tap, push the sinker up and away from the surface of the ball, as shown in the figure.
The sinker moves up and then falls back to the starting point.
This is how an object travels when it is projected at low speed straight up from the earth.
2. With a slight tap, push the sinker off the surface of the ball at an angle.
Show by a diagram what happens.
The sinker moves away from the ball and then falls back at some distance from the starting point.
The distance spanned depends upon the angle of launching and upon the forcefulness of the tap.
3. With a stronger tap, push the sinker of f the surface of the ball at an angle.
Make a diagram of the orbit.
The sinker moves away from the ball, circles it, and lands.
Evidently, a complete orbit passes through the starting point of the orbit.

8.3.11 Weight of a falling chain
Hold an end link of a linked chain vertically over a sensitive balance with end of the bottom link touching the pan of the scale.
Release the upper link and observe how the chain forms a heap on the pan.
Observe the maximum reading on the scale while the chain forms a heap and compare this value with the weight of the still heap.
The force exerted on the scale may be five times the weight force of the chain itself.
So the instantaneous force of a falling chain is much greater than the simple weight force of the object itself due to its momentum.
So where a long flexible object is dropped attached to suspension, an additional force is caused by the momentum.

8.3.3 Ball projected upwards from a moving cart
Fix a vertical spring in a cart.
The condensed spring is secured by a pin attached to a long string.
Attach the other end of the string to the leg of the table.
Put the cart on the floor and pull it away from the table with constant velocity.
At the length of the string from the table leg, the pin is pulled out to release the spring that projects the ball upwards to land again on the spring in the still moving cart.

8.3.4 Bullet projected upward from a stationary gun
A bullet fired at to a vertical angle would have a non-ballistic trajectory.
On falling, it would tumble, lose its spin, and fall at a much slower speed due to terminal velocity.
It is unlikely to be lethal on impact.
However, a bullet is fired at a lower angle would have a ballistic trajectory.
It would maintain its spin and will keep enough energy to be lethal on impact.
People have been injured and killed by bullets fired into the air.
Firing a gun into the air is illegal in most countries.
Do not fire a gun into the air, unless you can be sure where the bullet will land, e.g. on a rifle range or a hunting ground.

8.3.10 Velocity of an arrow
Resolve the velocity of an arrow fired up at angle α to the horizontal into a horizontal component, v cos α, and a vertical component, v sin α.
The range of the arrow = horizontal velocity × time of flight.
Time of flight = time to reach greatest height, t × 2 (up and down).
At greatest height, v = 0.
Use the equation: v = u + at.
(0 = v sin α - gt), So t = (v sin α / g) × 2 = 2v sin α / g.
For greatest height, h, use the equation: v² = u² + 2as.
0 = v² sin² α - 2gh, So h = v² sin² α / 2g.
Range = v cos α × 2t = v cos α × 2v sin α / g = 2v² cos αsin α / g.
[from trigonometry: sin α cos α = 1/2 sin 2 α].
So range = v² / g × sin 2 α.
R is maximum when sin 2 α = 1, i.e. when 2 α = 90^o, so α = 45^o.
Maximum range = v² /g, when angle to the horizontal = 45^o.

8.4.2 Balance with a metre stick
Balance with a metre stick, stationary meeting point, centre of mass, centre of mass
See diagram 8.146: Stationary meeting point.
See diagram 4.146: Uniform rod.
See diagram 4.146.1: Metre stick.
A body acts as if its mass is concentrated at a single point, the centre of mass.
Gravity acts through the same point, the centre of mass.
In a uniform gravitational field, the centre of gravity coincides with the centre of mass.
If a vertical line through the centre of mass of an object does not pass through its base, the object falls over.
An object, e.g. a motor car, will not roll over easily if it has a low centre of mass and a wide base.
The centre of mass of a metre stick or uniform rod is in the centre.
If two fingers support the rod and one finger moves towards the centre of mass the rod begins to tip towards that finger to increase the weight and increase the force of friction.
The other finger feels less wight and has less friction so the rod easily slides above it.
1. Support a metre stick or uniform rod over your two index fingers so that each finger is exactly 1 cm from the end.
The weight on the fingers feels exactly the same.
Keep the left finger in place, but slowly move the right finger towards the centre until it is half way between the centre and the end.
The metre stick feels heavier on the right finger than on the left finger.
Move the fingers together while keeping the metre stick balanced.
As your left finger moves towards the right finger, the metre stick feels heavier on it.
The weight on each finger feels about the same when the two fingers move together to be just each side of the centre of mass.
2. Repeat the experiment by moving one finger quickly and the other finger slowly.
Maintain the ruler in balance while moving the fingers.
If the metre stick remains horizontal, the two fingers always meet at the centre of the metre stick.
3. Repeat the experiment using two round smooth pencils on a level table instead of fingers.
Move the right pencil towards the middle of the rod while holding the left pencil in place.
As the right pencil approaches the middle of the rod the pencils have the same distance to the ends of the rod.
4. Repeat the experiment by hanging your hat on one end of the metre stick.
Note the new position of the centre of mass.
5. Repeat the experiment with a broom to find its centre of mass.
6. Slide two kitchen scales under a loaded beam.
Note the scale readings of the moving and stationary scales change in the same way that your fingers feel change in weight under the metre stick.
7. Put an empty drink-can on a rough wooden board.
Raise one end of the board until the drink-can falls over.
At that angle, a vertical line through the centre of mass of the drink-can passes outside its base.
8. Stand still then raise your right arm sideways.
Nothing happens.
Raise your right leg sideways.
If your upper body moves to the left, your centre of mass remains over your left foot so you remain stable.
If you keep your upper body rigid, your centre of mass moves to the right and is no longer over your left foot, so you fall over.

