School Science Lessons
(UNPh18) 2024-04-01
Rotation, Moments
Contents
18.3.4 Angular momentum
18.2.2.0 Complex systems
18.3.5 Gyroscope, precession
18.3.1 Moment of inertia
18.2.3.0 Moments, torque
18.3.2 Rotational energy
18.0.0 Rotational motion
18.3.6 Rotational stability
18.3.3 Transfer of angular momentum
18.2.2.0 Complex systems, yo-yo on a balance
18.2.3.1 Acceleration on a balance
18.2.3.4 Funnel of water on a balance
18.2.3.3 Hourglass on a balance
18.2.3.5 Reaction balance
18.2.3.2 Yo-yo on a balance
18.2.3.0 Moments, torque
18.4.2.0 Moments, parallel forces, couple
18.4.2.01 Torque, moment of a couple, torque beam
4.146 Balance with a metre stick
18.4.2.3 Balance with a see-saw (teeter-totter)
18.4.2.15 Grip bar
18.4.2.10 Loaded beam
18.4.2.6 Metre stick balance
18.4.1.0 Resolution of forces, inclined plane
18.4.2.8 Rolling spheres
18.4.1.2 Rope and three students
18.4.2.7 Tightrope walking
18.4.2.9 Walking the plank
18.3.4 Angular momentum
18.3.4 Conservation of angular momentum
18.3.4.12 Air rotator with deflectors, Feynman inverse sprinkler
18.3.4.10 Buzz button
18.3.4.4 Centrifugal governor, rotating stool and weights, "squeezatron", Watt's regulator
18.3.4.7 Counter spinning
18.3.4.2 Hero's engine, lawn sprinkler
18.3.4.9 Pocket watch
18.3.4.3 Pulling on the whirligig
18.3.4.5 Rotating stool
18.3.4.5a Rotating stool and wheel
18.3.4.11 Sewer pipe pull
18.3.4.13 Skiing
18.3.4.1 Spinning funnel, marbles and funnel
18.3.4.6 Toy train on a circular track
18.3.4.8 Wheel and brake
18.3.5 Gyroscope, precession
See: Gyroscope, (Commercial)
18.3.5.1 Boomerangs
18.3.5.0 Gyroscope, precession, flywheel
18.3.5.5 Gyro pendulum
18.3.5.4 Gyrocompass, gimbals mount
18.3.5.3 Gyroscope, bicycle wheel gyro, gyro in gimbals
18.3.5.3a Precessing gyro
18.3.5.2 Precession, spinning top, precessing ball
18.3.5.6 Ships' stabilizers
18.3.1 Moment of inertia, angular momentum
18.3.1.1 Inertia wands
18.3.1.5 Rattleback, wobblestone
18.3.1.4 Rigid and non-rigid rotations, parallel axis wheels
18.3.1.3 Race rings, discs and spheres down a smooth slope
16.1.4.1 Rotational inertia
16.1.4.2 Inertia of rotational solid
16.1.4.3 Spin dryer for clothes
16.1.4.4 Spinning ice skater
18.3.1.2 Torsion pendulum inertia
18.3.2 Rotational energy
18.3.2 Rotational energy
18.3.2.1 Adjustable angular momentum
18.3.2.2 Angular acceleration wheel
18.3.2.4 Faster than gravity, falling chimney, coins on a metre stick
18.3.2.3 Spool on incline, rolling spool
18.0.0 Rotational motion
Central forces, Loop the loop, Watts governor
18.3.4 Conservation of angular momentum
18.3.5 Gyros, gyroscope, precession
18.3.1 Moment of inertia, conservation of angular momentum
18.3.0 Rotational dynamics
18.3.2 Rotational energy
18.3.6 Rotational stability
18.3.3 Transfer of angular momentum
18.3.6 Rotational stability
18.3.6.0 Magnus effect
18.3.6.4 Football spin, spinning lariat (lasso)
18.3.6.1 Humming top, tippy top
18.3.6.3 Spinning coin
18.3.6.6 Static balance
18.3.6.7 Tides simulation, spinning glass of water
18.3.6.5 Tossing the book, tossing the hammer
18.3.6.2 Yo-yo, Chinese diabolo
18.3.3 Transfer of angular momentum
18.3.30 Transfer of angular momentum
18.3.3.1 Passing the wheel, pass bags of rice, catch ball on the stool
18.3.3.2 Satellite derotator
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: Uniform rod.
See diagram 4.146.1: Metre stick.
A body acts as if its mass is concentrated at a single point, the centre of mass.
Gravity acts through the same point, the centre of 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 weight and has less friction so the rod easily slides above it.
Experiments
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.
4. Repeat the experiment by hanging your hat on one end of the metre stick.
Note the new position of the centre of gravity.
5. Repeat the experiment with a broom to find its centre of gravity.
6. Slide two kitchen scales under a loaded beam.
Note the scale readings of the moving and stationary scales change in the same way that your fingers feel change in weight under the metre stick.
