School Science Lessons
8. Mass and weight, balances, centre of gravity
2012-01-15
Please send comments to: J.Elfick@uq.edu.au
Table of contents
8.1.0 Mass and weight
8.2.0 Centre of gravity
8.3.0 Gravity
8.2.0 Centre of gravity
8.2.0 Centre of gravity, balancing
2.23 Balance, See-saw balance (teeter-totter) (Primary)
4.146 Balance with a metre stick, stationary meeting point, centre of mass, centre of gravity
16.4.2.3 Balance with a see-saw (teeter-totter)
8.2.16 Balanced H-shape
1.21 Balanced mobile (Primary)
2.11 Balanced parrot (Primary)
8.2.2 Centre of gravity and stability of an object, balancing pins
8.2.3 Centre of gravity, stability of an object
8.2.4 Centre of gravity of a double cone, uphill roller, circular cone with two heads on a ramp
8.2.4.1 Upward rolling can
8.2.1.1 Centre of gravity of a map
8.2.5 Centre of gravity of a shape, wobbling circles
8.2.1 Centre of gravity of an irregular shaped object
8.2.6 Centre of gravity of man and woman
8.2.9 Centre of gravity of maps
8.2.11 Centre of gravity of pine trees
8.2.17 Centre of gravity of playing dice
8.2.8 Centre of gravity outside an object, U-shape cardboard
8.2.10 Centre of mass of biased bowls
15.4.3 Centre of mass, Motion of the centre of mass
8.2.7 Find centre of gravity
8.2.13 Hanging belt, "sky hook"
8.2.12 Hidden centre of gravity, weight under false bottom
8.1.1 Letter scale
3.17 Plumb bob, (vertical test), (Primary)
9.2.3 Roll-back jar, come back can, elastic potential energy and kinetic energy
16.4.1.9 Rubber band scale or spring scale
8.2.18 Stable, unstable and neutral equilibrium
8.2.14 Shared centre of gravity, stability brace
8.2.15 Tipping bottle
8.2.15.1 Tipping glass of water
8.3.0 Gravity
8.3.0 Gravity, gravitational field
36.108B Gravitational field of the Earth, g
36.109 Gravitational potential energy
8.3.1 Shape of a hanging cable or flexible chain, catenary curve
4.148 Acceleration of marbles down an incline
4.147 Ball bearings fall together
4.108 Ball projected upwards from a cart
7.111 Earth rotation and wind farms
4.104 Falling ball and paper
1.41 Falling parachutes (Primary)
4.154 Falling washers on a string
36.36 Foucault pendulum
7.110 Kepler's laws of planetary motion
36.108 Newton's universal law of gravitation, gravitational constant, G
4.152 Paths of projectiles, free fall
14.2.13 Path of projectiles, mid-air target, monkey and hunter
4.106 Satellite launcher
36.108C Satellite in stable orbit
4.153 Three holes can, 3-hole can, a vase with three holes, spouting cylinder
3.22 Throw up and fall down (Primary)
4.151 Time a falling body with a stopwatch
4.109 Velocity of an arrow
12.1.1 Weight and pressure
4.107 Weight of a falling chain
36.108A Weight of an object and g
4.105 Weightlessness
8.1.0 Mass and weight
8.1.0 Mass and weight, weighing devices (balances)
7.0 Balances, weighing devices, experiments (Primary)
16.4.2.3 Balance with a see-saw (teeter-totter)
12.3.3.1 Carbon dioxide has mass
36.110 Inertial mass and gravitational mass
17.5.0 Law of mass action, law of chemical equilibrium (Chemistry)
6.3.1.2 Mass, kilogram
7.9.39 Molecular mass (Chemistry)
8.1.0 Mass and weight, weighing devices (balances)
1. Mass
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.
36.108 Newton's universal law of gravitation, gravitational constant, G
2. Weight
Mass is commonly measured by using gravitation. The weight of an object is the force of gravity on it. Weight, W, = mg (mass X 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.
3. Weighing devices, often called balances, include the following
1. Inertial balances measure mass.
2. 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.
3. Electronic balances use electric force to measure mass.
4. Spring balances, force meter, top loading balance, compression balance, kitchen scale, bathroom scale, measure weight
4.1 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.
4.2 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.
4.3 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.
4.4 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.
See 34.5.02 Hooke's law, elastic limit, deforming force, stress and strain
4.146 Balance with a metre stick, stationary meeting point, centre of mass, centre of gravity
See diagram 8.146: Stationary meeting point | See diagram 4.146: Measurements on a uniform rod
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 gravity. If a vertical line through the centre of gravity 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 gravity and a wide base.
The centre of gravity 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 gravity 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 gravity.
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.
