School Science Lessons
(UNPh21)
2024-07-25
Machines
Contents
21.4.0 Gears
21.3.0 Inclined planes
21.1.0 Levers
21.6.0 Machines
21.5.4 Pile driver
21.5.0 Pulleys
21.2.0 Wheel and axle
21.4.0 Gears
21.4.01 Gears
21.2.2 Belt drives, transmission belts
21.4.1 Bicycle gears
21.4.2 Bottle top gears
21.4.3 Rolling coins
21.3.0 Inclined plane
21.3.1 Inclined plane
21.3.6 Mechanical jack, car jack
21.3.3 Paper inclined plane, paper screw thread
21.3.8 Propellers
21.3.4 Screw, nut and bolt, pitch of screw
21.3.5 Simple screw jack
21.3.2 Wedge
21.1.0 Levers
21.1.01 Levers
21.1.1 Class 1 levers
21.1.1.5 Bent class 1 lever, hammer
21.1.1.6 Drinking straw lever
21.1.1.7 Letter scale
21.1.1.3 Lever raises table
21.1.1.4 Pliers crush a match box
21.1.2 Class 2 levers
21.1.3 Class 3 levers
21.1.3.4 Arm joint
21.1.3.5 Break matchstick between fingers
21.1.3.3 Catch a fish with a rod and line.
21.1.3.1 Use chopsticks
21.1.3.2 Use forceps
21.6.0 Machines
21.0.0 Units of work and energy
21.04 Efficiency of a machine
21.01 Machines
21.02 Mechanical advantage (MA) of a machine
21.03 Velocity ratio of a machine
21.5.0 Pulleys, block and tackle
21.5.01 Pulleys, block and tackle, broomstick pulley
21.5.1 Pulleys, Systems of pulleys
21.5.2 Block and tackle
21.2.0 Wheel and axle
21.2.01 Wheel and axle, screwdriver, windlass, crank handle, steering wheel
21.2.1 Windlass, raising weight with rotary pencil sharpener
21.0.0 Units of work and energy
See diagram 21.0.0: Work and wheels.
1. Work, joule
Work done = force x distance in direction of the force, W = Fs.
The work done by a force F in moving its point of application through a distance s in the direction of the force by equation: W = FS
When a wheel is moved by a force, the work done = displacement x component of the force in the direction of the displacement.
2. Energy, joule
Energy is the capacity for doing work.
The joule, J, is the SI unit of work and energy.
A joule is equal to the amount of work done when the point of application of a force of one newton moves one metre in the direction of the force.
So 1 joule = 1 newton.metre.
3. Calorie
The calorie is the CGS (cgs) unit of heat.
The 15oC calorie is the quantity of heat required to raise the temperature of 1 g of water by 1oC at 15oC =4.1855 J.
However, the International Table calorie = 4.186 J.
Nowadays the SI unit the joule, J, is used.
1 calorie (cal) = 4.184 J, commonly, 4.2 joules.
The nutrition industry still uses the calorie, but that unit is too small for nutrition calculations so the kilocalorie is used, i.e. 1000 cal.
Popular information on diet and cooking in the daily press, cooking literature and women's magazines refers to "calories", are really kilocalories.
4. Kilowatt-hour
The kilowatt-hour, kWh, is the energy used when an appliance with the power of one kilowatt runs for one hour.
A power of one watt = one joule per second, so a kilowatt-hour = 3, 600, 000 J, about the energy used by one bar of an household electric heater.
A 40-watt bulb runs at 0.4 of a kilowatt, but a tumble clothes dryer runs at 3 to 4 kilowatts.
5. Kinetic energy
Work done on an object changes its energy that may be stored as potential energy or cause change in speed, kinetic energy.
Kinetic energy is the energy a body possesses, because of its motion and so can be measured by the work done in coming to rest.
If a body mass m moving with constant velocity v, and brought to rest by application of a constant force F, producing a constant negative acceleration a,
Work done = F x s, where s is the distance travelled in the direction opposed to F.
From the equations of uniformly accelerated motion:
v2 = u2 + 2as
0 = v2 = 2as (initial velocity v, and final velocity 0)
as = v2/2 (1/2v2)
mas = 1/2 m v2
F = ma
Fs = mas = 1/2 m v2
Kinetic energy of any mass, m, moving at speed, v = 1/2 mv2.
Change in kinetic energy, joule = work done, joule.
6. Potential energy
Potential energy is the energy a body possesses, because of its position, measured by the work it could do by passing from its position to some defined position.
The gravitational potential energy of a body at height h, above some level equals the work that must be done to raise it from that level to the given height.
The force required to lift a body mass m is equal to the weight of the body, mg.
So when the body is lifted vertically through height h, the work done = mg x h = mgh, so the increase in potential energy = mgh.
Potential energy is the stored energy that an object has due to the state it is in, e.g. steam compared with water, compressed spring compared with a relaxed spring, or its position, e.g. height compared to ground level above the earth.
Work can be done when potential energy is released from storage.
If an object falls, the gravitational potential energy lost, Ep = mgh = the kinetic energy gained, Ek, = 1/2 mv2.
21.01 Machines
Machines transmit or modify energy to allow us to do tasks.
It transmits, modifies or changes the direction of force to help us to do work.
With a machine, a small force can be used to overcome a larger resisting force.
Machines allow a force to be applied, the effort, E, to overcome other forces, load, L.