7. Put an empty drink-can on a rough wooden board.
Raise one end of the board until the drink-can falls over.
At that angle, a vertical line through the centre of gravity of the drink-can passes outside its base.
8. Stand still then raise your right arm sideways.
Nothing happens.
Raise your right leg sideways.
If your upper body moves to the left, your centre of 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.
18.2.3.1 Acceleration on a balance
Burn the string extending a mass on a spring on a taped platform balance.
18.2.3.2 Yo-yo on a balance
Hang a yo-yo from one side of a balanced critically damped platform scale.
18.2.3.3 Hourglass on a balance
Observe an hour glass running down on a taped critically damped balance.
Put a very large hour glass on a critically damped balance and note the deflection as the sand starts, continues, and stops falling.
The centre of mass is accelerating upwards during most of the process.
18.2.3.4 Funnel of water on a balance
Put a funnel full of water on a taped platform balance, release the water and collect in a beaker.
18.2.3.5 Reaction balance
Support one mass on an equal arm balance by pulleys at the end.
The balance is in equilibrium if the string holding the mass is not touched or pulled in uniform motion.
18.4.1.0 Resolution of forces, inclined plane
Resolution of forces, resolution into rectangular components, forces in cables, parallelogram law, resolving a force, inclined plane
See diagram: 18.4.1: Normal reaction on inclined plane.
If the angle of an inclined plane = a, then the component of the weight perpendicular to the inclined plane = W cos a, is balanced by the normal reaction of the plane.
The component of the weight that would cause the object to accelerate down the inclined plane, if no friction = W sin a, or (W sin a - force of friction).
18.4.1.1 Hanging block, suspended block
See diagram 18.4.1.1: Suspended block.
Show vector addition of forces, by attaching pulleys and string to a block of wood resting on an incline plane.
Taking the "support" board away from the suspended block reveals a force vector equilibrium.
The sides of the triangle are in the ration of 3: 4: 5.
The 1500g block rests on a stop which can be removed after the force parallel to the incline is balanced.
The block will be suspended in the same configuration as when on the incline.
The masses to balance the perpendicular and parallel forces are 1200g and 900g, respectively.
When the blocks are in this position on the inclined plane the inclined plane can be removed.
18.4.1.2 Rope and three students
See diagram 18.4.1.2: Rope and three students.
A single student can easily deflect a rope held very taut by two other students.
18.4.2.0 Moments, parallel forces, couple
See diagram 18.4.2.0: Moments.
1. The moment of a force about a point = force × perpendicular distance from the point to the line of action of the force.
The moment of a force about an axis = component of the force in a plane at right angles to the axis × perpendicular distance to the axis from the line of action of the component.
2. The size and direction of the resultant, R, for two like parallel forces, P and Q = P + Q, passing through point O between P and Q, where P × AO = Q × OB.
3. The size and direction of the resultant, R, for two unlike parallel forces, P and Q = P - Q, passing in the direction of the greater of the forces through point O beyond the greater force where P × AO = Q ×OB.
4. A couple refers to the application of two equal and opposite forces to cause rotation, e.g. pushing on the handle bars of a bicycle to change its direction.
The moment of a couple, i.e. the torque = the product of one of the forces and the perpendicular distance between the line of action of the forces.
18.4.2.01 Torque, torque beam
See diagram 16.165: Torque beam 1.
See diagram 18.4.2.01: Torque beam 2.
A torque is a twisting or rotating force, the moment of a system of forces causing rotation or changing the speed of rotation.
A torque wrench is a tool for setting and adjusting the tension of nut and bolts so that they are not too loose and not too tight.
A torque converter in the automatic transmission system in a motor vehicle changes the torque in the engine.
A couple consist of two equal and opposite parallel forces that have a turning effect, moment, called a torque = one of the forces × perpendicular distance between their lines of action.
The turning effect is about an axis that is normal to the plane of the forces.
The SI unit for the torque of the couple is newton metre.
(A torque converter in an automatic transmission system of a motor vehicle varies or multiplies torque.)
1. Different combinations of mass at different distances from the pivot can be used to show torques in equilibrium.
Distances from the pivot are integer multiples: r, 2r, 3r, 4r. Individual masses are identical.
2. To show different torque in equilibrium, use different combinations of masses at different distances from a pivot.
3. A uniform rod mass 500g and length 120 cm is supported horizontally by two vertical strings, T1 at one end A, and T2 at 30 cm from the other end B.
What is the tensions in the strings when a mass of 200 g hangs from end B? (see diagram).
4. Use a torque wrench to break aluminium and steel bolts.
18.4.2.1 Beam balance, moments, parallel forces in equilibrium
See diagram: 16.13: Beam balance.
The moment of a force is a measure of the turning effect, or torque, produced by the force acting on an object.
It is equal to the product of the force and the perpendicular distance from its line of action to the point, or pivot, about which the object will turn.
Its SI unit is the newton metre (Nm) If the magnitude of the force is F newton and the perpendicular distance is d metres then the moment is given by: moment = Fd.
