School Science Lessons
Physics - Motion in two dimensions, circular motion, SHM, pendulum, relative motion,
Updated: 2008-02-20 L
Please send comments to: J.Elfick@uq.edu.au
See also: Interesting websites
Table of contents
15.0.0 Vibration and circular motion
15.1.0 Oscillations, pendulum
15.2.0 Centripetal force and centrifugal motion
15.3.0 Angular acceleration
15.4.0 Motion in two dimensions
15.4.1 Displacement in two dimensions
15.4.2 Velocity, position, and acceleration
15.4.3 Motion of the centre of mass
15.5.0 Central forces
15.6.0 Centrifugal escape
15.7.0 Projectile motion
15.8.0 Relative motion, Coriolis effect, Coriolis force, Foucault pendulum
4.155.0 Inertia
18.3.0 Rotational dynamics
Spinning top (primary)
15.0.0 Vibration and circular motion
15.0.1 Circular motion velocity, circular path of object, frequency of revolution
15.0.1.1 Circle from straight lines
15.0.2 Oscillation or cycle
15.0.3 Frequency
15.0.4 Angular velocity, rigid body movement about axis, radian, rad, rad / sec
15.0.4.1 Axis of rotation, spinning cross, boomerang
15.0.5 Centripetal acceleration
15.0.6 Centripetal force
15.1.0 Oscillations, pendulum
15.1.1 Simple pendulum
15.1.1.1 Pendulum tells the time
15.1.2 Simple pendulum, find and prove the relationship of T to L
15.1.3 Coupled pendulums
15.1.29 Galileo pendulum, stopped pendulum, conservation of mechanical energy
4.147 Ball bearings fall together
4.149 Simple pendulum
4.150 Coupled pendulums
6.22 Pendulum tells the time (Primary)
15.2.0 Centripetal force and centrifugal motion
4.158 Centripetal force with a liquid
4.159 Centripetal forces with a fresh and hard-boiled egg
4.160 Centripetal force with a water bucket
4.161 Measure centripetal force
4.79 Model of the solar system
15.2.1 Coin on a coat hanger, centripetal force, revolving coin, sticky penny, "centrifugal force", clothes dryer
15.2.2 Centripetal force, frequency and radius of uniform circular motion
15.2.3 Forces with rotating liquid
15.2.4 Centrifugal force, clothes dryer, swinging a bucket
15.2.5 Whirling bucket of water, pail of water
15.2.6 Ball on a string
15.2.7 Whirligig
15.2.8 Carnival ride model, "wall of death"
15.2.9 Swinging up a weight
15.2.10 Aircraft on string
15.2.11 Conical pendulum
15.2.12 Balls on a propeller
15.2.13 Welch centipetal force, spring stretching
15.2.14 Hand rotator
15.2.15 Banked track
15.2.16 Ball in a megaphone
15.2.17 Rolling chain
15.2.18 Raise a ball without touching
15.2 19 Lifting with spin
15.2 20 Turning the corner
15.2.21 Candle flame on a turntable
15.3.0.0 Angular acceleration, alpha, tangential acceleration, at, and centripetal acceleration, ac
15.3.0.0 Angular acceleration (tangential acceleration and centripetal acceleration)
15.3.0.1 Rotational kinetic energy, 1 / 2I X omega2, and moment of inertia, I, kg.m2, kilogram metre2
15.3.0.2 Rotational and translational kinetic energy of the earth
15.3.0.3 Sliding and rolling
15.3.0.4 Torque and angular acceleration, line of action of a force, centre of gravity
15.3.0.5 Angular momentum, conservation of angular momentum,  ice skater
15.4.0 Motion in two dimensions
15.4.1.2 Ball in a tube
15.4.1.3 Cycloid generator
15.4.1.4 Spots on a globe
15.4.2 Velocity, position, and acceleration
15.4.2.2 Kicking a moving ball
15.4.2.3 High road low road
15.4.2.4 Passing the train
15.4.2.5 Galileo's circle
15.4.2.6 Brachistochrone track
15.4.2.7 Sliding weights on triangle
15.4.3 Motion of the centre of mass
15.4.3.1 Lifting
15.4.3.2 Throwing
15.4.3.3 Spinning, dry ice puck
15.4.3.4 Centre of mass disc
15.4.3.5 Air table centre of mass
15.4.3.6 Earth moon system
15.5.0 Central forces
15.5.1 Flattening earth
15.5.2 Empty jug by swirling
15.5.3 Water parabola
15.5.4 Rotating manometer
15.5.5 Balls in water centrifuge
15.5.6 Inertial pressure gradient
15.5.7 Observations in a turning motor car
15.5.8 Rotate a doll
15.5.9 Rotate a candle
9.124 Geotropism, clinostat
15.5.11 Paper saw
15.5.12 Rotating rubber wheel
15.6.0 Centrifugal escape
15.6.01 Disc circle with gap, broken ring
15.6.02 Crossing the river
15.6.03 Release ball on a string
15.6.04 Grinding wheel
15.6.05 Spinning disc
15.7.0 Projectile motion, projectiles (horizontal and angular projection)
15.7.1 Gun and tunnel
15.7.2 Simultaneous fall
15.7.3 Midair target
15.7.4 Range of a gun
15.7.5 Parabolic trajectory
14.2.13 Path of a projectile
14.2.14 Projectile paths
14.2.15 Parabolic trajectory of thrown chalk
15.8.0 Relative motion
15.8.1 Moving reference frames
15.8.1.1 Crossing the river
15.8.2 Rotating reference frames
4.146 Coriolis effect and Coriolis force
4.147 Plug hole experiments
4.148 Weather rotations
15.8.3.5 Coriolis force as an inertia force
15.8.3.6 Chalk line on a turntable
15.8.3.7 Rolling ball on turntable
15.8.3.8 Turning sheet
15.8.3.9 Walk on turntable
15.8.3.10 String through spinning globe
15.8.3.11 Deflected water stream
7.36 Foucault pendulum
15.8.3.12 Use Foucault pendulum
15.0.0 Vibration and circular motion
15.0.1 Circular motion velocity, circular velocity, v, uniform circular motion in a horizontal plane ac = v2 / r = 4 pi2r / T2
Circular motion velocity is always at a tangent to the circle. For circular motion at constant speed, the centripetal ("centre seeking") acceleration vector always points towards the centre of the circle. The centripetal force keeps the object moving in the circle and the direction of the force is towards the centre of the circle. Centripetal force comes from (a) gravitational force, for planets in orbit, the sun and satellites about the earth (b) tension in a string, for an object whirled in a horizontal circle (c) electrical force, for electrons orbiting a nucleus (d) friction, for a car turning, and from (e) air pressure, for an aircraft turning in a horizontal circle.