Machines transmit force or directs the application of force.
Machines allow a force called the effort to overcome another force called the load.
Machines may save labour, but do not save work, because force (effort) x distance (effort) = force (load) x distance (load), neglecting friction.
The functions of machines are to allow us to do the following:
Function 1. Amplify force
Function 2. Amplify movement and speed
Function 3. Change the direction of force
The six basic simple machines are as follows:
21.3.0 Inclined plane,
21.3.2 Wedge,
21.1.0 Levers,
21.2.0 Wheel and axle,
21.5.0 Pulley,
21.3.4 Screw, nut and bolt, pitch of screw
Inclined plane, wedge and screw are simple machines developed according to the inclined plane principle.
Lever, pulley and axle are simple machines developed according to the lever principle.
Function 1. To amplify force
See diagram 21.1.2.1: Matchbox and pliers.
See diagram 21.1.2b: Wheelbarrow.
Machines may allow a smaller applied force, the effort, to overcome a larger resistance force, the load.
A man with a strong hand shake can exert 20 kg force, but using pliers he can exert a force of more than 60 kg force.
Most people cannot crush an empty match box between their thumb and fingers, but crushing the matchbox with pliers needs little effort.
The effort force is multiplied, because the distance to where you grip the handles of the pliers is much greater than the distance to the end of the pliers' jaws.
A big stone may be too heavy to carry, but you can move it on a wheelbarrow.
In the ear, three tiny connecting bones pass on x 30 or more the force from sound waves.
Function 2. To amplify movement and speed
Increasing displacement and speed:
Sometimes a machine passes on only a fraction of the force that is applied to it, but it will then increase or amplify movement and speed.
A lever of this class is common in jib cranes.
Function 3. To change the direction of force
See diagram 21.254: Simple pulley.
See diagram 21.255: Single fixed pulley.
See diagram 21.256: Single moveable pulley.
The single fixed pulley just changes the direction of an applied force from up to down.
Sometimes it is more convenient to pull down than to pull up.
Machines let us overcome a resistance at one place with an effort by applying a force at another place to move a load.
Your fingers may be too big to pick up very small objects or pick up objects inside small spaces, but you can use tweezers.
21.02 Mechanical advantage (MA) of a machine
The mechanical advantage, MA (force ratio), of a machine is the ratio of the load to the effort, load / effort, L / E.
If an effort of 10 N applied to a machine can move a load of 25 N, the mechanical advantage (MA), of that machine = 25 / 10 = 2.5.
If MA > 1, i.e. heavier loads are moved by smaller efforts, then the effort must move further than the load.
So mechanical advantage is the ratio of the force provided by the machine to the force applied to it.
Mechanical advantage (MA) is the number of times the load moved by a machine is greater than the effort applied to that machine, i.e. MA = load / effort.
MA has no unit, as it is a ratio.
If MA > 1, load > effort, i.e. you an use a smaller effort to move a bigger load.
However, the effort must move further than the load.
Distance moved by the effort / distance moved by the load = velocity ratio.
Machines usually allow a smaller applied force, the effort, to overcome a larger resistance force, the load.
However, machines may be used to change the direction of a force or to handle small objects, e.g. tweezers (forceps).
21.03 Velocity ratio (VR) of a machine
Velocity ratio, VR, = distance travelled by the effort / distance travelled by the load.
The velocity ratio of a particular machine can be calculated without using it, but just by measuring its dimensions.
However, the mechanical advantage (MA) of a particular machine can be calculated only by using it, only by applying a force to a load.
The velocity ratio (distance ratio, gear ratio), VR, of a machine is the ratio of the distance travelled by the effort to the distance travelled by the load.
21.04 Efficiency of a machine
The efficiency of a machine is the ratio of the work done by the machine to the work supplied to the machine.
Efficiency of a machine = energy output / energy input, as percentage.
No machine is 100 % efficient, because always some energy is lost due to friction.
Work = force X distance.
Efficiency = work done on the load / work done by the effort = load X distance moved by the load / effort X distance moved by the effort.
If mechanical advantage (MA) = velocity ratio (VR), the efficiency of the machine = 1 or 100%, but efficiency is always < 1.
Mechanical advantage = velocity ratio X efficiency
Percentage efficiency for a simple machine = (useful work out / total work in) X 100.
Efficiency = Mechanical Advantage X 100 / Velocity Ratio
For a machine with efficiency = 1, i.e. 100%, MA = VR.
However, the efficiency of a machine is less than 100% cent, because some energy loss always occurs.
21.1.01 Levers
A lever is a simple machine consisting of a rigid rod pivoted at a fixed point called the fulcrum, used for shifting or raising a heavy load or applying force.
Examples of levers include the following:
Scissors, wheelbarrow, forearm, claw hammer to draw a nail, sugar tongs, boat oars, nut-crackers, pliers, can opener, bottle opener, crow bar.
Levers have a rigid beam supported a one point, the fulcrum, F, with a load force, L, applied at one point and an effort force, E, applied at another point.
The lever principle is load X length of load arm = effort X length of effort arm.
The lever principle states that motive force X the arm of the motive force = resistance X the arm of resistance.
Each side of this equation is a moment, i.e. force X perpendicular distance to the pivot.
So moments clockwise = moments anticlockwise.
The distance of the effort from the fulcrum is called the effort: arm and the distance of the load from the fulcrum, the load arm.