18.4.2.3 Balance with a see-saw (teeter-totter)
See diagram 16.4.13: See-saws.
See diagram 8.145: Balance with a see-saw.
See diagram 4.145: Balance with a see-saw.
M1 moments anticlockwise, M2 moments clockwise.
1. Use a strong board 3 m long and a saw horse to make a see-saw, or use a playground see-saw.
Select two students of similar weight.
Tell them to sit at either end of the board so that they balance and the see-saw is horizontal.
Measure the distance from the balance point, the fulcrum, to each student.
They are similar distances from the fulcrum.
For each student, calculate the moment by multiplying the distance rom the fulcrum by the student's weight.
The moments clockwise should equal the moments anticlockwise.
Select a heavier student and a lighter student and repeat the experiment.
Tell them to sit on the board so that they balance.
Measure the distance from the balance point to each student.
Multiply the distance by the student's weight to calculate the moments clockwise and moments anticlockwise.
For objects in equilibrium the moment in one direction is equal to the moment in the opposite direction.
2.1 Make a see-saw with 3 m board and a sawhorse for a fulcrum.
Use two students of equal weight.
Sit at either end of the board so that they balance.
Measure the distance from the fulcrum, balance point, to each student.
Multiply the distance by the weight of the student.
2.2 Select a heavier student and a lighter student.
Tell them to sit on the board so that they balance.
Measure the distance from the fulcrum to each student.
Multiply the distance by the student's weight.
2.3 Select a heavier student, weight m1, and a lighter student, weight m2.
Sit on the board so that they balance.
Measure the distance from the fulcrum to each student, d1 and d2.
Multiply the distance by the student's weight.
You will discover that m1d1 = m2d2.
2.4 Select a heavier student, weight m1, and two lighter students, weight m2 and m2.br> Sit on the board so that they balance.
Measure the distance from the fulcrum to each student.
Multiply the distance by the student's weight.
Add the products for the two lighter students.
m1d1 = m2d2. m1d1 = m2d2 + m3d3.
18.4.2.6 Metre stick balance
1. Hang weights from a beam that pivots in the centre on a knife edge.
2. Use a metre stick suspended at the centre as a torque balance.
18.4.2.7 Tightrope walking
Compare how much easier it is to balance a metre stick over the finger compare to a 10 cm ruler.
The reason is the difference in inertia of the two rules.
A tightrope walker may carry a long stick, often with a weight attached to each end to increase inertia.
Moment of inertia, I = mr2, where m = mass and r = distance.
The masses further away from the centre are harder to move.
So any deviations from the balanced equilibrium position happen slowly and give the tightrope walker ample time for correction.
18.4.2.8 Rolling spheres
If two spheres have same diameter and mass, but different densities of their material such that one sphere is solid and the other sphere is hollow, the hollow sphere rolls more slowly down a slope, because all of its mass is at a distance from the centre of the sphere.
18.4.2.9 Walking the plank
Place a 25 kg block on one end of a long plank, hang the other end of the lecture bench and walk out as far as you can.
18.4.2.10 Loaded beam
Put large masses on a board resting on two platform balances.
Move a heavy toy truck across a board bridge supported on two platform scales then two spring scales.
18.4.2.15 Grip bar
See diagram 16.164: Grip bar 1.
Suspend a 1 kg mass from hooks on a bar at 5 different distances from the handle grip.
The further from the handle grip the more difficult to keep the bar level or rotate it upwards using wrist strength.
18.3.0 Rotational dynamics, rotational motion
Rotational motion refers to a situation when a rigid body is rotating about a fixed axis.
The average angular velocity of the body, ωav = θt radians per second, where θ is the angle turned, i.e. the angular displacement, during the time t.
So for one revolution at constant angular velocity, ω, the angle turned = 2π radians during the time period, T.
So angular velocity, ω = 2 πT radians per second.
The average angular acceleration, αav = change in angular velocity.
Time taken = (ωf - ωi)t radians per second2, where ωf = final angular velocity and ωi = initial angular velocity.
Table 18.3.0
Linear equations of motion
1. v = u + at
2. s = (u + v) t2
3. s = ut + at22
4. v2 = u2 + 2as
Rotational equation of motion
1. ωf = ωi + α t
2.θ = ( ωf + ωi) /2 × t
3. θ = ωit + 1/2αt2
4. ωf2 = ωi2 + 2αθ
Linear equations of motion where u = initial velocity, v = velocity after time t, s = distance travelled in time t, a = constant acceleration, (v, s, and a are positive in the direction of u).
18.3.1.1 Inertia wands
See diagram 18.3.1.1: Inertia wands.
1. The two wands have the same mass, but have the mass distributed differently.
One has the mass concentrated in the middle, the other has the mass concentrated at the ends.
Instruct two students to rotate back and forth as fast as they can to see the difference.
2. Twirl two equal mass wands with the mass at the ends and with the mass at the middle.
Use hollow cylinders containing hidden weights.