15.0.1.1 Circle from straight lines
Make a cardboard rectangle 2 cm X 10 cm. Put the rectangle on a piece of paper on the desk. Push a drawing pin through the centre of the cardboard rectangle. Move the cardboard from place to place drawing a line along the end of the ruler at each place. Where the lines cross draw a dot. Join the dots to draw a circle
15.0.2 Amplitude, A and period, T
An oscillation or cycle is one complete to-and-fro movement of a vibrating object. The maximum displacement of the vibrating object from its rest or equilibrium position is called the amplitude of the oscillation, with symbol A. For any particular vibration, the time for one complete oscillation is called its period, with symbol T.
15.0.3 Frequency, f
The number of oscillations or cycles in one second is called its frequency, f. The relationship between the period and frequency of an oscillating object: T = 1 / f. An oscillation may be very complicated, including more than one frequency. So the oscillation of interest may comprise other oscillations. If the oscillation can be described by a harmonic function with a single frequency, it is called simple harmonic oscillation. An object moving at a constant speed in a circular path is undergoing uniform circular motion. Calculate the speed or linear velocity of the object by dividing the length of the object's path by the time taken to complete the path. For a complete circle the time taken is called the period, T. Linear velocity (tangential velocity), v = 2 X pi X r / T, where r = radius of the circle and T is the time period for one revolution with unit ms-1 and is always at a tangent to the circle.
15.0.4 Angular velocity, angular measure, axis of rotation
See diagram 15.04: Rigid body motion
Circular motion can be described using angles. An object moving with circular motion has an angular velocity, w or omega = change in angular position of the object / time. Angular velocity, w = angle omega in radians / t, where omega is the angular displacement measured in radians, with unit rads-1. There are 2 X pi radians in one complete revolution, so 360o = 2 X pi rad, so 1o = 0.01745 rad and 1 rad = 57.30o. Angular velocity is commonly measures as revolution per minute, rpm. 1 rpm = 0.105 rad / sec. The angular velocity of an object moving with uniform circular motion is the ratio of its linear speed and the radius of its path, so v = omega r, where omega is expressed in radians. A 12 inch (12 X 2.54 cm = 30.48 cm) Record turntable, radius of revolution, gramophone record rotates at 331 / 3 rpm = 3.50 rad / sec.
The earth spins once in 24 hours, 86,400 seconds, so angular velocity = 4.16 X 103 degrees per second, i.e. 7 X 10-5 radians per second.
15.0.4.1 Axis of rotation
The axis of rotation of an object is a line of particles, or imaginary particles, that do not move. All other particles in the body move in a circle, V = omega, so the further particles are from the axis of rotation the faster they move, although they all have the same angular velocity. A point on the rim of a gramophone record rotating at 331 / 3 rpm moves with velocity, v= omega X r = 3.50 rad / sec X 0.5 X 30.48 = 53.34 cm / sec.
15.0.5 Centripetal acceleration, ac
Since an object undergoing uniform circular motion is constantly changing direction, you say that it is accelerating. For circular motion the direction of acceleration is towards the axis of rotation, i.e. centre of the circle. This acceleration is called centripetal acceleration, ac= v2r / T, replace 2pr / T for v, ac = 4 X pi2X r / T2.
15.0.6 Centripetal force, Fc
Newton's First Law states that an object at rest or at constant velocity will remain in that state unless acted upon by an external force. An object in circular motion, at constant speed v, must have a force acting on it to keep it in a circular path. This centripetal force can be expressed as Fc = mv2 / r, = m4 X pi2X r / T2, which is a form of Newton's Second Law, F = ma. When you observe an object undergoing uniform circular motion from a position either above or below the object's path you see the circular path. However, when the object is observed from a position in the same plane as the circular path, you see the object moving back and forth in an oscillatory motion that corresponds with Simple Harmonic Motion.
15.1.0 Pendulum, oscillations, physical pendula
A simple pendulum is a mechanical system that displays oscillatory motion. It consists of a mass, called a bob, suspended by a light cord that does not stretch. For angles less than 20o, the period of oscillation is dependent only on the length of the cord. T = 2p(sqrt)l / g, where l is the length of the pendulum and g is gravitational acceleration. The period is independent of the mass of the bob. Pendulums are used as timekeepers and to measure the acceleration due to gravity.
15.1.1 Simple pendulum
See diagram:15.1.1 Pendulum calculation
Period = 1 / frequency, Hertz, Hz, or cycles per second, cps. The period, T, of a simple pendulum depends on its length, l, and the acceleration of a falling object, g. T = 2 x pi x sqrt (l / g). The period, T, is not affected by the angular amplitude, or the mass of the pendulum bob, m. The simple pendulum does not display simple harmonic motion precisely and therefore is dependent on the angular amplitude. However, for angles less than 20o, the dependence on angular amplitude can be ignored.