To arrange the lever so that a small effort would lift a big load the effort arm must be as long as possible and the load arm as short as possible.
Application of levers in everyday life
See diagram 21.1.2: Different levers.
See diagram 21.1.2.1: Matchbox and pliers
Compare the convenience of using the hand only, with using a tool.
Look for levers used at everyday life and classify them according to the class of lever, e.g. Chopsticks belongs to the third order lever.
Using it needs greater effort, but can prevent food from slipping away.
The three classes of lever depend upon the relative positions of F, L, and E.
Classify levers into 3 classes according to where the effort is applied, and the load moving force developed, in relation to the position of the fulcrum.
21.1.1 Class 1 lever
See diagram 21.252.1: Class 1 lever.
See diagram 21.1.1: Class 1 levers.
Class 1 levers have the fulcrum between the load and the effort (E F L), so the load and effort are on opposite sides of the fulcrum.
Examples of class 1 levers include the child's seesaw (US teeter totter), and the beam balance.
Also, scissors, tin snips, bolt cutters, pliers are two class levers with the fulcrum as a pivot.
When cutting paper or cloth with scissors, the effort < load, because you want a long length of scissors blade and cloth does nor require much force to cut it.
Try using a pair of scissors as tin snips to feel the difference.
21.1.1.3 Lever raises table
Experiment
Use a board the same height as a table.
Place a stick across the board and use it as a lever to raise the table.
Note that the longer end of the stick moves farther than the shorter end.
The force exerted by the shorter end, the load, is greater than the force used to move the longer end, the effort.
21.1.1.4 Pliers crush a match box
Experiment
Close a wooden match box and try to crush it between your thumb and fingers.
You cannot do it.
Hold the match box in the jaws of a pair of pliers.
You can easily crush it by squeezing the handles together.
21.1.1.5 Bent class 1 lever, hammer
See diagram 21.1.1.5: Hammer.
Experiment
Hammer a nail into a big piece of wood.
Try to pull the nail out with your fingers.
You cannot do it.
Use a claw hammer to pull out the nail.
The load is the force of the nail on the claw.
The fulcrum is the round part of the hammer head.
The effort is your pull on the handle.
You are using the hammer as a bent class 1 lever to pull out the nail.
21.1.1.6 Drinking straw lever
See diagram 21.1.3.2: Drinking straw lever.
Experiment
Lift a bottle with a drinking straw by turning it into a lever.
Push the drinking straw down into a bottle to make a sharp bend at the bottom of the bottle and the
bottom end pushes against the side of the bottle.
The shorter piece of drinking straw is between the bend and the side of the bottle.
The longer piece of drinking straw must extend outside the bottle.
The fulcrum of the lever is where the drinking straw is sharply bent.
The section of the drinking straw between the fulcrum and where it pushes against the side of the bottle is the load arm.
The effort arm is the section of the drinking straw up from the fulcrum.
Grab the top end of the drinking straw and pull up to lift the bottle as the drinking straw acts as a lever.
21.1.1.7 Letter scale
See diagram: Letter scale: Class 1 lever.
Experiment
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.
21.1.2 Class 2 lever
See diagram 21.252.2: Class 2 lever.
See diagram 21.1.2: Class 2 levers.
Class 2 levers have the load between effort and fulcrum (F L E), so the load and effort are on the same side of the fulcrum, with the load nearer the fulcrum.
Examples of class 2 levers include nutcrackers, wheelbarrow, biceps muscle on upper arm.
Experiment
Use a metre stick with a hole drilled in the centre near one end.
Hammer a nail horizontally into the side of a table.
Suspend the metre stick at one end by the nail through the hole and attach a spring balance to the other end.
Tie a loop of string to a weight.
Pass the bar through the loop so that the bar can support the weight.
Move the loop to any position along the bar.
21.1.3 Class 3 lever
See diagram 21.252.3: Class 3 lever.
See diagram 21.1.3: Class 3 levers.
Class 3 levers have the effort between fulcrum and load (L E F), so the load and effort are on the same side of the fulcrum, but the effort is nearer the fulcrum than the load.
Since a class 3 lever has the effort between the fulcrum and the load, the effort is always greater than the load, M A < 1.
Examples of class 3 levers include forceps, tweezers, tongs, chopsticks.
Use the same apparatus as for Class 2 levers, but put the weight, load, at the end of the bar and suspend the bar by a loop of string attached to a spring balance, effort.
Experiments
21.1.3.1 Use chopsticks
21.1.3 Class 3 levers
1. For chopsticks the fulcrum is the angle between your thumb and forefinger.
The force you apply with your fingers, effort, is greater than the force exerted by the ends of the tweezers or chopsticks, load.
To use chopsticks in a Chinese restaurant, hold one the thicker end of one chopstick in the crook of the hand, i.e. where the first finger and the thumb join on the hand.
Let it rest on the end of the middle finger.
This lower chopstick never moves during the eating motion.
Hold the upper chopstick between the end of the thumb and the end of the forefinger (index finger, second finger) with the thumb touching a thicker part than the forefinger.
The thinner ends of the two chopsticks should meet so that you can pick up piece of food with a pincer motion.
Tap the ends of the chopsticks on the table to even them.
Grasp a piece of food by closing the end of the upper chopstick down onto the end of the lower chopstick.
The action of the upper chopsticks is that of a class three lever.
