1a. Standard meter, Kg and seconds pendulum can be shown.
1b. A cube 10 cm on an edge to show liter.
1c. One Kg and two pound weights are compared, circulated to students.
2a. Vector Cube. A metal cube is used as the origin for a number of arrows that serve as vectors. The three cartesian axes, a vector and its projections onto the three axes, and the three unit vectors can be shown.
2b. Cross product model, the result of the cross product and the direction in which the cross product is taken can be shown.
2c. Triangle of forces using weights and pulleys: symmetrical or unsymmetrical. Lines of board give actual angles; no allowances need be given for pulley radius, etc. Both simple and accurate. See file card for diagram.
2d. Sailboat tacking into the wind: toy sailboat and electric fan, about 45 degrees is best performance.
2e. 2-D vector addition using magnet-supported arrows on the chalkboard.
3a. Constant velocity by qualitative observation of moving cart on a level air track. Level air track using single foot; lock with thumb screw. Newton's first law.
3b. Level air track --- if speed is above 20 to 30 cm/sec, then measuring time for equal distances work well; below these speeds, track slopes are too variable.
3c. Inclined air track --- Level track using the single foot.
Then insert special spacer under single foot to give 10 cm/sec2
acceleration of carts. Markers (black masking tape, e.g.) are placed so cart
released from rest reaches each marker at 1 second intervals. Metronome gives
one second timing. Works well, needs some practice in releasing at same time
metronome gives a tick. We are working on computer graphing and analysis of
this experiment. Sonar measurements of cart positions is available.
3d. Average versus instantaneous velocity --- Use inclined air
track (above, part c) over 100 cm; V0 = 0, a = 10 cm/sec2. Use
two
timing gates (one at 100 cm, one variable location). Measure D t over the D x
values of 100, 80, 50, 40, 30, 20, 10 cm; plot (Dx/Dt) vs. Dt; extrapolate to D
t = 0. Final measured (D x) (D t) approaches 44 cm/sec, vtheory = SQUAREROOT(2ax)
= 44.7 cm/sec.
3e. Instantaneous velocity --- photo gates can be set to give
time for cart to travel 1 cm after accelerating down the track.
3f. A car is released from rest on an inclined air track and
simultaneously a clock is started. The time for the car to go a distance D is
measured, and then compared with the time for 2D, 3D, 4D, etc. Many variations
on the basic experiment are possible, including measuring the time for the car
to go a given distance at various places along the air track.
3g. As in c, d, e, but the acceleration is produced by a system of
pulleys, hidden or visible, the air track being level.
4a. Coin and feather --- enclosed in evacuable tube. Show
relative rates of fall for evacuated and room pressure cases.
4b. Balls of copper, ivory, cork, and plastic are
simultaneously released from a stand about one meter above the bench. They are
all heard to hit the table or floor at the same time.
4c. Falling body --- two timing gates are placed at distances
of four feet and one foot below a ball held on an electromagnet. The timing
system starts when the ball is released. The time is noted and the other gate
is then used in a repeat of the experiment. Using g = 32 ft/sec2 the
times of fall can be predicted (.25 sec and .50 sec), or g can be determined
from times.
4d. (In preparation) A jet of water produces a parabolic
trajectory ending in the sink. Vectors can be mounted magnetically
on
a large board behind the trajectory. Constant water head (up to 2 meters) gives
steady trajectory.
4e. A toy monkey is suspended by an electromagnet about 3
meters from a blowgun. As ball reaches end of the gun barrel (aimed by use of
laser), it switches off the electromagnet. Question is, how should the gun be
aimed so ball reaches the falling monkey? Takes a bit of practice, works well.
4f. A ball is dropped at the same time as one is fired
horizontally from a spring gun. Both balls are heard to hit the floor at the
same time.
4g. Projectile from moving car is launched straight up relative
to the car. The ball is then caught by a small funnel on the car.
4h. y versus t for freely falling body --- (in preparation) can
use strobed polaroid camera (display on TV) or can record on video
tape along with a clock (display is slow motion or stop motion). Use black and
white meter stick(s).
5a. A rotating bike wheel has r, v, a, { vectors fastened to it
to show their relationships in circular motion.
5b. A centripetal force measurement using a special apparatus
with rotating pulley. View and measure r from above using TV, measure omega
using a timer. (In preparation.)
5c. Rotating deformable objects --- two fluids in a glass
globe, a model of earth's equatorial bulge, a governor. All show centripetal
acceleration. Best not to attempt detailed explanation; emphasize that
distortions occur to produce required centripetal forces.
