School Science Lessons
18. Rotational motion, rigid body motion, angular measures, torque, moment
of inertia, precession, gyroscope, centrifuge
2012-05-04 SPwP
Please send comments to: J.Elfick@uq.edu.au Table of contents 18.0.0 Rotational motion
18.3.7 Central forces, 1. Loop the loop, 2. Centrifuge 3. Watts governor 18.3.4 Conservation of angular momentum 18.3.5 Gyros, gyroscope, precession 18.3.1 Moment of inertia, angular momentum, conservation of angular momentum 18.3.0 Rotational dynamics, rotational motion 18.3.2 Rotational energy 18.3.6 Rotational stability, dynamic stability 18.3.3 Transfer of angular momentum 18.3.4 Conservation of 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 18.3.4.11 Sewer pipe pull 18.3.4.5 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
18.3.5.0 Gyroscope, precession 18.3.5.1 Cardboard boomerang 4.101 Ear structure and function, See
4. Semicircular canals See pdf:
Flip Over Top, "Tippe Top", gyroscopic forces 18.3.5.5 Gyro pendulum 18.3.5.4 Gyrocompass, gimbal mount 18.3.5.3 Gyroscope, bicycle wheel gyro, gyro in gimbals, air bearing gyro 36.23 Obliquity of the ecliptic, precession and nutation, (astronomy) 36.42.1 Precession of the equinoxes, (astronomy) 18.3.5.2 Precession, spinning top, precessing ball, precession of the equinoxes 18.3.5.6 Ships' stabilizers 18.3.1 Moment of inertia, angular momentum, conservation of angular momentum 18.3.1.1 Inertia wands 18.3.1.4 Rigid and non-rigid rotations, parallel axis wheels 18.3.1.3 Rolling bodies on incline 18.3.1.2.1 Spinning eggs, forces with a fresh egg and hard-boiled egg 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 machine, 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 down an incline, rolling spool 18.3.6 Rotational stability, dynamic stability
18.3.6 Rotational stability, dynamic stability 18.3.6.4 Football spin, spinning football, spinning lariat (lasso) 18.3.6.1 Humming top, tipped top, tippy top 18.3.6.3 Spinning coin 18.3.6.6 Static balance, dynamic balance 18.3.6.7 Tides simulation, spinning glass of water 18.3.6.5 Tossing the book, tossing the board, tossing the hammer 18.3.6.2 Yo-yo, Chinese diabolo 18.3.3 Transfer of angular momentum
18.3.3 Transfer of angular momentum 18.3.3.1 Passing the wheel, pass bags of rice, catch the ball on the stool 18.3.3.2 Satellite derotator 15.3.0.5 Spinning ice skater, angular momentum 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
= θ / t 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 π / T 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.
Linear equations of motion
Rotational equation of motion
v = u + at
ωf
= ωi + α t
s = (u + v) t / 2
θ = ( ωf
+ ωi) /2 × t
s = ut + at2 / 2
θ = ωit
+ 1/2αt2
v2 = u2 + 2as
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
Twirl 2 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 torsional pendulum. 18.3.1.2.1 Spinning eggs,
forces with a fresh egg and hard-boiled egg
1. Use your fingers to spin a fresh egg and a hard-boiled egg end-on. The
hard-boiled egg spins longer because the inertia of the fluid contents of
the fresh egg brings it to rest sooner.
2. Use your fingers to spin a fresh egg and a hard-boiled egg end-on, i.e.
about its short axis. The hard-boiled egg spins for a long time but the raw
egg soon slows because of the inertia of liquids and friction within the
egg. The spinning raw egg damps more quickly than a boiled egg due to internal
friction. However, if a spinning hard-boiled egg is touched it stops spinning,
but if a spinning raw egg is touched it stops spinning then weakly continues
to spin when the finger is removed,. The raw egg starts spinning again because
internal liquids are still moving when the outer shell has been stopped by
the finger.
Centripetal forces are different with spinning fresh and hard-boiled eggs. 18.3.1.3 Roll down an incline,
bearing Inertia, racing discs, weary roller, rolling bodies on incline, rolling
cylinder, rolling hoop, rolling ball, restored force in a rolling can
Roll a ring and sphere of the same diameter down an incline. Roll a set
of discs and hoops of different diameters down an incline. Roll an empty
and full coffee can down an incline. Roll down an incline 2 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. Load a roller with fine dry sand or powdered tungsten
or iron then roll down an incline. 18.3.1.4 Rigid and non-rigid rotations, parallel
axis wheels
Spin with a falling weight 2 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. 18.3.2 Rotational energy
Other experiments: bike wheel angular acceleration, bike wheel on incline,
hinged stick and ball, penny drop stick 18.3.2.1 Adjustable angular momentum
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 bike 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 bike 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 bike 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.m2 / s. 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.m2 / s. kilogram
metre2 / second (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
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: Coin on wire coat hanger
| See diagram 15.2.2: Rotating objects 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
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 bike wheel while on a turntable or merry-go-round. Turn
yourself around on a turntable by variation of moment of inertia. 18.3.4.5 Skiing
Go skiing while holding a bike 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. 18.3.4.6 Toy train on a circular track
Use a clockwork HO gage train running on a track mounted on a bike 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. 18.3.5.0 Gyroscope, precession See diagram 18.3.5.0: Gyroscope Order online: Gyroscope, conservation
of angular momentum, precession
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.
Angular velocity of precession from formula L = Ι ω Ω,
where L = torque about the axis of torque, Ι (upper case iota) = moment of inertia about the axis
of spin, ω (lower
case omega) = angular velocity of
spin, Ω (upper case omega) = angular velocity
of precession
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. 1. 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.
2. 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.
3. 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. 18.3.5.1 Cardboard boomerang 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. Draw a boomerang
on the cardboard in the shape. Cut it off. Hold the centre of the boomerang.
Bend two wings gently to make wings slightly turn upward. Hold the edge in
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. 18.3.5.2 Precession, spinning top, precessing
ball, precession of the equinoxes
Behaviour of a spinning top with a round end spinning on a surface with
friction. The regular motion of the inclined axis of the spinning top around
the vertical is an example of precession. Spin a cardboard on a pencil
inserted in a hole at the centre and touch a finger to the rim to cause
it to precess. Spin a phonograph record or aluminium disc on a nail at the
end of a wood dowel and predict which way the record will turn when touched
with a finger. Put a ball on a rotating table to precess about the vertical
axis with a period 7 / 2 of the table. A rubber band provides a torque to
a gyro framework hanging from a string causing precession of the equinoxes.
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 bike 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 bike 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. 18.3.5.4 Gyrocompass, gimbal mount
A gimbal 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 and without
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.
However, 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 Rotational stability, dynamic stability
Other experiments: spinning rings, spinning stone bounce on water, spinning
Earth, spinning top, bullet, satellite, sports ball, Magnus effect, stable
and unstable axes of rotation, spinning rod and hoop on wire, static / dynamic
balance 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 Order online: Euler's Disk, conservation
of rotational momentum.
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
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.
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. because the high tide bulge is dragged ahead by friction
between the water bulge and the Earth.