Use weights taped to metre sticks.
18.3.1.2 Torsion pendulum inertia
Use the period of a torsion pendulum to find the moment of inertia.
Put objects on a trifilar-supported torsion pendulum.
16.1.4.1 Rotational inertia
(Rugby footballs and American footballs are "spin stabilized").
They make them to spin about the axis of their direction of motion then do not tumble end over end and be retarded by extra air resistance.
A children's top falls over when place on its tip, but a rotating top remains upright until it loses all its angular momentum to friction between the tip and the ground and some air resistance.
16.1.4.2 Inertia of rotational solid
Observe the inertia of an object at the state of rotational motion.
Rotate a coin with your middle finger and thumb on a slippery tabletop.
When leave the fingers off the coin, it does not stop rotating at once.
It keeps moving for a long time then falls down.
Use a 25 cm piece of string.
Fasten one end of the string to a clip.
Hold the other end of the string to quickly rotate the clip differently at upright, horizontal and inclined planes.
When your hand holding the string stops suddenly, the string and clasp always keep rotating several circles before they stop.
16.1.4.3 Spin dryer for clothes
Half fill a spin dryer with wet clothing.
Turn on then turn off the spin dryer and count the number of rotations and record the time until it stops.
Observe the arrangement of the clothing in the stopped spin dryer.
Repeat the experiment with the spin dryer 3 / 4 full of wet clothing and make the same observations.
The more wet clothing in the spin dryer, the more rotations when you turn it off, because of the greater rotational inertia.
The clothing dried off is always distributed equably with more on the outside.
However, if the spin dryer contains only pieces of wet clothing, the spin dryer barrel does not rotate normally, because of unequal distribution of mass.
16.1.4.4 Spinning ice skater
Go to an ice stadium or watch on television the actions of an athlete rotating at high speed.
Observe the positions of the body, arms and legs of the athlete starting to rotate, rotating, stopping rotating, the changes in position, the relationship of the changes to the velocity and time of the rotation.
The positions of the body, arms and legs of athletes affect their rotational inertia through affecting the distribution of their mass.
An ice skater can start a spin on one toe with one leg extended and both arms extended (I is large and ω is small), but when the ice skater brings both legs and both arms together (now I is small and ω is large), the moment of inertia decreases and speed of spin increases, the skater spins much faster due to conservation of angular momentum.
18.3.1.3 Race rings, discs and spheres down a smooth slope
See diagram 18.3.1.3: Ring, disc and sphere race.
Release rings, discs and spheres at the same time and record which one gets to the bottom first.
The moment of inertia of a body, I, depends on its distribution of mass.
Let M = mass and R= radius.
Rings, hoops or cylindrical shells, e.g. hoola hoops, tyres, bicycle wheels
Moment of inertia, I = MR2.
Discs or solid cylinders, e.g. checkers, drink coasters, plastic plates, coffee cans
Moment of inertia, I = 1/2 MR2.
Solid spheres, e.g. ball bearings, billiard balls, golf balls, marbles, solid balls (not ping-pong balls)
Moment of inertia, I = 2/5 MR2.
So moment of inertia of rings > discs > solid spheres.
The speed of an object at the bottom of a slope depends on its moment of inertia and the larger the moment of inertia the slower the speed.
So solid spheres beat discs, beat rings, down the slope.
The weight of solid spheres is the most closely distributed around its centre of gravity.
Experiments
Roll down a smooth slope.
1. Rings, discs and spheres with same radius, different radius, same mass, different mass.
2. Coffee tins empty, loaded with fine dry sand or powdered tungsten or iron
3. Two wooden discs, same mass, diameter, and weight, weighted in the centre, and, weighted at the rim.
The discs have different moments of inertia, but have the same kinetic energy at the bottom.
18.3.1.4 Rigid and non-rigid rotations, parallel axis wheels
Spin with a falling weight two masses on a horizontal bar fixed to a vertical shaft, so that you can lock or free the masses to rotate in the same plane as the vertical shaft.
Measure with the wheel spinning or locked, the period of a bicycle wheel suspended as a pendulum.
18.3.1.5 Rattleback, wobblestone
"Rattleback", wobblestone, semi-ellipsoidal mass distribution (toy product)
A rattleback is a semi-ellipsoidal toy that rotates only in one direction.
If spun in the other direction, it rattles, and reverses its spin.
18.3.2 Rotational energy
Other experiments: bicycle wheel angular acceleration, bicycle wheel on incline, hinged stick and ball, penny drop stick.
18.3.2.1 Adjustable angular momentum
See diagram 18.3.2.1: Adjustable angular momentum.
1. The moment of inertia can be changed by sliding, the masses on the stick in or out.
Three different pulley sizes offer three different torques.
2. Hang various weights from the axle of a large wheel and time the fall.
A falling weight on a string wrapped around a spindle spins objects to show Newton's second law for angular motion.