5d. Centripetal force on pendulum --- a small mass is connected
to a larger one over a pair of pulleys, such that the arc swept out by the
smaller weight does not intercept the string to the larger. The large weight is
arranged to be in contact with the table at one edge of the table. When the
smaller weight is released to swing in a large arc the larger weight can
be seen to lift slightly when the small one swings through the bottom of its
arc. It can be made more visible by using the TV system.
5e. Radian and degree --- using a disk of radius R and string
of length R, the relationship between the degree and the radian is shown.
5f. Linear-angular relationship --- two wheels of dissimilar
radii are connected by a continuous belt. The relationship of the angular
displacement of one to the angular displacement of the other in terms of the
radii of the wheels is demonstrated.
5g. Earth shape simulator --- a spring hoop of steel is
arranged to rotate around an axis through its diameter.
One end of the hoop of steel is firmly attached to the shaft. As
the system is rotated, the hoop deforms. (See 5-c).
6a. Cart on a level air track to illustrate first law in a
qualitative way.
6b. Cart accelerated by weights --- on a level track,
accelerate various mass carts (.2, .4, .8 Kg) by small masses (5, 10, 20 g.)
hung over a pulley at end of the track; measure times taken to travel 1 m, for
example. May want to neglect small masses in calculating resultant mass being
accelerated.
6c. Inclined air track --- measure a by timing over known
distance. Compare with a = g sin (θ) = H/L. Provided spacer gives a =
10cm/s2. See file card for diagram.
6d. Accelerating elevator --- can be simulated with a mass
taped to a spring scale. By accelerating up or down, reading will increase or
decrease, respectively. Not terrific, but effect can be seen.
6e. Tension demonstration --- using spring scales and two
masses of 2 Kg each (see file card for diagram). Cover faces of scales before
rolling out; ask for predictions. Can use two or more scales to add to
confusion.
6f. Static Friction - force needed to move a wooden
block horizontally is proportional to weight of the block plus added weights.
Mass hung over the edge of table (via a pulley) works well.
6g. Kinetic versus Static friction forces --- on a shellacked
piece of wood, a large mass is placed, and attached to a spring balance. Upon
SLOWLY increasing the pull on the balance, it will be seen that the force
increases to a maximum, then drops to a constant value if the mass moves at
constant speed. Adding weights on top of the block increases the force needed
to move the block.
6h. µ=tan (θ) --- blocks of various materials
(wood,
stone, etc.) are placed on an incline whose angle with the horizontal may be
varied. The angles at which the blocks start to move are noted, and the
coefficients of friction can be calculated.
6i. Car on an incline --- a car is placed on a 30°
incline with wires attached parallel and normal to the plane and going over
pulleys to weights. When the weights are such that they counteract the normal
and parallel components of the weights of the car, the incline may be removed
with no effects.
6j. Table cloth --- set table with plate, cutlery, a cup. Pull
cloth (no seams!) down sharply. With practice, setting remains. A card and ball bearing variation also available.
6k. Cord breaking --- a mass is suspended by a string from a
strong support, with a second string hanging off the bottom. The top string
will break if the bottom string is pulled slowly. However, the bottom string
will break if the pull is rapid
7a. A conical pendulum --- (1) Simply uses a ball on end of a
string to demonstrate conical pendulum and the forces acting.
7b. A conical pendulum --- (2) uses a vertical tube as a
support. The string goes through the tube to a second weight which provides the
needed centripetal and vertical forces.
7c. A conical pendulum --- (3) A one meter string is anchored
so pendulum bob can go in a circle (takes a bit of practice). Estimate H of
string relative to vertical by estimating the circle radius. Make quick
measurement of time for ten cycles. Compare result with theoretical
T = 2π (L cos (θ /g))½
7d. Simple pendulum --- force on bob, see 5.d
7e. Biological form of Principal of Equivalence --- about two
weeks before it is needed, wheat seeds are planted in a tray that is rotated
continuously at 78 rpm on a turntable with 12" diameter. Provided they are
watered frequently, the seeds will germinate and the plants will be about six
inches high in two weeks. It will be seen that the outermost plants are
inclined at about 45 degrees towards the center. Geotropic plants interpret
centripetal acceleration as gravitational force.
7f. Air table demonstration --- the Ealing air table is set up
between two tables and the center plug removed. A string with a small weight
attached is passed through the hole and attached to a heavy puck upon the table.
The motion of the puck when it is given a push can (with practice) be made
circular. This is viewed using the closed circuit TV system.