The end of the thumb is the fulcrum.
The end of the index finger supplies the effort.
The resultant force of the food on the ends of the chopsticks is the load.
The action is similar to the action of tweezers or forceps.
Note the mechanical disadvantage, but the social advantage of being able to select and pick up the small pieces of food and dip them into sauces, a practice characteristic of Chinese cuisine.
21.1.3.2 Use forceps
Experiment
Pick up something heavy with tweezers, forceps.
They consist of two Class 3 levers joined at the fulcrum.
Class 3 levers are convenient for picking up small things.
21.1.3.3 Catch a fish with a rod and line.
Experiment
The load is the pull of the fish.
The effort is your pull on the rod.
The fulcrum is where you hold it lower down or where the rod touched the ground.
21.1.3.4 Arm joint
See diagram 9.232: Arm joint.
Experiment
Keep your upper arm vertical and your forearm horizontal in front of the body.
Put a heavy stone in the palm of your hand and move it up towards your mouth without moving the upper arm.
The load is the weight of the stone.
The effort comes from the shortening of the biceps muscle in your upper arm.
The fulcrum is the elbow joint.
21.1.3.5 Break matchstick between fingers
See diagram 21.1.5: Matchstick between fingers.
See diagram 21.1.6: Flexing a finger.
Experiment
Place a wooden matchstick across the back of the middle finger and under the first and third finger at the joints, near the finger nails.
Try to break it with fingers.
The arrangement is the same as a class 3 lever.
Move the matchstick to the base of the fingers.
It is easier to break, because the arrangement is the same as a class 2 lever.
21.2.01 Wheel and axle, screwdriver, windlass, crank handle, steering wheel
See diagram 21.1.01: Wheel and axle.
See diagram 21.253: Wheel and axle (pencil sharpener).
Experiment 1. A set of wheel and axle consists of two wheels having different radii.
The large wheel has radius R and the small wheel an axle, has radius r, such that R > r.
Wind one rope around the wheel and another rope in the opposite direction around the axle.
Pulling on the wheel rope supplies the effort.
The wheel around the axle bears the load.
When you pull on the wheel so that it make one complete turn, a point on the circumference of the wheel has moves through 2 π R and a point on the circumference of the axle has moved through 2 π r.
So velocity ratio = 2 π R / 2 π r = R / r.
Taking moments about the centre of the axle, effort X R = load X r so R / r = load / effort = mechanical advantage.
2. Tie one end of a string to books.
Grab the other end of the string, pull up the books and feel their weight.
Remove the cover from a pencil sharpener and tie the end of a string around the end of the shaft.
Turn the handle of the pencil sharpener to raise the books.
The force needed to turn the handle is much less than the force needed to pull up the books by grabbing the string.
21.2.1 Windlass, raising weight with rotary pencil sharpener
See diagram 21.2.1: Windlass
Experiment
Remove the cover from a pencil sharpener and tie a string tightly around the end of the shaft.
When you turn the handle you find the force needed to turn the handle is much less than the force of gravity on the books.
Feel the magnitude of the force lifting the heavy weight.
Lift the heavy weight directly.
Compare the magnitudes of the forces at two conditions.
21.2.2 Belt drives, transmission belts
See diagram 21.2.2: Simple transmission belt.
Experiment
1. Drive two long nails into a block of wood.
Place spools, one larger than the other, over the nails so that these can be used as axles.
Slip a rubber band over both spools.
Rotate the larger spool through one turn and note whether the smaller spool makes more or less than one full turn.
In which direction does the small spool turn?
Try crossing the rubber band and observe the result.
2. Use several spools with different diameters, a wooden block, two long nails, a piece of elastic.
Nail the two nails on the block.
Cover the two spoons on the two nails to make the nails as axles.
Cover the elastic on the two spools.
Tighten the elastic at fit degree, not too loose and not too tight.
Rotate the spool with a larger axle a circle and meanwhile observe the small spool's rotating amount and direction.
Again cover the elastic across on the two spools.
Repeat the experiment and observe the small spool's rotating amount and direction again.
Compare the above two conditions and find the difference.
3. Do the experiment again using two spools with the same diameters and using two spools with very different diameters.
Compare and analyse the experiment data to find the relationship of the spool's diameter and the way of covering the spool with elastic to the rotating amount and direction.
21.3.1 Inclined plane
See diagram 21.3.1: Inclined plane.
See diagram 21.3.0: Inclined plane, triangle of forces.
See diagram 21.3.01: Pitch of a screw
Inclined plane is a slope that allows a load to be raised gradually using a smaller effort than would be needed if it lifted vertically upwards.
So it is a force multiplier.
The ratio of the height of the top point of the inclined plane to the length of the plane is called the gradient.
The smaller the gradient the more force is saved.
See 1. 1. To raise the load L through a vertical height h, the smaller effort E moves a greater distance d equal to the length of the incline.
Velocity ratio = length of incline / height of incline = d / h
Assume work done on the load = work done on the effort, so L X h = E X d
Mechanical advantage = L/E = d/h<.br> The action of a surfer on a surf board is the same as a block sliding down an inclined plane. Experiments
See diagram 21.3.06: Pull trolley up inclined plane
1. Use a smooth board at an angle of 30o to the table.
Weigh the trolley by suspending it from a spring balance.
This is the effort needed to lift the trolley from the table to the top of the board.