18.3.2.2 Angular acceleration machine, angular acceleration wheel
Measure the angular acceleration of a bicycle wheel due to the applied torque of a mass on a string wrapped around the axle.
Use a spring scale to apply a constant torque to a bicycle wheel and measure the angular acceleration.
18.3.2.3 Spool on incline, rolling down an incline, rolling spool
Roll a large spool down an incline on its axle.
When it reaches the bottom it rolls on the diameter of the outer discs showing conservation of linear momentum.
Roll a bicycle wheel rolls down an incline on its axle with the axle pinned to the wheel or free.
Time a roller as it rolls up an incline under the constant torque produced by a cord wrapped around over a pulley to a hanging mass.
18.3.2.4 Faster than gravity, falling chimney, coins on a metre stick
A ball at the end of a falling stick jumps into a cup faster then gravity.
A hinged inclined board with a ball on the end jumps into a cup a short distance down the board as the incline drops.
Line a meter stick with coins and drop one end with the other hinged.
18.3.3 Transfer of angular momentum
Corresponding to linear momentum, if an object is in rotational motion, it will have a quantity of motion angular momentum, with symbol L, and with SI unit kg.m2s.
For rotation about a fixed axis, the angular momentum is the product of the rotational inertia of the object about the axis and its angular velocity:
L = I × kg.m2s. kilogram metre2second (kg m2 s-1).
Rotational inertia is a quantity describing rotational state, with symbol I or J, and with SI unit kg.m2.
18.3.3.1 Passing the wheel, pass bags of rice, catch the ball on the stool
See diagram 18.3.3.1: Passing the wheel.
1. Tip the spinning tire half way and hand it to a student on a turntable.
This student tips it another half way and hands it back.
Repeat until the spinning student it turning to fast for the hand off.
You can add or subtract from the angular momentum depending on which way you tip the wheel.
2. Pass a spinning bicycle wheel back and forth to a person on a rotating stool or small merry-go-round.
Stand on a rotating stool or small merry-go-round and holds out 5 kg bags of rice and drop them.
Sit on a rotating stool or merry-go-round and catch a heavy ball at arms length.
18.3.3.2 Satellite derotator
Heavy weights fly off a rotating disc carrying away angular momentum.
18.3.4 Conservation of angular momentum
See diagram 15.2.1: Whirling coat hanger.
See diagram 15.2.2: Whirling stopper.
18.3.4.1 Spinning funnel, marbles and funnel
A funnel filled with sand spins faster as the sand runs out to show conservation of angular momentum.
The angular speed of marbles increases as they approach the bottom of a large funnel.
18.3.4.2 Hero's engine, lawn sprinkler
Cylindrical boiler pivots on a vertical axis with tangential pressure relief nozzles.
Suspend a round bottom flask with two nozzles so that the flask rotates on a horizontal axis.
Use a gravity head of water to drive a Hero's engine lawn sprinkler.
18.3.4.3 Pulling on the whirligig
Attach balls to either ends of a string that passes through a hollow tube so you can set one ball twirling and pull on the other ball to change the radius.
Shorten the string of a rotating ball on a string.
18.3.4.4 Centrifugal governor, rotating stool and weights, "squeezatron", Watt's regulator
See diagram 18.3.4.4: Centrifugal governor.
1. Spin a small governor on a hand crank.
Spin on a rotating stool or merry-go-round with a dumbbell in each hand so you can extend and retract your arms while rotating on a stool.
Expand or contract a fly ball governor by squeezing a handle showing the pirouette effect of ice skaters.
Spin and turn a bicycle wheel while on a turntable or merry-go-round.
Turn yourself around on a turntable by variation of moment of inertia.
2. Demonstrate the principles behind the Watts governor.
The Watts centrifugal governor has two heavy balls connected via pinions to a rotating axle.
Turn the handle to fling out the balls caused by centrifugal forces.
The balls are attached via linkage arms and a spring to a collar on the rotating axle.
This collar is connected via some mechanism to the input of the engine controlling the rotation of the axle to constitute a feedback control mechanism.
The governor can be used in the motor control gear for the lifts.
3. Engines supplying motive power usually have to run at a fairly constant speed.
This is managed by means of a governor, which controls the supply of steam, gas, or oil, by opening or closing a valve.
The governor usually consists of two revolving balls mounted on two arms attached to a shaft driven by the engine.
The balls move out to a larger radius of rotation if the speed increases, and move inwards if the speed falls, and this movement controls the supply valve.
The forces acting on one of the balls are its weight, W, the pull of the arm, T, and centrifugal force, P.
If the ball is rotating at a steady speed it will keep at a constant radius from the axis, and P, W and T will balance.
18.3.4.5 Rotating stool
See diagram 18.3.4.5: Rotating stool.
1. Start a student rotating on the platform with the masses close to their body.
Watch the change in spin as the student moves the masses further away.
2. A student sits on a rotating stool while holding a dumbbell weight in each hand close to the body.
Another student rotates the stool slowly.
The student on the stool outstretches the arms and the redistribution of mass requires a change in rotational speed to conserve angular momentum.