1. Tie a two metre long string to a heavy object. Swing the pendulum bob through a maximum angle of 20o between the pendulum and a vertical line at the bottom of the swing. Count the number of swings per minute. Swing the pendulum bob through a smaller arc. Count the number of swings per minute. Note that the length of the arc does not affect the time of vibration of a pendulum. Keep the length of the pendulum the same but change the weight of the bob. Count the number of swings per minute. Note that the change in weight does not affect the time of oscillation of the pendulum. Make the pendulum one metre long. Count the number of swings per minute. The number of swings per minute increases. The length of the pendulum does affect the time of oscillation of the pendulum.
2. To verify the relationship between period of a simple pendulum and (a) angle of the swing (b) mass of the pendulum bob, and (c) length of the pendulum. The period is the time for a complete cycle of motion, i.e. forwards then back to where it started from. The angle of the swing is the angle between its farthest position from the vertical and the vertical. The length is the distance from the centre of the spherical bob and the point of suspension.
3. Use about 2 metres of fishing line or thread. Tie one end to a heavy circular bob. Wind the other end around a metal cylinder. Clamp the cylinder horizontally. Turn the cylinder so that the length of the pendulum is exactly 1 metre. Draw back the bob 10o and set the pendulum in motion. Time how long it takes for the pendulum to complete 10 oscillations. Calculate and record the period. Pull the bob 20o and calculate the period. Let the pendulum swing in a smaller arc by drawing the bob back 5o and calculate the period. Observe that the period is the same in each case, thus period is not dependent on size of swing.
4. Exchange the bob with another of different mass and repeat the procedure. Observe that the period is the same. Thus period is not dependent on the mass of the bob. Construct a pendulum 2m long. Using a protractor, draw back the bob 15o, release the bob and calculate the period. Now reduce the length of the pendulum by 50 cm and repeat the procedure and calculate the period. Repeat this procedure reducing the length of the pendulum each time. Calculate 2 pi X sqrt l / g.
Mass of bob, m (kilograms, kg) Length of pendulum, l (metres, m) Angular amplitude

(angle of swing), theta (oC)

 No. cycles (complete swings) Time (for 10 cycles) (seconds, s) Average time

(Time taken / 10) (seconds, s)

 Period T (seconds, s) (calculate:

T = 2 pi sqrt l / g)
1 kg 1 metre 15oC 10 - - -
1.5 kg * 1 m 15oC 10 - - -
1 kg 1.m 10oC* 10 - - -
15.1.2 Simple pendulum, find and prove the relationship of T to L
To make a simple pendulum, hang a weight from a soft cord more than 2 m length. Draw back the object 15o then let it swing in a vertical plane. Record the time for a complete swing, forwards then back. Let the pendulum swing in a smaller arc and record the time for a complete swing. Compare the times of the two angles of swing. The length of the arc does not affect the time of the swing. Change the weight of the object and let the pendulum swing again with the same angles and length of cord. Record the times of swing. The weight of the object does not affect the time of swing. Change the length of the cord. Repeat the experiment with the same angles of swing and the same weights. The length of the pendulum affects the time of the swinging. Make a form using the data. List the length of the pendulum l in turn according to the length at the first line. List the corresponding vibrating period T (T = 60 / the time of swinging per minute) at the second line. You will find out the relationship between vibrating period T and the length of the pendulum, L.
15.1.3 Coupled pendulums
See diagram 15.1.3
1. Fill two same size bottles with water, stopper and suspend the bottles with string as pendulums from a rod. The strings must be the same length. Hold one pendulum still then start the other swinging, then release the first one to hang at its zero point. Soon the swinging pendulum slows, and the other pendulum takes up the swing. Hang the pendulums from a stationary support such as a doorway but join the pendulum cords with a third string tied between them about one eighth of the way down the cords.
2. To observe the energy transfer of a simple pendulum use two full identical bottles; several pieces of string; a stick; two chairs. Fill the two bottles with water or sand. Put the chairs back to back. Lay the stick across the backs of the chairs. Cut two pieces of string with the same length. Tie the bottlenecks with string then hang the bottles from the stick. When the bottles are at rest, hold the first bottle and draw back the second bottle. Let the second bottle swing in the plane upright to the stick. Let the first bottle go too then observe the movement of the two bottles. The amplitude of the second bottle will decrease gradually. The first bottle will gradually swing from rest and its amplitude will increase gradually. Finally the two bottles will swing with the same amplitude and velocity.
15.1.29 Galileo pendulum, stopped pendulum, conservation of mechanical energy
See diagram 15.1.20
1. Suspend a 5 cm diameter wooden ball by a 100 cm fine thread from a nail in the wall or from a nail above the blackboard. Draw a horizontal line CD 50 cm above the wooden ball such that the wooden ball can swing from C to D. Release the wooden ball at C so that it swings through the arc CBD to reach D. The momentum at B carries the wooden ball through the arc BD to the horizontal CD. In fact the ball does not quite reach the horizontal CD because some energy is lost due to resistance of the air and the string.
2. Place a peg as a stop at some position E between the nail and CD. Release the wooden ball at C. The ball again travels through arc CB, hits the peg E then travels though arc E E1 reaching the same height as CD.
3. Replace peg E with peg F between E and level CD. Release the wooden ball at C. The ball again travels through arc CB, hits peg F then travels through arc FF1 reaching the same height as CD.
4. Replace peg F with peg G such that the remainder of the thread below G cannot reach level CD. Release the wooden ball at C. The thread leaps over peg G and twists around it.