Put the trolley on the smooth board.
Pull it slowly up the board noting the reading on the spring balance.
The effort will be about its weight.
The smooth board is twice as long as it is high at the top.
By taking a longer path, the slope will be less and the effort less.
2. Attach a heavy toy car or a roller skate to a spring balance and pull it up an inclined plane.
Note the force required to move the car and compare it with the force needed to lift it vertically.
Note also that in moving up the inclined plane, the force is exerted over a greater distance than when the car is lifted vertically to the same height above the table.
Neglecting friction, the work required is the same in both cases and this is also true for other simple machines.
The force P is the effort required to move the weight, Mg, up the slope.
A movement of the weight a distance X along the incline will result in a vertical displacement of X sin θ.
If the incline is frictionless the force P required to move the weight up the slope = Mg sin θ.
Velocity ratio = 1 /Sin θ.
3. When deciding how to move a roller onto a truck, the roller may be too heavy to lift vertically so question is how to find a suitable plank to act as an inclined plane.
4. Put a roller skate or a heavy toy car on the table, attach a spring balance with a string.
See diagram 21.3.03: Pull roller skate up inclined plane
Steadily raise the spring balance and note the weight of the roller skate.
Pull the roller skate up an inclined plane with constant speed and note the force required.
You need less force to pull up the roller skate up the inclined plane than to lift it vertically.
However, you must apply the force for a greater distance up the inclined plane than when the roller skate is lifted vertically through the same vertical height.
5. Using an inclined plane to lift weights can save force, and the relations between saving force and inclined angle, saving force and the distance.
Use a piece of elastic band, a piece of black thread and a small plastic bottle.
Tie an elastic band to the neck of the bottle containing water.
Use a smooth long wooden board supported at one end several books.
Adjust the height of the inclined plane by pushing the books towards the lower end of the plane, not by changing the number of books.
Tie a black thread marker to the elastic about 20 cm from the neck of the bottle.
Place the bottle at the bottom of the inclined plane at minimum inclined angle.
Pull the bottle to the top of the inclined plane and note the distance from the black thread marker to the neck of the bottle.
Repeat the experiment by increasing the slope of the inclined plane five times.
In each experiment try to maintain the same speed of pulling.
Observe which situation the deformation of the elastic band is the smallest and longest.
21.3.2 Wedge
See diagram 21.3.2: Uses of a wedge
A wedge is two inclined planes, base to base.
Wedges are used to split logs and to split off blocks of stone in quarries.
A wedge has two inclined planes and a third surface that may be hit to apply force to lift an object (under one leg of an unsteady table) or lock (door jamb) or split (woodcutter's wedge) or cut (axe).
Many tools are in the shape of a wedge, e.g. knife, chisel, the spade, drill, awl.
The mechanical advantage of an inclined plane increases as the angle decreases.
Also, the sharper edge of a wedge allows a small force to produce a large pressure.
Experiment 1. Pile of heavy books.
Try to put your small finger inside the lowest book of a pile of heavy books.
Fold cardboard then open it to make an angle.
Hold the folded cardboard in your hand and push the thin edge between the pages of the bottom book.
Put your small finger in this angle and take the book out of the pile.
2. Block on a wedge
See diagram 21.3.1.1: Block on an incline
Weigh a wedge, m1, on a block, m2.
Total weight = m1 + m2.
Place the block on the surface of the wedge inclined at angle θ.
Let the block accelerate down the slope of the inclined plane of the wedge.
Total weight = m1 + m2 cos θ (ignoring frictional forces).
21.3.3 Paper inclined plane, paper screw thread
See diagram 21.3.3: Paper imitation screw thread, inclined plane
A screw thread is an inclined plane wound around a cylinder.
Make an imitation of a screw thread by wrapping a right angled triangle around a pencil, starting with the shorter of the two sides about the right angle parallel to the axis of the pencil.
The hypotenuse represents the path of the screw thread.
The pitch of a screw is small compared to its circumference, so the slope is very gentle and the gain in force is high.
A further gain in force is added when a lever, the spanner, is used to turn the nut.
For one complete turn of the nut, it advances by the distance of one pitch.
Experiments
1. Make a simple screw thread.
Cut a piece of white paper to make a right angle triangle 50 cm hypotenuse and 30 cm along its shortest side.
Use a round rod 40 cm long and roll the triangular piece of paper on the rod, beginning at the short side and rolling towards the point of the triangle.
Keep the base line of the triangle even as it rolls.
Observe that the inclined plane, the hypotenuse, spirals up the rod as a screw thread.
A screw thread is an inclined plane.
2. Make a simple screw thread.
Cut out several vertical triangles from a sheet of paper, each triangle has a different size.
Draw an obvious line in texts along the edge of the hypotenuse in each triangle.
Roll each triangle cardboard around a pencil to make it like a screw.
Compare the distance of the screw of each nail, analyse the difference between them.
The principle of the screw nail is similar to the inclined plane in which a weight is pushed upward.
Although a screw is a spiral inclined plane, it acts like a second class lever.
The screw's point is the fulcrum, where the thread meets a substance such as wood is the load, and the effort is applied to the head of the screw.
21.3.4 Screw, nut and bolt, pitch of screw
See diagram 21.3.01: Pitch of a screw
1. A screw is a simple type of machine, acting like an inclined plane rolled up in a helix.
The pitch of the screw is the distance between the threads.