If initially the student has the hands outstretched when the weights are brought in toward the body, the increase in rotational speed can be enough to make it almost impossible for the student on the stool to stay seated.
18.3.4.5a Rotating stool and wheel
See diagram 18.3.3.1: Passing the wheel.
A student rotating on the platform can tip a spinning wheel in order to spin.
Tipping the wheel in the opposite direction spins the student the opposite direction.
18.3.4.6 Toy train on a circular track
Use a clockwork HO gage train running on a track mounted on a bicycle wheel rim.
The train and track move n opposite directions.
18.3.4.7 Counter spinning
An induction motor is mounted so both the frame and armature can rotate freely.
No torque is required to tilt the direction of axis of rotation unless either the frame or armature is constrained.
18.3.4.8 Wheel and brake
Brake a horizontal rotating bicycle wheel attached to a large frame and the combined assembly rotates slower.
18.3.4.9 Pocket watch
Suspend a pocket watch by its ring from a sharp edge.
18.3.4.10 Buzz button
Pull on a twisted loop of string threaded through two holes in a large button to get the button to oscillate.
18.3.4.11 Sewer pipe pull
Put O rings around a section of large PVC pipe to act as tyres.
Place on a sheet of paper and pull the paper out from under it.
When the paper is all the way out the pipe stops.
Pull a strip of paper horizontally from under a rubber ball.
As soon as the ball is off the strip it stops.
18.3.4.12 Air rotator with deflectors, Feynman inverse sprinkler
Run an air sprinkler then mount deflectors to reverse the jet.
Place an air jet Hero's engine in a bell jar and pump out some air.
The inverse sprinkler moves in a direction opposite to that of a normal sprinkler.
An inverse sprinkler made of soda straw in a carboy shows no motion due to conservation of angular momentum.
18.3.4.13 Skiing
Go skiing while holding a bicycle wheel gyro so that by conservation of angular momentum you turn yourself with the gyro.
Stand on a rotating turntable or merry-go-round with skies on to show the upper part of the body turning opposite the lower part.
18.3.5.0 Gyroscope, precession, flywheel
See: Gyroscope, (Commercial).
ZeCar Flywheel Car (toy product)
See diagram 18.3.5.0: Gyroscope.
See diagram 18.3.5.01: Gyroscopes.
See diagram 18.3.5.03: Bicycle wheel.
Experiments
1. A body is free to rotate about three mutually perpendicular axes.
If when rotating about one axis (axis of spin) a torque is applied about another axis (axis of torque), the body will rotate about the third axis (axis of precession).
Use the right hand rule (thumb, first and second finger mutually at right angles), where first finger represents axis of spin, second finger represents axis of torque and thumb represents axis of precession.
2. Angular velocity of precession from formula L = Ι, where L = torque about the axis of torque, Ι (upper case iota) = moment of inertia about the axis of spin (lower case omega) = angular velocity of spin (upper case omega) = angular velocity of precession.
3. A bicycle wheel weighted with lead on its rim is spun by a motorized wheel spinner and supported by a wire connected to a protruding axle.
The gravitational torque causes precession of the wheel.
Give the spinning bicycle wheel is handed to a student seated on a stool.
When the student raises the spinning wheel overhead, the student rotates through 90o.
When the student lowers the wheel to its original position, the student rotates in the other direction.
4. An electrically driven gyroscope may be mounted so that weights may be added to the protruding axles.
If equal weights are placed equidistant from the rotating mass, no precession occurs.
When an unbalanced torque exists, the gyroscope rotates perpendicular to both the rotation axis and the couple axis.
5. A car you are driving has a flywheel revolving in an anticlockwise direction with reference to you.
When you turn a corner to your right you are applying a torque to turn the flywheel about a clockwise axis (looking down on the flywheel).
The axis of spin is away from you, the axis of torque is vertically up and the flywheel precesses, so that the axis of spin follows the axis of torque
with the top of the flywheel tending to move towards you and the bottom of the flywheel tending to move away from you.
So during the right hand turn the axis of precession is anticlockwise to the left horizontally.
6. A gyroscope is a spinning wheel or disc that spins around it axis where the direction of rotation remains the same and the direction of spin remains
the same.
The gyroscope is mounted in two rings, gimbals that allow the axis of spin to remain opined in the same direction matter how the gyroscope is held.
The gyroscope will resist movement in direction of the input axis or output axis, but remain spinning at right angles to the spin axis.
7. The pull of gravity on the gyroscope is countered by the force of precession, i.e. the tendency of spinning bodies to move at right angles to a force that tends to change its direction of rotation.
8. A gyroscope can be used in navigation, because it resisted changes in its direction of rotation so it can show the direction of movement compared to the original direction of movement when first set spinning.
So a gyro compass shows the direction of north not by using terrestrial magnetism, but because it was originally set to point true north before the start of the journey.
Boomerangs
Boomerangs University of Cambridge, UK
See diagram 18.3.5.1: Boomerang.