15.2.0 Centripetal force and centrifugal motion
Even though an object is travelling in uniform circular motion at a constant speed, it is still accelerating. The direction of the velocity changes, so there is an acceleration, called centripetal acceleration. Centripetal force keeps an object in a circular path and changes the direction of the velocity towards the centre of the circle. Centripetal force, F = mv2 / r, m = mass of the object, r = radius of the circle, v = linear, i.e. tangential, velocity. The centripetal force keeping the earth in its orbit around the sun is the gravitational force. The centripetal force keeping an object sitting on a rotating turntable is friction. If you stand on a rotating platform you feel as if you are being forced in a direction away from the centre of the circle. This is called "centrifugal force" but it is the result of inertia. Your body would like to travel in a straight line according to Newton's First Law. As the platform floor accelerates towards the centre and you try to continue on your straight line motion, it feels as if a force pushes you to the outside. However, there is no force. Centrifugal force is a "fictitious" force that is observed from within accelerating frames of reference. Of course as centripetal force disappears, the object will move along a path that is tangent of the circle at the point where the centripetal force vanished. A simple pendulum consists of a cord whose mass can be ignored and a ball whose volume is very small called a "bob". When the swinging angle is less than 5o, the simple pendulum moves with a constant period, T. T is not affected by the mass of the bob. T = 2pi X sqrt l / g) where l = length of the cord and g = acceleration of gravity.
15.2.1 Coin on a coat hanger, revolving coin
See diagram 15.2.1
Balance a coin on the hook of a coat hanger. Twirl the coat hanger around your finger and the coin does not fly off. Pull the middle of the longest straight part of a wire coat hanger out and shape it into a long narrow shape. Use a file to flatten the end of the hook. Let the hanger hang on your index finger and place a coin on the filed end of the wire. Start to swing the hanger. Swing slowly at first back and forth, and then do full loops. Swing it slower when ending the swing and try to catch the coin when it falls off the end of the hanger. The spinning of the penny gives it a "centrifugal force" holding the coin against the end of the hanger. The faster the spinning, the larger this centrifugal force; the slower the spinning, the smaller the force. The force keeps the coin up against the hanger until the circular motion is too slow. Centrifugal forces are applied in the automatic cloth washer, where the drum filled with clothes and water spins fast to separate the water from the clothes. The tendency to stay moving in a straight line causes this force of inertia or centrifugal force. [This experiment may need some practice before demonstrating to a class].
15.2.2 Centripetal force, frequency and radius of uniform circular motion
See diagram 15.2.2
Research the relationship among the centripetal force, frequency and radius of uniform circular motion. Use a 15 cm length of glass, inner diameter 1 cm. Burn its end over Bunsen burner into round and smooth. Twine two layers plastic adhesive tape or list on the surface of the glass to prevent the glass slip from your hand. Use a piece of nylon fish line. Use one end of the fishing line to tie a rubber stopper with 2 holes. Let the fishing line go through the two holes and make a tie at the centre of the stopper. Let the other end of the fishing line go through the glass then hang 6 iron gaskets, outer diameter 1 cm or 2 nuts, M10, and finally tie the line in a knot. Let a clip go through the last tie to prevent the gaskets slipping away. Adjust the length of the line, from the rubber stopper to the top of the glass, to 1 m. Now hold the glass and on its top swing the rubber stopper in a level circle. Here centripetal force keeping the stopper moving in a circular motion is the line's strain. The force is caused by the weight of gaskets or nuts. You may need to practice several times before demonstrating this experiment. Record the amount of the original gaskets and the frequency of the uniform rotation. Change the length of the level line so that the rotating radius goes up 0.5 times of the original size at first then down 0.5 times. Differently count the time of the rotations per minute then exchange into frequency and record. You will find how the centripetal force affects the frequency of the rotation.
15.2.3 Forces with a rotating liquid, forces with a fish bowl
See diagram 15.2.3
Observe liquid's circular motion and related phenomena. Use a small plastic goldfish bowl or the lower part of a plastic drink bottle with 2 holes drilled each side of the rim. Fasten a 1 m piece of string to the rim then hang it on the end of a spoon. Fill it to 1 / 3 with water containing tea and tea leaves. Hold the jar to make it immovable and twist the spoon in a level plane to make the strings twist together. Raise the spoon to raise the bowl above the ground or table then take your hand off the jar. Observe the rotation of the bowl and the rotation of the tea and the tea leave in the bowl. The tea rises close to the side of the bowl. Do the experiment again after the water is rest completely. This time mostly observe the water's movement when the jar just begins to rotate and just stops. Do the experiment once again. When the jar rotates fastest brake it with your hands. Observe the water's movement this time.
15.2.3.1 Swinging a bucket
Half fill a plastic bucket with water. Hold it by the handle and swing the bucket in a vertical circle. If you swing fast enough the water does not spill out because the water is pressing against the bottom of the bucket. Keep swing and then let go the handle. The bucket and water moves at a tangent to the direction of spin. This action is the same as throwing a stone with a sling shot to throwing a lasso to land over the head of a runaway cow. A pilot, making a sharp turn feels himself pressed against the front of the plane's cockpit. In some fairgrounds people are spun around in a "Wall of death". The floor drops down but the people remain pressed against the spinning wall. A centrifuge spins a test-tube very high speed to separate substances according to their relative density. A milk separator is used to separate the milk from the cream. Similarly water can be separated from fuel.
15.2.4 Centrifugal force, clothes dryer, swinging a bucket
See diagram 15.2.4
Clothes dryer: Use an empty container such as a can without lid or a part of drink bottle being cut from middle of it, use the one which has bottom. Punch several holes around the container, then punch three holes equidistant from each other around the top of the container to suspend it with thread or spring band, see the diagram. Place wet clothes in the container and make sure that the whole container is in a state of even mass distribution. Spin the container slowly in one direction to make thread gradually turn around. Remove your hand suddenly, the container can spin rapidly and you will find that water is thrown from the container. As the container spin rapidly, the adhesive force between water drop and clothes is smaller than centripetal force that water drop needed to move in a circular line. Then water drop leave the clothes to do centrifugal motion. As a result, it is thrown out from the clothes. The household dryers are made according to the principle. Note what relationships between the speed of spinning of the container and the amount of the water. to strengthen the effect of the experiment, you may add more water.