So for one revolution of the screw, it moves laterally through a screwed nut a distance equal to the pitch.
The screw may be turned by a force applied tangentially at one end of an arm, with the other end of the arm is fixed to the screw.
The screw advances or withdraws through a screwed not or similar screwed appliance.
For one complete revolution the screw moves laterally through a distance = the pitch of the screw.
In a perfect machine, work done by the effort = work done on the load.
Work done by the effort = Effort, E X distance travelled by the effort = E X 2 π X r, where r = radius of the screw.
For one complete revolution by the effort, the load moves by distance = pitch of the screw.
Work done on the load = load X pitch = L X p
L X p = E X 2 π r
L/E = 2 π r/p
Mechanical advantage = L / Et = 2πr / p
For a vertical screw of pitch p lifts a mass of m when a tangential force is applied at the end of an arm X in length.
If efficiency = 80%, what force is required?
If force E is applied then the screw moves through 2πr, while load rises in height = pitch, p
Work done = change in potential energy
E X 2πr = L X p
E = Lp / 2πr X 100/80.
2. Philips screw heads, like a '+' sign, are being replaced by a six-sided 'torx' screw head, like a '*' sign, or by an Allen head, a regular hexagon.
Philips screws were designed to strip under high torque to protect the tool from damage.
21.3.5 Simple screw jack
1.See diagram 21.3.07 Simple screw jack
A screw jack is a mechanical device that can increase the magnitude of an effort force.
A screw jack is like an inclined plane wrapped around a cylinder.
A simple screw jack has a bench-mounted base and a turntable fitted with screw jack thread.
A cord is wound around the periphery of the turntable.
The free end of the cord is threaded over a pulley and then hangs vertically to support a load hanger.
Weights added to the load hanger produce a torque on the system.
Also, weights can be applied to the top of the turntable.
To find the pitch, turn the platform around once and measure the change in vertical distance between the platform and the base.
Increase the effort until the platform just starts to turn.
The mechanical advantage, MA = load / effort.
Experiments
1. Make a simple lifting jack.
Bore a hole through a block of wood to fit a carriage bolt.
Select a bolt that is threaded nearly its entire length.
Sink the head of the bolt in the wood, so that it is flush with the surface and nail a piece of board over it.
Over the projecting threads put a nut, then a washer and short piece of metal pipe.
The inside diameter of the pipe must be slightly larger than the diameter of the bolt.
By turning the nut with a wrench the device acts as a lifting jack.
2. To make a model jack use two wooden blocks of about 10 cm length.
Drill a hole at the centre of the smaller block.
Use a screw of more than twice of the block thickness.
Make a groove with a knife at the centre of the other block so that the screw can be placed in the groove.
Use a screw with a six angle head or flat-head.
Forcibly twist the screw through the hole on the smaller block.
Put the groove on the larger block on the screw cap.
Put the two blocks together then tightly nail them together with small nails.
Make sure that the screw does not rotate and it is not loose.
The contact surface between the two blocks is flat and level.
If the screw is loose, because the hole is too large, twist a nut to fix it.
If it is very difficult to make the groove that can contains the screw cap, place cardboard or iron gaskets between the two blocks instead of the groove.
The iron gaskets should be able to be just passed by the nails nailing the two blocks.
Place the device upside down.
Twist a nut on the screw and put an iron gasket on the nut and cover a piece of sawed iron tube on the gasket.
The inner diameter of the iron tube should be slightly larger than the crew and its outer diameter should be smaller than the gasket so that it can stay on the gasket, not to slip away.
The iron tube may is slightly shorter than the pole of the screw appearing.
Rotate the nut and the tube will go up or down.
When use the device to lift a heavy object, use a spanner to screw the nut and underlay the tube with a rigid board to reduce the pressure of the tube.
A real screw jack is upright like the model and when the screw is turned through one full turn the heavy is lifted up or down a distance equal to the pitch of its thread
2.See diagram 21.3.05: Screw thread and pitch
A screw may transfer between translation and rotation.
The screw thread is a ridge in the form of a helix on a cylindrical core.
The ridge may be triangular (V-shape) or square or round.
If the screw thread is triangular, greater friction may be used.
If the screw thread is square, larger loads may be lifted.
The distance between the adjacent threads of a screw or bolt is called pitch.
When you turn a screw through one full turn, it moves up or down a distance equal to the pitch of its thread.
Different standard screw threads include the following:
2.1 pre metric British Standard Whitworth (BSW) with an angle of 55o,
2.2 Sellers or USS screw thread (pre metric USA standard) with an angle of 60o,
2.3 and various metric screw threads.
Screws turning in bolts are used to fasten things together.
Screw mechanisms are used to adjust the focus of camera lenses.
A screw thread in a clamp movers the jaws of a the clamp together.
21.3.6 Mechanical jack, car jack
See diagram 21.3.04: Automobile screw jack
In a car jack a screw passes through a nut carrying an arm that fits into the chassis of the automobile.
Effort is applied to the lever above the screw.
When the lever completes one turn, the load (chassis) rises a distance equal to the pitch of the screw, the distance between successive threads.
Velocity ratio, VR = circumference of circle made by effort applied to the end of the lever / pitch of the screw.
The effort force for a screw jack, F = (Q p) / (2 π R), where F = effort force at the end of the arm or handle, Q = weight or load, p = pitch distance or lead of thread in one turn, r = pitch radius of screw, R = lever-arm radius.