7g. Air jet rotator --- an air jet, fed from the high pressure
airline, is mounted on a rotatable horizontal bar. The angle of the air jet and
its distance from the center of rotation are variable. The angular acceleration
of the system for varying conditions of jet angle and jet radius are noted
qualitatively.
7h. Rolling races on inclined plane --- a cylinder of mass M
and a ring of mass M are rolled down an inclined plane. It is found that the
cylinder always beats the ring and that all rings and spheres (independent of
radius) take the same time. The experiment may be extended to include rings of
various radii, disks of various radii and spheres. It will be found that any
sphere will always beat any disk, will always beat any ring. Some small rings
bounce and/or slide thus making them run slow. Some objects don't have center
of mass at geometrical center. Check out before using.
7i. Rotating dumbbell --- a dumbbell system is made to rotate
around a vertical axis by a mass and pulley arrangement. The distance of the
masses from the center of rotation is a variable and, hence, the moment of
inertia is the variable. For a given force caused by a given mass on the
pulley, the angular acceleration is noted for two separate radii for the two
masses on the dumbbell.
7j. T =Iα --- the relationship T = Iα is demonstrated using the
apparatus of 7-i. The torque is varied either by varying the mass on the pulley
or the lever arm of the pulley on the vertical bar. The moment of inertia is
varied as in 7-i by varying the radii of the masses and the angular acceleration
is either observed qualitatively or measured quantitatively, timing for ten
revolutions.
7k. Angular momentum stool --- a person, either a student or a
faculty member sits on the rotatable stool, holding in each hand a large mass.
With the weights held close to the body, the stool is set in rotation. The
person then allows the weights to swing out as far as possible, the decrease in
angular velocity being noted. When the weights are once again pulled back in
towards the body, the angular velocity increases. If weights are dropped, then
angular momentum is conserved until the weights hit the floor.
7l. Stool and rotating wheel --- a person sits on the rotatable
stool holding the bicycle wheel which is rotating as fast as possible. Upon
inverting the axis of the bicycle wheel, the entire system of person, stool and
bicycle will be found to rotate around the vertical axis.
7m. Stool and weights --- a person sits on the rotatable stool
holding a bucket of weights. After being set in rotation, the person on the
stool passes the bucket of weights in front of and behind himself from hand to
hand. As the angular momentum of the system is "absorbed" by this mass rotating
around the vertical axis, the angular velocity of the person with respect to the
ground progressively decreases until all the angular momentum of the system is
stored in the rotating mass. The person may remain stationary so long as he
continues to pass the weights around himself. If, however, he should place the
bucket of weights on his lap, it will be found that the stool once again starts
to rotate.
7n. Gyroscopes --- of many forms are available, ranging from a
small seven centimeter gyroscope to a large, massive, twenty-five cm gyroscope.
We also have a Norden Bomb Sight that works; features a gyroscope stabilized
platform. Very impressive for after lectures. The gyroscopes can be set up on
a movable table and the constancy of the direction of the angular momentum
vector shown as the table is rotated or translated. A hand held motor is used
to get gyroscopes up to speed.
7o. Precession --- a bicycle wheel is started and balanced on
one finger. The axis of rotation will immediately start to precess. Alternate,
from hand to hand the side of the axis that is being supported. This will
change the direction of the precession.
7p. Precession again --- a bicycle wheel is suspended from its
axis by a rope from the ceiling. With the axis horizontal, the bicycle wheel is
set in rotation. The unbalanced torque on the bicycle wheel about the
suspension causes the bicycle wheel's axis to precess. The rate of precession
may be varied by adding weights to the axis on either side of the suspension
point.
7q. Yo-yo --- the yo-yo converts gravitational potential energy
into transnational and rotational energy. Two forms of yo-yo are available; a
toy yo-yo mainly used to demonstrate the principles (and also for
amusing the students), and a second form, a large, plexiglass yo-yo
used for detailed discussion of the forces involved in rotation and lines of
action.
7r. Angular momentum conservation --- two masses are arranged
to be rotatable around a vertical axis. The diameter of rotation of the two
masses may be varied by pressing on an external lever system. When the system
is set in rotation with some angular velocity and lever pulled, the radii
decreases and the angular velocity increases rapidly. When the handle is
released, the balls return to their original radius and the angular velocity
decreases.
7s. Wrap-around motion --- a ball and string are attached
firmly to a vertical post. The ball is swung so as to rotate around the post
and progressively wind up. As the radius decreases, the angular velocity of the
ball increases. Angular momentum is conserved.