8.4.3Balance with a see-saw
See-saw, seesaw,( US: teeter-totter or teeterboard)
See diagram 21.13: Beam balance and moments.
Experiments
1. Use a strong board about three metres long and a sawhorse to make a see-saw.
Use two students of equal weight.
Sit at either end of the board so that they balance.
Measure the distance from the balance point to each student.
Multiply the distance by the student's weight.
2. Select a heavier student and a lighter student.
Tell them to sit on the board so that they balance.
Measure the distance from the balance point to each student.
Multiply the distance by the student's weight.
3. Select a heavier student, weight m₁, and a lighter student, weight m₂.
Sit on the board so that they balance.
Measure the distance from the balance point to each student, d₁ and d₂.
Multiply the distance by the student's weight.
You will discover that m₁d₁= m₂d₂.
4. Select a heavier student, weight m₁, and two lighter students, weight m₂and m₂.
Sit on the board so that they balance.
Measure the distance from the balance point to each student.
Multiply the distance by the student's weight.
Add the products for the two lighter students.
In each experiment the ratio of the masses is the reciprocal of the ratio of the distances from the balance point or centre of mass for the system,
m₁d₁ = m₂d₂
m₁d₁ = m₂d₂ + m₃d₃

8.4.4 Balanced forks on glass rim
See diagram 8.2.22: Balancing coin and forks.
See diagram 8.2.22a: Balanced forks and spoon.
See diagram 8.2.23: Balanced forks and coin (photo).
1. Overlap the prongs of two forks.
Insert a coin between the prongs to hold the forks together.
Balance the structure on the rim of a glass.
Heavy handles of the forks curve towards the glass, shifting the centre of mass of the structure to a point directly below where the coin rests on the rim of the glass.
2. Insert a coin between the prongs of two forks.
Hold a fork in each hand and place the edge of the coin flat on the rim of a drinking glass.
Carefully move the coin with the corks until the coin and the forks balance on the rim of the drinking glass.
When they balance, the centre of mass of the combined forks and coin is a point on the circumference of the rim of the glass.
The balance depends on the weight of the forks on each side of the coin and their distance from the centre of mass.
3. Pick up the glass with the forks and coin still balancing.
Fill the raised glass with water.
Turn the glass to pour out the water.
During the pouring motion the forks and coin remain balanced on the rim of the glass.
So the forks and coin are in a state of stable equilibrium.
4. Observe a fork and spoon in moment equilibrium on the glass rim.
Observe the burning stops at the point that is the boundary of the two objects.
Attach the spoon to the fork by pushing it in between the teeth.
S oone tooth is held out by the convex surface of the spoon and other teeth are in the concave surface of the spoon.
Place a toothpick between two of the forks.
The toothpick should be in the same plane with normal axis of the handle of the spoon and fork.
Adjust the angle between fork and toothpick, as the fork is above the glass.
Once the spoon and fork are in balance on the glass rim, burn the end of the toothpick or match inside the glass.
As the heat of the flame is absorbed by the glass, the temperature drops below the wood's ignition temperature and the burning of the toothpick stops exactly at the glass rim.
Burn the other end of the toothpick.
The burning wilts at the top of the fork and the heat of the flame is absorbed by the metal.
Observe the equilibrium of the fork and spoon about the toothpick on the glass rim.

8.4.5 Balanced H-shape
Select two identical coins and draw two H shapes on thin cardboard
Make the widths of the vertical arms and cross arms of the H-shape slightly longer than the diameter of the coins.
Place the two coins at the bottom of the two vertical arms of one of the H shapes.
Put the other H-shape on top then join top and bottom H-shape with adhesive tape.
The usual centre of mass of an H-shape would be along the line bisecting the cross arm.
However, this H-shape can balance on a string held below the arms containing the coins.

8.4.6 Balanced mobile
Centre of mass of a balanced mobile
See diagram 8.14: Balanced mobile.
See diagram 8.14.1: Centre of mass of two objects.
2.0.2a Balanced circle with circular cutout, golden ratio (GIF)
A mobile has two masses, m1 and m2, suspended from it, at distances x1 and x2 from one end.
The centre of mass is along the supporting rod at distance X from the same end.