As the boomerang flies in the air, it does two movements, a spinning motion and a general forward motion.
The spinning produces two effects, on one hand the angular momentum of spinning should be maintained unchanging, so the speed and plane of spinning are all unchanging; on the other hand spinning changes the direction of the flying boomerang, i.e. the boomerang does not fly in a straight line, but a curved line.
When the boomerang moves to the farthest point, its momentum has used up and it will drop acted by the gravity.
However, as the spinning of the boomerang, the line that the boomerang goes will also a curved line, return to the man who threw it at first.
Experiments
1. Draw a boomerang on the cardboard in the shape and cut it off.
Hold the centre of the boomerang.
Bend two wings gently to make wings slightly turn upward.
Hold the edge in the centre of the boomerang between index finger and thumb of your left hand, flick it away with the middle finger of your right hand to make it fly inclined upward.
The boomerang will fly along a arc line and return by itself.
2. Look at the different types of boomerangs in terms of the forces acting on them and how they change the motion.
Give each student a wooden craft-cut boomerang and sandpaper of different grades.
Tell the students to alter the shape of the boomerang and test it to see if they can get it to return to them.
If they add Blu Tak adhesive to the mix the students can observer at the effect of centre of mass.
The students should do the throwing tests on a sports oval in a clearly-delineated launch area.
18.3.5.2 Precession, spinning top, precessing ball, precession of the equinoxes
Precession is the rotation of the axis of a spinning body about another axis, caused by a torque acting to change the direction of the first axis.
This motion is like the movement of the Earth's axis in the precession of the equinoxes.
The regular motion of the inclined axis of the spinning top around the vertical is an example of precession.
Spin a child's spinning toy top and note how the axis of the top gradually moves in a circle, because of precession.
When the spinning of the top slows, the circle of the precession increases, until the top wobbles and falls over.
Some children can whip a spinning top to increase the velocity of spin and decrease the circle of precession until the axis is almost vertical.
A spinning top has gyroscopic inertia in that it stays spinning on its axis at the same angle, and so it is difficult to push it over.
18.3.5.3 Gyros, gyroscope, bicycle wheel gyro, gyro in gimbals, air bearing gyro
Mount a bicycle wheel on a long axle with adjustable counterbalance.
Support a spinning bicycle wheel with two handles by a loop of string around one of the handles and push the ends of the handles horizontally in opposite directions.
Make a gyro out of an auto wheel and tyre big enough to sit on.
Push a cart with a gyro around the room.
Spin a flywheel hidden in a suitcase and turn around with it.
1. Hold a heavy gyro outfitted with handles.
Use a motorcycle as a gyro.
The handlebars are twisted, but not moved in the direction opposite to the turn to lay the machine over.
Tip to one side a hand spun bicycle wheel on a front fork.
2. Separate a bicycle wheel from a bicycle.
Hold it in front of you by holding each end of the axle.
Spin the bicycle wheel very fast.
While still holding the bicycle wheel in font of you try to twist the spinning wheel by pushing down with the left hand.
The wheel will move forwards at right angles to the source of pressure, i.e. towards the left.
3. Observe bicycles mounted upside down on cars.
These bicycles may be taken to sports events or are owned by families who ride them during family holidays.
Note how the wheels start to spin when the car accelerates or turns a corner.
18.3.5.3a Precessing gyro
See diagram 18.3.5.3a: Precessing gyro.
A high quality gyroscope with a counterweight is used to show the fundamental precession equation with fair precision.
18.3.5.4 Gyrocompass, gimbals mount
See: Gyroscope, (Commercial).
A gimbals mount is a bearing for supporting an object to keep a horizontal position allowing for rotation about 2 perpendicular axes, e.g. nautical compass, gyroscope, chronometer used on a ship.
A gyroscope in gimbals is deprived of one degree of freedom A slight change of direction will cause a spin flip.
In an aircraft turn indicator the gyro precesses about the axis of the fuselage.
A ship stabilizer is like a gyro on a trapeze.
18.3.5.5 Gyro pendulum
Swing as a pendulum a gyroscope hung from one end of its spin axles by a string.
18.3.5.6 Ships' stabilizers, anti-roll stabilizers
See diagram 18.3.5.6: Ship with stabilizers.
Most ships, especially passenger ships, are fitted with two stabilizers about 5 m long on each side of the ship.
They are retractable and are normally housed in compartments below the hull when the ship is in narrow waters or in port, or when sea conditions are calm.
The stabilizers direct the flow of water to create lift in a similar manner to aeroplanes.
The created lift can counteract 90% of the rolling motion of the ship.
The fin angle of the stabilizers is adjusted automatically by a system linked to a gyroscope to detect the motion of the ship.
The first ships stabilizers were fitted in the 1930s by the Denny-Brown shipbuilding company.
However, although stabilizers can counteract roll, they can do little or nothing to counteract lift, the fore-aft motion of ships.