15.2.5 Whirling bucket of water, pail of water
Swing a bucket of water in a vertical circle over your head. Swing a bucket of nails in a tin bucket. The nails stay in the bucket but you can hear the bottom layers of nails dropping away from the bottom of the bucket
15.2.6 Ball on a string
Tie a lightweight ball to a sting and twirl around in a vertical circle. Use a glass tube for the holder and rubber stoppers for the masses and twirl the masses around your head. Arrow on a disk: Mount an arrow tangentially on the edge of a rotating disk.
15.2.7 Whirligig
Attach 1 kg and 100 g ball to the ends of a 1 m string passing through a hollow tube. Twirl one ball then the other ball around your head.
15.2.8 Carnival ride model, "wall of death"
Toy person stands on the inside wall of a rotating cylinder.
15.2.9 Swinging up a weight
A swinging weight picks up a much heavier weight.
15.2.10 Aircraft on string
A model plane flies around on a string defining a conical pendulum.
15.2.11 Conical pendulum
Use a ceiling mounted bowling ball pendulum as a conical pendulum. The main bearing of a conical pendulum is from a bicycle front wheel axle where the string tension is set by a counterweight.
15.2.12 Balls on a propeller
Balls sit in cups mounted on a swinging arm at 0.5 and 1.0 m Calculate the period necessary to keep the ball in the outer cup and swing it around in time to a metronome. Ride on a twirling device in an amusement park.
15.2.13 Welch centipetal force, spring stretching:
Compare the angular velocity with the mass needed to stretch a spring a certain distance.
15.2.14 Hand rotator:
Mount two 2000 g spring balances on a rotator. Attach equal masses to each spring balance and read them at some rotational velocity.
15.2.15 Banked track
Roll a steel ball down an incline into a funnel so that it reaches an equilibrium level where it revolves in a horizontal plane.
15.2.16 Ball in a megaphone
Throw a ball into a megaphone and it turns around and comes out the wide end.
15.2.17 Rolling chain:
Spin a loop of chain on and released to maintain rigidity and roll down the bench as a rigid hoop. The spun flexible chain acts as a solid object
15.2.18 Raise a ball without touching
Put a marble or a small ball made of Plasticine (modelling clay), and an empty jar on a table. Ask a friend to raise the ball without touching it. Place the jar over the marble or ball. Twist the jar very quickly so the ball is running around the entrance to the jar. If you lift the jar the ball also rises while still spinning around the inside of the jar.
15.2.19 Lifting with spin
Make two holes in each side of a plastic pot then join the holes with a string to make a handle. Put marbles or stones in the plastic pot. Tie one end of a 50 cm string to a weight. Thread the other end through a ball pen case then tie it to the centre of the handle. Twist the ball pen case so that the weight is spinning very fast. Raise the pen case slowly. The spinning weight will raise the pot of stones.
15.2.20 Turning the corner
Put a coin in the centre of a smooth wooden tray. Move the tray forwards sharply. The coin stays in the same place or moves forwards depending on the friction between the coin and the tray. Move the tray forwards and to the right sharply. The coin still moves forwards when the tray moves to the right and may hit the front of the tray. The coin is like people in a motor car without seat belts when the car swerves sharply to the right to avoid hitting something.
15.2.21 Candle flame on a turntable
Mount a candle on a turntable, e.g. a record player. Light the candle. Put a large tall jar over the candle with space under the jar to allow air to come it and keep the candle alight. Turn on the turntable. The candle flame points towards the centre of the turntable. Inside the jar the more dense air moves outwards because of centripetal force so the less dense gases of the candle flame move inwards. The centipetal force moves the more dense air more. The flame with less mass than the air around it is accelerated more by the centripetal force.
15.3.0.0 Angular acceleration, alpha, tangential acceleration, at, and centripetal acceleration, ac
See diagram 15.3.0
If angular velocity changes from omega 1 to omega 2, angular acceleration, alpha = (omega 2 - omega 1) / time, t
v = omega X r, so v1 = omega 1 X r and v2 = omega 2 X r, so alpha = (v2 / r - v1 / r) / t = (v2 - v1) / t X r
linear acceleration of a body tangent to its path, at = (v2- v1) / t, so alpha = at / r, and at= alpha X r, i.e. tangential acceleration = angular acceleration X radius centripetal acceleration ac = v2 / r, towards the centre of the circle, so at and ac are at right angles
15.3.0.1 Rotational kinetic energy, ½ X I X omega2, and moment of inertia, I, kg.m2, kilogram metre2
See diagram: 15.3.0.1 Moments of inertia of bodies about axes
Rotational kinetic energy
v = omega X r, so for particle mass m, KE = ½ X m X v2= ½ X m X omega2 X r2, so for all particles in an object (All particles in the object have the same angular velocity.) KE sum of 1 / 2mv2 = ½ X (sum of m X r2) X omega2
Moment of inertia of object, I = sum of m X r2 (Moment of inertia is a sort of rotational analogue of mass.)
KE of object, moment of inertia I, rotating with angular velocity omega, = ½ X I X omega2
In theory, for particles m1, m2 and m3 at radius r1, r2, r3 in an object, I = sum of m X r2 = m1 X r12 + m2 X r22+ m3 X r32.kg m2
Moment of inertia is difficult to calculate for irregular bodies, however, for regular bodies about axes, it can be easily calculated.