3. Repeat the experiment by hanging your hat on one end of the metre stick. Note the new position of the centre of gravity.
4. Repeat the experiment with a broom to find its centre of gravity.
5. 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.
6. 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 gravity of the drink-can passes outside its base.
7. 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 gravity remains over your left foot so you remain stable. If you keep your upper body rigid, your centre of gravity moves to the right and is no longer over your left foot, so you fall over.
8.1.1 Letter scale
Use adhesive tape to attach a heavy coin to the top right hand corner of a picture post card. Punch a hole in the bottom left hand corner of the post card and insert a wire paper clip through the hole. Attach a second paper clip to the first paper clip. Push a thick pin or nail through the top left hand corner of the post card. Push the pin into a vertical board and let the post card hand down from the pin as pivot. Hang a letter with exact weight, e.g. 50 g, from the second paper clip and then mark the position of the top right hand corner of the post card on the wall. Use a second letter of exact weight to make a second mark on the wall. Now you can weigh letters and decide what stamps to stick on them. This letter scale is a first order lever. The left hand edge of the post card is the load arm. The pin is the fulcrum. The upper edge of the post card is the force arm, effort arm. The letter scale measures small differences in weight because the force arm is longer than the effort arm.
8.2.0 Centre of gravity, balancing
Centre of gravity of geometric shapes
Gravity and centre of gravity, statics of rigid bodies, finding centre of gravity and exceeding centre of gravity
Centre of gravity, centre of mass, is the point in or near an object from which its total weight, or mass, appears to originate and act, the point through which the line of action of the weight always passes, the point at which the weight of the body may be considered to act. The gravitational force acted on an object by earth goes through a point, called the centre of gravity of the object. If the pulling force or supporting force acted on an object goes through the centre of gravity and are equal to gravitational force in size, the object will be in equilibrium. The centre of gravity of an object is the point around which all its mass is balanced. For a regularly shaped object made of homogeneous material, the centre of gravity 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, such as 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. In a uniform gravitational field, centre of gravity and centre of mass are in the same place. A tight rope 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 gravity of the tight rope 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 gravity into a position vertically above the rope.
8.2.1 Centre of gravity of an irregular shaped object
See diagram 8.2.1: 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 gravity 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 gravity of the cardboard shape. Test that the point is the centre of gravity, by supporting the cardboard shape with the end of sharpened pencil below the point. The cardboard shape will remain in equilibrium.
8.2.1.1 Centre of gravity of a map
See diagram 8.169: Centre of gravity of a map
A plumb line is a ball of lead attached to a string used to define a vertical line. 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 gravity.
8.2.2 Centre of gravity 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 then 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 that 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 becomes less stable because the centre of gravity of the system approaches the level of the pivot. By slanting the knitting needles with the rubber stoppers pushed down, the centre of gravity 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 gravity and prevent the racing car rolling over during a turn.
8.2.3 Centre of gravity, stability of an object
Every object has a centre of gravity. The centre of gravity does not change if the distribution of mass in an object does not change. An object in water is acted on by water buoyancy due to the displacement of the weight of water. Buoyancy also has its own centre of action, called the centre of buoyancy, determined by the mass of water displaced by the object. The shape of the displaced water determines the position of the centre of buoyancy, if the density of water is constant. The centre of gravity may not be in the same position as the centre of buoyancy. Only when the two positions are under a certain condition can the object maintains a stable equilibrium state. If the state of the object changes, the stability of equilibrium of the object is lost.
8.2.4 Centre of gravity of a double cone, uphill roller, circular cone with two heads on a ramp
See diagram 8.2.4: Uphill roller
1. A disc with a non-uniform mass distribution is placed on an incline so it rolls uphill. 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.
2. As a double cone moves up an set of inclined rails its centre of gravity lowers. The double cone appears to roll uphill. A double cone rolls up an inclined track.
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 gravity is on the line the connecting of the two tops, so the height of the tops is the height of the centre of gravity 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 gravity. While the cone rolls up to the top of the ramp due to the wider width of the top, the centre of gravity of the cone is lower. Thus the centre of gravity of the cone is higher at bottom, the centre of gravity 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 gravity 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.
8.2.4.1 Upward rolling can
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.
8.2.5 Centre of gravity of a shape, wobbling circles
See diagram 8.2.5: Spinning card
All flying objects spin around their centre of gravity. Cut cardboard into a kidney shape. Support it on your finger to let it balance and find its centre of gravity then mark the centre of gravity of it by a pencil. Draw concentric circles around the centre of gravity on one surface of the cardboard, called A surface, by a compass and thicken the circles 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 gravity. 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 gravity. On the A side of the cardboard the concentric circles are around the centre of gravity, so they stay steady as they spin. When you see the other side B, the circles wobble, because it is off the centre of gravity and the circles spin around a point outside the centre of gravity making a wobbling motion.