21.3.8 Propellers
See diagram 21.266: Propellers
A propeller is used as a rotary propelling device in ships and aeroplanes.
It has blades (inclined planes) radiating from a central hub to be inclined to the plane of rotation as they drive a helical path water or air.
The two parallel propellers of a ship turn outward when the ship moves "ahead'".
The propellers in the fore and aft side of a modern passenger ship are called thrusters.
They can be used to turn a ship in a circle, but are used to speed up docking and undocking without the need for assistance from tug boats.
However, the authorities in some ports require tug assistance to be available for all ships entering or leaving the harbour.
Experiment
1. Make a rotor from the lid of a drink-can.
Roll the outer edge to avoid cuts.
Draw the three blades on the lid.
Make cuts first along the thick lines and then along the dotted lines.
Remove the smaller sections leaving three blades.
Put the drink-can lid on a block of wood and cut out the shape with a chisel.
Drill at the centre two 5 mm diameter holes 5 mm apart, then remove the little bridge of metal between them to make a central slot.
Use a twisted strip of thick metal 1 cm × 25 cm to fit the above slot or use two strong wires.
To twist the wire, fold a 60 cm length in half with a large loop at the bend.
Slip rod B through the loop and clamp the free ends close together in a vice.
Then twist up the doubled piece to give a long uniform twist of angle about 20o to the axis.
The holes in the rotor may need a little trimming so that it will spin freely up and down the twist.
Use a short tube made from tin plate that slides easily along the wire.
Twist the blades so that the angle of the rotor blades gives lift when the rotor is spun by being pushed off the wire.
The assembly has 3 parts:
* the wire, held vertically,
* the tin tube, which should rest on the loop at the foot of the twisted wire, and,
* the rotor, which should rest on top of the tin tube.
To launch this flying saucer, hold the arrangement steady above your head by the tube and strongly pull the wire twist down with the other hand.
Use different blade angles, or different numbers of blades to get the best flight effect.
Use a rubber band drive as a source of power for your propeller in a model aircraft or model boat.
21.4.01 Gears
Gear wheel, gear train, motor vehicle gears, servo-mechanism, transmission systems, clutch, differential gear, automatic gear change
See diagram 21.4.0: Motor vehicle gears
See diagram 21.3.023: Worm gear uses screw threads to transmit the rotary movement of one shaft to another shaft at right angles to it.
Gear is a toothed wheel that transmits the turning movement of one shaft to another shaft.
Gear wheels may be used in pairs, or in threes if both shafts are to turn in the same direction.
Gear with different shaped teeth or shaft may rotate at different angle.
The gear ratio, the ratio of the number of teeth on the two wheels, determines the torque ratio, the turning force on the output shaft compared with the turning shaft on the input shaft.
The ratio of the angular velocities of the shafts is the inverse of the gear ratio.
Gears mesh directly into each other without any chain.
A car has a box of gears that can be changed.
Imagine a large gear wheel and a small one.
When the larger gear wheel is attached to the engine, and the small gear wheel is attached to the back axle, we say the car is in top gear, because we get maximum revolutions of the car wheels per revolution of the engine.
If a smaller gear wheels is attached to the engine and a larger gear wheel is attached to the back axle, we say the car is in low gear.
If a toothed gear wheel with 10 teeth is meshed with a toothed gear wheel with 20 teeth.
For each complete revolution of the 20 teeth wheel the 10 teeth wheel makes 2 revolutions.
If the effort is supplied to the 10 teeth wheel to drive the 20 teeth wheel, the velocity ration is 2 = number of teeth on the driven gear (load) / number of teeth on the driving gear (effort).
21.4.1 Bicycle gears
Examine the gearing of a bicycle by counting the teeth on the sprocket wheel attached to the back wheel and the teeth on the large sprocket wheel to which the pedals are attached., e.g. 16 and 48.
Each time you push the pedals round once, you pull around enough chain to cover 48 teeth.
However, there are only 16 teeth in the back sprocket, so it is turned more than once, 48 / 16 = 3 times.
So each time you turn the pedals once, the back wheels rotates three times.
When you reach a hill your bicycle will be easier to push if you used a back sprocket with more teeth or you used a pedal sprocket with fewer teeth, so that one revolution of the pedals would give you fewer revolutions of the wheel.
Experiments 1. Turn a bicycle upside down.
Turn the pedal wheel exactly one turn and note the number of turns made by the rear wheel.
Examine the gear mechanism.
2. Study the speed of rotation and transmission of roller chain.
Use a bicycle without chain box.
Turn the bicycle upside down and let it stand on its saddle and handle.
Count the amount of the teeth of the large gear at the middle shaft and that of the small gear at the back shaft.
Forcibly rotate a pedal of the bicycle a circle then differently count the amount of the circles the large gear and the small gear rotating.
For convenience, mark with colour on some tooth of each gear beforehand.
Calculate the ratio of the amounts of their circles and teeth and find the relationship of the two ratios.
Observe how to transmit the large gear' rotation to the small one.
Observe the direction of the large gear and that of the small one.
Find the relationship of the direction of transmission to the directions of rotations of the large gear and small one.
21.4.2 Bottle top gears
See diagram 21.260: Gear wheels
Punch holes exactly in the centres of bottle tops.
Put two of the bottle tops on a block of wood so that the tooth like projections mesh.