15.3.0.2 Rotational and translational kinetic energy of the earth
See diagram: 15.3.0.1 Moments of inertia of bodies about axes
Mass of the earth = 6 X 1024 kg. Radius of earth = 6.4 X 106 metres.
If earth is a sphere, I = 2 / 5MR2 = 2 / 5 X 6 X 1024X 6.4 X 106 = 9.8 X 1037 kg.m2
Angular velocity of earth =7.3 X 10-5 rad / sec (one rotation per day)
Rotational kinetic energy, KE (Earth turns around its axis.) = 1 / 2Iomega2 = ½ X 9.8 X 1037 X 7.3 X 10-5 = 2.6 X 1029j
Translational kinetic energy, KE (Earth revolves around the sun at average orbital speed + 3 X 104 metres / sec.) = ½ X m X v2= ½ X 6 X 1024 X 3 X 104= 2.7 X 1033j
So the kinetic energy of orbital motion is much greater than the kinetic energy of rotational motion.
If mass of the earth = 6 X 1024 kg, and radius of earth = 6.6 X 106 metres, moment of inertia = 0.4 X M X R2 = about 1 X 1038 kg m2.
15.3.0.3 Sliding and rolling
A cylinder mass m and radius R is at the top of an inclined plane, PE = mgh
1. The cylinder slides down the slope with no friction.
It travels with velocity v at end of slope. PE = KE, mgh = 1 / 2mv2(translational kinetic energy at the bottom), so v = sqrt 2gh.
2. The cylinder rolls down the slope.
Moment of inertia of cylinder that rolls and does not slip = I = ½ X m X R2. Angular velocity omega = v / R, so omega2 = v2 / R2
PE = KE, mgh = 1 / 2mv2 (translational kinetic energy at the bottom) + ½ X I X omega2 (rotational kinetic energy at the bottom) = 1 / 2mv2 + ½(1 / 2mR2) (v2 / R2) = 1/2mv2 + 1/4mv2 = 3/4mv2
so v = sqrt 4/3 gh
3. At the bottom of the inclined plane the rolling cylinder moves slower than the sliding cylinder because some energy is absorbed by rotation.
15.3.0.4 Torque and angular acceleration, line of action of a force, centre of gravity
See diagram 15.3.0: Torque
A force f acts on an object mass m attached to the end of a string and moving in a circular path. F = ma, and angular acceleration, alpha = a /r, so F = m X r X alpha, and multiplying both sides of the equation by r: Fr = m X r2 X alpha. Fr = torque (tau) of the force F about the axis of the particle's rotation and its moment of inertia, I, = mr2. So torque = I X alpha (analogous to F = Ma). Angular acceleration, alpha is proportional to the torque and inversely proportional to the moment of inertia of the object. When a force acts on an object such that the line of action of the force passes through the centre of gravity of the object, the object will accelerate according to f = ma. However, if the line of action of the force is not through the centre of gravity, the object will have both linear and angular acceleration.
15.3.0.5 Angular momentum, conservation of angular momentum,  ice skater
Angular momentum, L = I X omega. When there is no net torque on an object its angular velocity and angular momentum are constant. Conservation of angular momentum: When the sum of torques acting on an object = 0, the angular momentum is constant. The greater the value of angular momentum, the more torque is needed to change its direction so bullets, rockets and even footballs can be aimed more accurate if spin stabilized. Similarly a spinning top stays upright until friction causes it to lose angular momentum. An ice skater can start a spin on one toe with one leg extended and both arms extended (I is large and omega is small) but when the ice skater brings both legs and both arms together (now I is small and omega is large) the skater spins much faster due to conservation of angular momentum.
Angular momentum of the earth = moment of inertia X angular velocity = (1 X 1038) X (7 X 10-5) = 7 X 1033 newton metre (Nm)
15.4.0 Motion in two dimensions
15.4.1.2 Ball in a tube
Attach a ball to a string at the bottom of a vertical tube and hold the string while moving the tube horizontally.
15.4.1.3 Cycloid generator
Roll a hoop with a piece of chalk fastened to the circumference along the chalk tray. Join large and small cylinders coaxially. A spot on the larger cylinder moves in a cycloid when the smaller cylinder is rolled on its circumference.
15.4.1.4 Spots on a globe
Spin an inclined globe with spots rotated in an orbit while not spinning, and both rotated and spun. The spots form parallel lines perpendicular to the angular velocity vectors.
15.4.2 Velocity, position, and acceleration
15.4.2.2 Kicking a moving ball
Kick a moving ball to score a goal!
15.4.2.3 High road low road
Race two balls, one down an incline the other down same slight incline that includes a valley.
15.4.2.4 Passing the train
Let a ball accelerate down an incline to pass another ball moving at constant velocity on a horizontal track or pass a striped rope moving at constant velocity in the background and note where the two balls, or the ball and striped rope, have the same instantaneous velocity.
15.4.2.5 Galileo's circle
Simultaneously release small balls to roll down guides that form chords of a large inclined circle to make a single click marking simultaneous arrival.
15.4.2.6 Brachistochrone track
See also 2.0.5: Conic sections, parabola | See also 2.0.6: Parabola equation
Release balls at any height on the brachistochrone track to reach the middle at the same time. Let balls roll down an incline brachistochrone and a parabola so that the ball on the brachistochrone wins. (A brachistochrone track follows a curve that joins two points such that a bead travelling along it under the influence of gravity takes a shorter time than along any other curve.) Release two balls on opposite sides of a cycloid to always meet in the middle. The ball on the cycloid always beats the ball on the incline. (A cycloid is a curve traced by a point on the circumference of a cycle as the circle rolls along a straight line. Oxford English Dictionary.) Note the use of brachistochrones and cycloids in amusement parks. Physics of "Amusement Parks" and associated phenomena loop the loop type of problems.