So even the largest passenger ships in the world, e.g. Queen Mary II, will lift and drop when directed into a swell with little or no counteraction
from the stabilizers.
Some passengers have claimed that during a storm the captain of a passenger ship has retracted the ship's stabilizers to avoid damage from the stormy conditions.
Ships stabilizers, gyrostabilizers, in a passenger ship, are adjusted to make the ship float in a more upright position and counteract the rolling motion.
They consist of fins mounted on each side of the ships and controlled by a motor-driven gyroscope.
The fins can be adjusted by a computer to produce maximum upward lift as with an aeroplane wing.
A toy gyroscope can remain balanced on any object and will remain in that position in the original direction as long as the wheel keeps spinning above a certain speed of rotation.
Ships stabilizers decrease side-to-side rolling motion, but have little or no effect on the up and down pitch, i.e. fore and aft, motion.
Princess Cruise Line
Automatic stabilizers operated by gyroscopic control were incorporated in the design of large passenger ships following their introduction by the shipbuilding firm of Denny-Brown (UK), during the 1930s.
Gyro-operated stabilizers in large ships are retractable into compartments inside the hull below the waterline and are thus stowed while the ship is in narrow water, in port, or when the sea conditions are calm.
A stabilizer has the form of a pivoted fin or horizontal rudder like those used for effecting fore and aft trim in a submarine.
As the vessel begins to roll and thus deviate from the fixed plane of the gyroscope, the stabilizing mechanism comes into play and the angle of the fin is made to vary against the ship's tendency to roll.
18.3.6.0 Rotational stability, dynamic stability, Magnus effect
See diagram 18.3.6: Magnus effect.
When a rapid spinning sphere or cylinder is moving through a fluid in a direction at an angle to its axis of spin, it experiences a sideways force at right angles both to the direction of motion and the axis of spin.
A student on top of the Hoover Dam dropped a basketball and it fell almost vertically to the ground.
However, when the basketball spinning inwards was dropped the basketball at first fell vertically, but as it gained speed downwards it moved away from the dam wall.
The axis of spin was from one student's hand to the other.
The direction of motion was down.
The sideways force was away from the Hoover dam.
When a tennis ball is hit with topspin, the axis of spin is at right angles to the direction of movement of the ball, the ball is rotating forwards, the sideways force is down, i.e. in addition to the downwards force of gravity on the ball.
18.3.6.1 Humming top, tipped top, tippy top
Pump up a toy humming top.
Spin a tipped top on smoked glass to show the path of the stem.
The tipped top spins in the opposite of the expected direction when inverted.
18.3.6.2 Yo-yo, Chinese diabolo
See diagram: 15.0.4.0: Rigid body motion.
Throw with a string to show rigid body rotational motion.
18.3.6.3 Spinning coin
"Euler's Disk", conservation of rotational momentum (toy product)
Euler's disc demonstrates a spinning disk on a flat surface, e.g. a spinning coin.
Wobbling by coins, bottles, and plates when they are spun on horizontal flat surfaces.
18.3.6.4 Football spin, spinning football, spinning lariat (lasso)
Spin a rugby or gridiron football on its side and it rises onto its pointed end.
Put an iron slug in the shape of a football on a magnetic stirrer.
Use a hand drill held vertically to rotate loops of rope or flexible chain.
18.3.6.5 Tossing the book, tossing the board, tossing the hammer
Throw a book or bread board with 3 different dimensions up in the air and spin it about its three principle axes.
Measure the moments of inertia about the three axes before tossing the book.
The hammer flip as an example of a centrifugal force in a rotating reference frame.
18.3.6.6 Static balance, dynamic balance
Dynamic tyre balancing for motor cars.
18.3.6.7 Spinning glass of water, tides simulation
1. Put a drinking glass of water on the centre of a hand-rotated circular table, e.g. "Lazy Susan", round table at centre of Chinese restaurants.
Spin the table and observe that the water level in the glass does not change.
Move the glass away from the centre of rotation and spin the table again.
The water in the glass heaves up against the side of the glass away from the centre in a parabolic shape, i.e. on a line joining the centre of the drinking glass to the axis of rotation.
2. Extend a line passing through the axis of rotation and the centre of the glass to the edge of the table opposite to the drinking glass.
Near this edge place a round ball of modelling clay (Plasticine) to represent the Moon.
Rotate the table and note that the ball and heaped up water stay on the same line.
When the round table is rotating with constant velocity the heaped up water just stays where it is and does not move to the left or right or heap up more or less.
The heaped up water represents a high tide on the opposite side of the Earth to the Moon.
So there are two simultaneous high tides twelve hours apart.
The high tide nearest the Moon caused by the gravitational attraction between the Moon and the ocean and the high tide on the opposite side of the Earth to the Moon.
The Earth rotates in the direction as the Moon's orbit around the Earth.
High tide is at 40 minutes after the Moon reaches the highest point in the sky, i.e. about 10o west of the meridian, the high tide bulge is dragged ahead by friction between the water bulge and the Earth.