8a. Torques on meter stick --- a meter stick is suspended at
its center by a knife edge. Weights are hung along the meter stick using metal
hangers. Conditions for equilibrium may be discussed. An antique equal-arm
balance can be shown for historical reasons.
8b. Unsymmetrical meter stick --- as is 8-a, except that the
two arms of the balance are made unequal. Under this circumstance, the mass of
the meter stick itself enters into calculations of equilibrium conditions.
8c. Ladder against wall --- a bar is used to simulate a ladder
leaning against a wall. The forces exerted by the ladder on the floor and on
the wall are measured by spring balances. A load may be placed anywhere along
the length of the bar thus simulating a person on the ladder, the change in the
forces being noted on the spring balances.
8d. Simulated bridge --- a pair of spring scales and two meter
sticks are used to simulate a bridge. A load is placed at some convenient
distance from one end and forces transmitted to either end of the bridge by the
beams are noted. A convenient distance is 33 1/3 centimeters from one end with
the spring balances reading m/3 and 2m/3, respectively. The mass should be as
large as possible (2 or 3 kg) so that the weight of the meter sticks does not
introduce an error in the weighing.
8e. Center of mass --- a meter stick is supported between the
index fingers of the demonstrator, one at about 10 cm, the other at 60 cm (these
are arbitrary). The class is asked where, if the two hands are moved towards
each other, will the fingers be when they touch. (Since the force of friction
depends on the normal force, the hands will meet in the middle of the stick.)
8f. Odd looking C.M. --- a wire is strung between two vertical
bars on one of the mobile benches. A wire walker consisting of a pulley and bar
whose center of mass is below the pulley, is placed on the wire. The wire
walker will run backwards and forwards tilting vehemently from side to side, but
will remain on the wire. A plastic horse and rider whose center of gravity is
arranged to be much lower than the feet of the horse may be balanced on any
convenient bar. The system is stable even though it may be made to oscillate
with large amplitude.
8g. The leaning tower of Pisa demonstrations --- are used to
show stability when the center of mass of an object is within the bounds of the
face of an object. Three versions are available. The first is
a small wooden odd-shaped device with a removable cap. With the cap removed,
the system is stable; with the cap added, however, the center of gravity is
outside the base and the system tilts. The second version is similar to the
first except that the top section rotates and contains a large mass of metal.
With the large mass of metal innermost, the system is stable; but, rotating the
cap 180° places the mass such that the center of mass of the system
is
outside the base. The third version is a large wood and metal model
consisting of three planes of wood, the center of gravity being marked by a
plumbob. As the system is progressively tilted to one side or the other, the
system will be seen to be stable until the plumbob is just over the edge of the
base at which point the system tilts.
8h. C.M. of an irregular object --- a sheet of plexiglass has
four holes; suspend by any 3 and draw a vertical line. They will intersect at
fourth hole, the C.M. Suspension by fourth hole gives stability at any
orientation.
8i. Mechanical advantage (pulleys) --- many systems of pulleys
can be set up, using the pulleys available in the storeroom. With any
arrangement it is possible to measure the theoretical mechanical advantage by
the method of displacement. It is also possible to measure the actual
mechanical advantage by bringing the system to equilibrium with weights on
either side of the pulley system. Mechanical advantages in the ranges 1 to 12
are possible.
8j. Equilibrium types --- ball placed on flat, convex, and concave surfaces are used to represent the three types of equilibrium , namely: neutral, unstable, and stable.
A wooden cone placed on a horizontal plane can show these equilibrium
states.
8k. Wheel and arm --- will show stable and unstable equilibrium, depending upon setting (in preparation).
9a. Work --- the work done as a block is slid across a bench at constant velocity is demonstrated.
9b. Work on nail --- work done on a nail driven into a block of wood using a hammer can be shown, an estimate of the forces involved
being made from the distance the nail moves, and the weight of the hammer and
the height from which it is moved.
9c. Pile driver --- is used to make an order of magnitude or better estimate of the work necessary to drive a nail into a block of wood. Different types of wood may be used. The height through which the large metal driver falls, the distance the nail moves and the weight of the driver are used
in the calculation.
9d. Compare work done --- on a weight as it is: first, placed on a rigid surface and pushed downward; second, hung from a spring and pushed downward.
9e. Pony Brake --- which consists of a string passing from one spring balance to another, over a rotating pulley. It is used to get an estimate of the brake horsepower of an electric motor. The difference in the scale readings with the motor rotating, combined with the radius of the pulley
and the rotational frequency are used to make the calculation.
7. ROTATIONAL DYNAMICS AND ANGULAR MOMENTUM
(see circular
motion, Sect. 5.)