15.4.2.7 Sliding weights on triangle
Chose different lengths and angles of a wire frame triangle so that two beads sliding down the wires traverse each side in the same time.
15.4.3 Motion of the centre of mass
15.4.3.1 Lifting
Stick unequal size corks on each end of a knitting needle, place a cord under at the centre of mass and jerk it into the air
15.4.3.2 Throwing
1. Throw a light disc with attached heavy slug that you can move from the centre to side. Mark the centre of mass before you throw the disc.
2. Throw a slab of Styrofoam with lights placed at the centre of gravity and away from the centre of gravity.
3. Mark the centre of gravity of a hammer with a white spot. Throw it in the air. Attach it to a hand drill to show it rotating smoothly.
4. Tie together a bunch of junk and throw it across the room.
15.4.3.3 Spinning, dry ice puck
Put markers on a large block of wood at and away from the centre of mass. Place the block on a large sheet of paper and hit off centre with a hammer. Use a pool cue to hit a dumbbell double dry ice puck on or off the centre of mass. Use an Earth Moon system hanging from a string to show the earth's wobble. Rotate the Earth Moon system from a hand drill on and off the centre of gravity. Attach two unequal masses at the ends of a rigid bar, spin the system about holes drilled in the bar at and off the centre of mass.
15.4.3.4 Centre of mass disc
Use a pool cue to hit a dumbbell double dry ice puck on or off the centre of mass.
15.4.3.5 Air table centre of mass
Stick unequal size corks in knitting needle place, a cord under at the centre of mass and jerk it into the air
15.4.3.6 Earth moon system
Use an earth moon system hanging from a string is used to show the earth's wobble. Rotate the earth moon system from a hand drill on and off the centre of gravity. Attach two unequal masses at the ends of a rigid bar, spin the system about holes drilled in the bar at and off the centre of mass.
15.5.0 Central forces
15.5.1 Flattening earth
Spin a globe made of flexible brass hoops, or sponge rubber ball, until the hoops flattens and becomes oblate. Spin a deformable clay or glycerine ball until it flattens or bursts.
15.5.2 Empty jug by swirling
Empty a jug when contents are swirled then not swirled to show that the swirled jug empties faster.
15.5.3 Water parabola
See also: 3.8.0 Conic sections, parabola
Rotate a rectangular transparent box partially filled with coloured water until you see the parabolic shape. Spin a glass half filled with coloured water on a rotating table.
15.5.4 Rotating manometer
Rotate tubing constructed in an E-shape on its back that is partly filled with water. Note the level of water in the arms. Rotate a U- shaped manometer with one of its arms coincident with the axis of a rotating table.
15.5.5 Balls in water centrifuge
Spin cork and steel balls in a curved tube filled with water. Spin a glass bowl containing water, a steel ball and a cork. Spin a semicircular tube filled with water containing two corks. Spin one cork is tied to the bottom of a cylinder and one ball is tied to the top of a cylinder full of water at the ends of a rotating bar.
15.5.6 Inertial pressure gradient
Spin a tube containing a bubble in water until the bubble goes to the centre when whirled in a horizontal circle.
15.5.7 Observations in a turning motor car
Observe a candle, a carbon dioxide balloon and a hydrogen balloon on strings, in a motor car driven in uniform circular motion.
15.5.8 Rotate a doll
Rotate a doll dressed in a full skirt and a doll dressed in a short skirt.
15.5.9 Rotate a candle
Place a lighted candle in a chimney lamp on a rotating table until the flame points to the centre. Lighted candles in chimneys are rotated about the centre of mass.
15.5.10 Geotropism
9.124 Geotropism, clinostat
Grow corn or wheat on a rotating turntable, axis horizontal.
15.5.11 Paper saw
Spin typewriter paper at high speed to cut through tougher paper and cardboard.
15.5.12 Rotating rubber wheel
Rotate a rubber wheel so that the radius stretches.
15.6.0 Centrifugal Escape
15.6.01 Disc circle with gap, broken ring
Roll a ball around a circular hoop with a gap. Where will the ball go when it reaches the opening?
15.6.02 Crossing the river
Pull a long sheet of paper along the bench while a toy clock-work car crosses the paper. The toy car moves across a sheet moving at half the speed of the toy car. A small stick placed on the top tread of a toy caterpillar tractor moves twice as fast as the toy tractor.
15.6.03 Release ball on a string
Release the string while swinging a ball overhead. Be careful! Use a slingshot A David and Goliath type slingshot
15.6.04 Grinding wheel
Observe direction of sparks flying off a grinding wheel.
15.6.05 Spinning disc
Red drops fly off a spinning leaving traces tangent to the disc. Place erasers on a disc at various radii and rotate until they fly off. Falling off the merry-go-round: A turntable is rotated until objects slide or tip over. See amusement park horizontal turn table. Line up coins radially on a rotating platform and spin at varying rates until all falloff.
15.7.0 Projectile motion, projectiles (horizontal and angular projection)
15.7.1 Gun and tunnel
A spring loaded gun on a cart shoots a ball vertically and after the cart passes through a tunnel the ball lands in the barrel. A ball fired vertically from cart moving horizontally falls back into the barrel.
15.7.2 Simultaneous fall
Two balls simultaneously dropped and projected horizontally hit the floor together. Drop one billiard ball and shoot another out simultaneous. One ball is projected horizontally as another is dropped simultaneously. Instructor rolls a ball off the hand while walking at a constant velocity.
15.7.3 Midair target
A hunter shoots a compressed air at a target released when the gun is fired The ball hits the target in midair.
15.7.4 Range of a gun
1. Fire a spring loaded gun at various angles. Use a tennis ball serving machine to find muzzle velocity and range of a gun.