Fasten them to the wood with small nails, but make sure that they still turn easily.
Turn one of the bottle tops and note the direction that the other turns.
Add a third bottle top and note the direction that each turns.
21.4.3 Rolling coins
Put two identical coins edge to edge and flat on the table.
Hold the first coin in a fixed position while you roll the second coin around it.
The second coin turns twice around its own axis when rolling once around the first coin.
If the radius of the coins = r, and the circumference = 2 X π X r, the centre of the second coin travels a distance of 2 X π X 2r.
21.5.01 Pulleys, block and tackle, broomstick pulley
See diagram 21.5.0: Types of pulleys
See diagram 21.254: Simple pulley
See diagram 21.255: Single fixed pulley
See diagram 21.256: Single movable pulley
Observe the tension in a string or rope.
Tie the upper end of a string to a support, and tie a brick to the lower end.
The string will be tight, i.e. have tension all along it.
21.5.1 Systems of pulleys
See diagram 21.5.2.1: First and second system of pulleys
See diagram 21.5.2.2: Third system of pulleys
1. First system of pulleys, MA = 2n (where n == number of pulleys).
The effort is lessened by the calculated effective mass of the pulleys.
In the diagram T3 = 1 / 2 T2 = 1 / 4 T1 = 1 / 8 mg
The mechanical advantage of a single fixed pulley = 1, if no friction.
The mechanical advantage of a single free pulley = 2, if no friction, pulley is weightless and strings are parallel to the load.
2. Second system of pulleys, the block and tackle has equal tension through the flexible string.
so T = E.
4T = L, so 4 E = L,
so MA = L / E = 4
The mechanical advantage = number of supporting strings if no friction and lower block is weightless.
The block and tackle is the most useful system of pulleys.
3. Third system of pulleys requires the determination of the line of action of parallel forces, T3, T2, T1 which should go through L, if the system is to be in equilibrium.
Experiments
1. Simple pulley
See diagram 21.254: Simple pulley
Use a wire clothes hanger and a cotton reel.
Cut the hanger wire 20 cm each side of the hook.
Bend the cut ends until horizontal then slip the ends into a cotton reel.
Push the cut ends through the cotton reel then turn them down where the come out of the other side.
2. Single fixed pulley
See diagram 21.255: Single fixed pulley
The single fixed pulley allows the use of a downward force, the effort, E, to lift a load, L.
It is just a convenient way to lift something by pulling down instead of pulling up.
The tension in the rope is equal to the weight of the body supported.
Mechanical advantage = 1, because E = L (ignoring friction).
Velocity ratio = 1, because distance of pulling down = distance body moves up.
Hang masses at A to find how much force you need to lift 50, 100 and 200 g placed at B.
You need the same force.
Pull 20 cm down on A and measure the distance moved by a mass at B.
The distances are the same.
3. Single movable pulley
See diagram 21.256: Single movable pulley
The single movable pulley has a fixed pulley to change direction and a moving pulley.
If the effort at the end of the rope = E, the total upward force on the moving pulley = 2E, because it is supported by two parts of the rope.
So the load = 2E (ignoring friction) MA = 2.
To raise the load by 1 m, the rope on each side of the moving pulley must shorten by 1 m, so 2 m of rope must be taken from the pulling end, VR = 2.
Use a spring balance to measure the weight of three books.
Suspend two pulleys on a string and use the books for a load.
Attach a spring balance to the end of the string and pull down on the ring at the end of the spring balance.
The force needed to lift the books is equal to half the weight of the books, ignoring friction and the mass of the pulley.
However, you must pull down at twice the distance needed to raise the books using a single fixed pulley.
Friction in the pulleys and the weight of the movable pulley lowers the efficiency of this system of pulleys.
4. See diagram 21.256.1: Wire coat hanger single movable pulley
See diagram.
One pulley
The string passes up from the load and then over the pulley - the effort is down.
The distance moved down by the effort equals the distance moved up by the load.
Two pulleys
The string passes up from the load then over the top pulley, then under the bottom pulley - the effort is up.
The distance moved by the effort is twice the distance moved by the load.
However, the effort required is about half the weight of the load.
21.5.2 Block and tackle
See diagram 21.5.2: Single fixed pulley, single free pulley, block and tackle
This pulley system is used in cranes and lifts.
In a car garage, the mechanics can lift a car engine out of a car by hand using a block and tackle.
You will notice that they pull down a long way while the engine block moves up a short way.
In the diagram the pulleys have been separated here to show the path of the rope more clearly.
Find the gain in force from the number of strings supporting the load.
The tension in the string remains constant and is one fourth of the upward pull on the load.
21.5.4 Pile driver
See diagram 21.184: Laboratory pile driver
A pile driver is a machine for driving piles into the ground usually by repeatedly dropping a dense heavy mass on the pile.
Raise the mass to the desired height and let it fall freely.
If the mass is raised twice as high, the nail will be driven twice as deep.
The maximum distance the mass can fall is about 86 cm.
A laboratory device has a 4 kg mass that can slide up and down on two vertical rails attached to a metal base.
It is used to find the average force needed to:
1. drive a nail through a block of wood,
2. crush a metal drink-can.
Each time the mass falls the nail moves a distance down, or the drink-can is squashed a distance down.
Calculate the work needed to crush a soda can or drive a nail.
Work done, W = FS.