8.2.6 Centre of gravity of man and woman
See diagram 8.2.6: Woman and man lifting
1. The centre of gravity of a woman is usually about 2-3 cm lower and further back than the centre of gravity 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 gravity is usually above the waist. When a man bends forward like a snow skier his centre of gravity is above his toes but a woman's centre of gravity is above her heels.
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 rest 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 gravity.
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 gravity 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. The lower centre of gravity of women is supposed to help them to be better dancers than men.
8.2.7 Find centre of gravity
1. Find centre of gravity 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 gravity.
2. 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 gravity.
3. Place a block on an incline and raise the incline until the block tips.
8.2.8 Centre of gravity outside an object, U-shape cardboard
See diagram 8.4.9: U-shape
1. To verify the existence of a centre of gravity 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 gravity. 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 gravity of an object that is empty in the centre may not be on the object. You cannot see the centre of gravity in space 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 gravity. 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.
8.2.9 Centre of gravity of maps
To find the centre of gravity 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 gravity is where the plumb lines intersect.
8.2.10 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.
8.2.11 Centre of gravity 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 gravity 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 gravity 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 gravity.
8.2.12 Hidden centre of gravity, weight under 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. However when the smuggler carries the suit case through the customs office, it tilts to one side so that the centre of gravity, now at the side, is vertically below the suspension point of the suitcase handle.
8.2.13 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 gravity of the system lies vertically below your finger tip.
8.2.14 Shared centre of gravity, 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 gravity. Within construction towers and scaffolding, braces between vertical elements form stable triangles with two brace sides and one vertical element side.
8.2.15 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 gravity is supported by the surface of the window sill.
1. and 3. have a higher centre of gravity 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 gravity is supported.
8.2.15.1 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 id 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 gravity is still in the same place in the glass of water
8.2.16 Balanced H-shape
Select two identical coins and draw two H shapes on thin cardboard with the widths of the vertical arms and cross arms of the H-shapes 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-shapes with adhesive tape. The usual centre of gravity 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.
8.2.17 Centre of gravity 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 gravity 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.
8.2.18 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 gravity object.
6. A tightrope walking toy unicycle rider carrying a balancing pole travels along a string. A model of a tightrope walker shows the centre of mass moves up with tipping.
7. Wires form a support at the centre of gravity of a laboratory stool. Construct a stool so that wires crossed diagonally will intersect at the centre of gravity. The stool can be oriented in any direction.
8. Hide heavy weights in the ends of a stool's legs so it will balance on a vertical rod placed under the seat.
9. Spread the bristles and a straw broom will stand upright.
10. Stick the neck of a wine bottle through a hole in a slanted board and it stands up.
11. Exceeding centre of gravity, tower of Lire. Stack a set of eight blocks or similar boobs blocks until the top block sticks out beyond any part of the bottom block and over the edge of the table.
12. Use adhesive tape to fix a weight in the corner of a cardboard box, e.g. a shoe box. Ask someone to push the box towards the edge of the table until it is about to fall down and so locate the centre of gravity of the box.
8.3.0 Gravity, gravitational field
Gravity is the attractive force between two masses or celestial bodies or a body and the earth. Gravitation refers to the attractive force between any two particles of matter. 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 X 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.
Gravitational force is said to be propagated by gravitational waves through space but nobody has yet detected them.
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 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 equation
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. 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.
The cable of a suspension bridge hangs in a parabola if the droppers allow the load to be suspended horizontally. 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.
4.104 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.
4.105 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 which 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.
2. 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 but his 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.
3. 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, Newton's first law. Outside the space-ship a man could, if he were completely free to move, push himself of f 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.
4.106 Satellite launcher
See diagram 36.106: Football satellite
Materials required are a bucket, a football, a coat hanger, or other suitable wire, 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 of f 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.
4.107 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.
4.108 Ball projected upwards from a 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.
4.109 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 X time of flight.
Time of flight = time to reach greatest height, t X 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) X 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 X 2t = v cos α X 2v sin α / g = 2v2 cos α sin α / g [from trigonometry: sin α cos α = 1/2 sin 2 α]
So range = v2 / g X sin 2 α. R is maximum when sin 2 α = 1, i.e. when 2 α = 90o, so α = 45o.
Maximum range = v2 /g, when angle to the horizontal = 45o.
Extra
Gravitation, weight, falling, S = ½ gt2, measuring g, distance / time graphs, Accelerated Reference Frames, earth's gravitational field strength, g = 9.8 N / kg
Newton's Law of Universal Gravitation. qualitative understanding of the inverse square law, quantitative treatment of the law F = Gm1m2 / d2, gravitational field strength, g = force per unit mass