2. Projectile motion: Two angles result in the same launched distance. Select an angle, e.g. 45 degrees + 15 degrees. Fire the projectile and observe where it lands. Then set the angle to 45 degrees- 20 degrees. Fire the projectile and observe where it lands. It lands at the same place.
15.7.5 Parabolic trajectory
See also: 3.8.0 Conic sections, parabola
Throw a piece of chalk so it follows a parabolic path drawn on the chalk board. Roll ink dipped balls down an incline onto a tilted stage on an overhead projector. Roll a tennis ball covered with chalk dust across a tilted blackboard. A stream of water matches the position of balls of lengths 1, 4, 9, 16, at all angles of elevation.
15.8.0 Relative motion
15.8.1 Moving reference frames, relative velocity, independence of horizontal and vertical velocities
15.8.1.1 Crossing the river
Pull a long sheet of paper along the bench while a toy clock-work car crosses the paper. The toy car moves across a sheet moving at half the speed of the toy car. A small stick placed on the top tread of a toy caterpillar tractor moves twice as fast as the toy tractor.
15.8.2 Rotating reference frames
15.8.3 Coriolis Effect
15.8.3.5 Coriolis force as an inertia force
Observe the motion in a spinning frame of reference from a static frame of reference. Coriolis force is a kind of inertia force, i.e. caused by non-inertial system. For an observer who stand in a frame of reference, spinning at angular speed sa, the object that is moving relative to the frame will be acted by not only an inertial centrifugal force but also an inertial force that makes the object not to leave the motion orbit, i.e. Coriolis force. The direction of the Coriolis force depends on cross product of the speed vector V relative to the spinning frame and angular speed of the frame itself, always vertical to V and opposite to acceleration of tangent direction. The magnitude of the Coriolis force is 2mVsa. Due to the spinning of the earth, the moving object relative to earth is seen acted by a Coriolis force by an observer on earth. Many phenomena on the ground and in the air in nature are caused by it. For example, the cold air in high latitude of north, south sphere drops, due to the spinning of the earth, form a southeast and northeast wind in each sphere. Due to the air of the equator rises, the two winds can meet near the equator to form a typhoon in the region of south Pacific Ocean.
15.8.3.6 Chalk line on a turntable
The Coriolis force operates independently of the direction in which an object is travelling. However in demonstrations where you draw a line radially across a slowly spinning turntable it works only when the line is drawn towards or away from the centre.
In the demonstration a chalk line is drawn radially on a slowly spinning turntable. The track of the chalk is a straight line but the line drawn on the turntable is curved. So, in the inertial frame of the classroom, the track was straight, while in the rotating frame of the turntable the track is curved. It is easy to draw the line radially, but very hard to match the velocities of the turntable and chalk closely enough by hand to draw a similar line tangentially.
15.8.3.7 Rolling ball on turntable
Roll a ball rolls across a slowly rotating turntable to show the Coriolis effect.
15.8.3.8 Turning sheet
Turn a nearly vertical sheet as a drop of ink is running down it.
15.8.3.9 Walk on turntable
Walk in a straight line on a turntable or a merry-go-round and feel a very strange force.
15.8.3.10 String through spinning globe
Thread a ball on a string through the pole of a spinning globe, then pull on the string and the ball moves to higher latitudes and crosses the latitude lines spinning.
15.8.3.11 Deflected water stream
Mount a can with a hole above a rotating table so that as the table turns the stream of water is deflected. Stretch a flexible rubber tube with water flowing in across a turntable which can be rotated so that the tube deflects when rotated.
15.8.3.12 Use Foucault pendulum
Build a simple model of the Foucault pendulum and show the Coriolis effect by the curved trace method.
2.243 Centripetal force with a liquid in a plastic goldfish bowl
See diagram 2.243
Tie a wire securely around the neck of a plastic goldfish bowl. Attach a string to the wire. Clamp a hook in a hand drill chuck and attach it in the centre of the string. Put water coloured with ink in the bowl. Put a plastic "fish" in the bowl. Turn the drill handle to spin the bowl and water or whirl the bowl around you by hand. The water begins to climb up the sides of the bowl due to the centrifugal, not centripetal, force on the water. Observe the effects of inertia of the water when starting and stopping spinning the bowl. When starting spinning the water tends to remain stationary. When stopping spinning the water tends to continue spinning inside the bowl.
2.244 Forces with a hard boiled egg
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.
2.245 Centrifugal, not "centripetal", forces with a water bucket and the water in glass trick
Almost fill a small bucket with water. Swing it around rapidly at arm's length in a vertical circle. The water does not spill because centrifugal, not centripetal, force acts on the water to keep it in the bucket. b. Some people can do the water in glass trick by turning a glass of water quickly through 360o, without spilling a drop!
2.246 Measure centripetal force with different radii and frequency
See diagram 2.246
Use a metal or wooden tube about 15 cm long. Tie a two holed rubber stopper to the end of 1.5 m of string. Pass the other end of the string through the tube and attach iron washers for weights. Adjust the string so that the distance from the top of the tube to the cork is 1 m. Grip the tube as a handle and swing it in a small circle above your head so that the rubber stopper moves in a horizontal circle. The force of gravity on the washers provides the horizontal force needed to keep the stopper moving in a circle. Attach a clip to the string below the tube. Swing the tube such that the level of the clip remains constant and so the circular motion is constant. To find the frequency, record the number of revolutions per minute. Record the number of washers when the stopper moves in a path of radius 1 m at constant revolutions per unit time. If you increase the number of washers you must increase the speed of the stopper to keep the washers at the same height, F = mg = m (v2/r) If you halve the radius of the stopper you must decrease the speed of the stopper to keep the washers at the same height, F = mg = m (v2/r).