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  • A2-01: CARTESIAN COORDINATE AXES

    A2-01
    Illustrate a standard right-handed cartesian coordinate system
    Used to show students what a cartesian coordinate system looks like in three dimensions and to remind students of the relationship between the directions chosen for the three axes. The axes are labeled x (red, pointing right), y (white, pointing up), and z (blue, pointing out, toward camera).

    As students: Why does the orientation matter? What is the difference between rotating axes and reflecting them? (Compare L2-05)
    A2

  • A2-02: XYZ AXES ON EXTENDED HANDLE

    A2-02
    Aid in discussing angular momentum
    Hold the model out and down toward the floor as you rotate. The colors on the three axes are useful in identifying the three vectors, e.g. L, r, and v. Also helpful in discussion coordinate transformations, or introducing the right-hand rule.
    OS9

  • A2-11: CIRCUMFERENCE VS. DIAMETER - CHAIN ON CYLINDER

    A2-11
    Show the relationship between the diameter and the circumference of a circle
    A chain, originally wound around a cylinder, is unwound. The diameter of the cylinder can be compared with the length of the chain wound off one turn, so the value of pi may be approximated. This is valuable to introduce students to the concept of radians, a unit common in physics formulae but rarely encountered in daily use.

    This can also be used to introduce concepts related to estimation and dimensionality. Encourage students to make predictions about the relationships of the lengths, then try them out hands-on so they can test and analyze their predictions.
    A2, D4

  • A2-12: SLATED SPHERE

    A2-12
    Demonstrate rotation and spherical geometry

    This device can be used to demonstrate rotation and spherical geometry, as well as for astronomical illustrations.

    Rotating metal sphere has green chalkboard surface. Axis is tilted as is that of the earth.
    A2

    en

  • A2-15: HYPERBOLOID FROM STRAIGHT LINES

    A2-15
    Approximate the hyperboloid

    Twist the two spools in opposite directions to produce a changing hyperbola. Hold heavy string taut across hyperbola as shown in the photograph. This can help students to relate concepts of multivariate calculus and three-dimensional geometry.

    This device can also be used as a model of light propagation in relativistic space-time.
    A2

  • A2-21: MAGNETIC VECTOR BOARD

    A2-21
    Demonstrate properties of vectors

    Various lengths of magnetic vectors can be affixed in arbitrary positions to a magnetic board. The lengths of the three longer vectors are in the ratio 3:4:5, so that they can form a right triangle. Illustrate vector addition, subtraction, motion of vectors, and commutative properties.

    Note:This board is for use in smaller classrooms; chalkboards provide magnetic base in lecture halls (Demonstration A2-22).

  • A2-23: VECTOR PRODUCT

    A2-23
    Illustrate the product of two vectors
    The directions of the two vectors A and B, along with their vector product AxB can be illustrated using the model. The cross product can be adjusted in length, but not reduced to zero; its minimum length is shown in the photograph above.

    This is particularly valuable for understanding concepts such as torque or the Lorentz force.
    A2

    vec

  • A2-32: HEIGHT MEASUREMENT BY TRIGONOMETRY

    A2-32
    Determine the height of a student using trigonometry
    Determine the height h of a student by measuring the distance x of the student from the protractor and the angle a of the top of the student from the floor: h = x tan a. Compare the experimental value with a direct measurement using the two-meter stick.
    A2

    g

  • A2-61: EXPONENTIAL INCREASE - CHESSBOARD AND RICE

    A2-61
    Illustrate exponential increase from 1 through 2<sup>63</sup>

    The transparent chessboard serves as the starting point for the progression. Rice can be put on the chessboard for 1 through 25 (32) grains. Use beakers for 26 through 217 grains. (Note: 1 ml = 64 grains, approximately.) The approximate volume of rice for successive powers of two are:

  • B1-07: DOUBLE CONE - SMALL

    B1-07
    Demonstrate a center of mass paradox
    When a double cone is placed on the narrow end of a V-shaped rail, the cone will roll towards the wider end of the rail when released. The cone appears to be rolling uphill (from the narrow end to the wider end), but in reality the center of mass is moving down. Upon special request, we can also provide a cylindrical rod of the same length as the double cone to roll along the rails to show their actual slope. Challenge students to predict whether the cylindrical rod travel in the same direction as the double cone.

    Note: B1-06 is a larger version of this demonstration, suited to large classrooms.
    B1

    t

  • B1-09: CENTER OF MASS - TRIANGLE

    B1-09
    Demonstrate the process of locating the center of mass

    TA triangular shaped object can be suspended from any corner by a nail at the top of the support. The center of mass will be directly below each corner, as defined by the plumb bob. Lines of the vertical position of the bob’s string is drawn on the object. The intersection of all the lines is the position of the center of mass.

    See also demonstration B1-01: CENTER OF MASS - DISC SECTION. These two demonstrations use the same stand and are often taught together.

    Ask students: What would happen if the triangle was suspended at its center of mass?
    B2

  • B1-11: BALANCE MAN

    B1-11
    Illustrate stable equilibrium
    When set on a pedestal, the Balance Man will rock back and forth without falling off no matter the position it is in. The center of mass and the center of gravity is along the line of symmetry of the figure, which is a short distance below the legs. Since the Balance Man’s center of gravity always remains below its pivot point, when it is pulled to one side, gravity will always exert a restoring force which will pull it back to an upwards position. Question: Will the Balance Man be in stable equilibrium if it is set on one leg? (Yes, it will always be in a stable equilibrium since the center of gravity is below the pivot point). Why does the toy rock back and forth instead of falling off the pedestal?
    B1

  • B1-12: CENTER OF MASS - HINGED STICK PARADOX

    B1-12
    Illustrate the center of mass in a surprising way
    To demonstrate center of mass, find how the hammer is supposed to be positioned in order to have it hanging from the loop on the bottom of the board so that the board sticks out horizontally. Question: How can you hang the hammer from the loop on the bottom of the hinged board so that the board will stick straight out horizontally? (A: Insert the handle into the loop so that the head of the hammer is pointing back toward the hinge.)
    B1, tools

    ct

  • B1-17: CENTER OF MASS - STICKS

    B1-17
    Illustrate the center of mass in a surprising way
    There are two sticks identical in appearance, but one is uniform in density and the other is weighted at one end. Since the center of masses for each stick will be different, they will have different balance points. The professor can mark the sticks either at their center of masses or where it appears to be, leaving the student to explain the peculiar behavior of the weighted stick when it is balanced. Questions: Which stick will balance at its center and why?

    These sticks also do double-duty as part of the demonstration D2-04: MOMENT OF INERTIA RODS.
    B1

  • B1-21: BALANCE BEAM - VARIATION ON THE TIGHTROPE WALKER

    B1-21
    Show how arms or a pole are used to obtain greater stability
    As someone walks across the balance beam without a pole, tell the student to take notice how the person’s arms are used to maintain balance in order to keep the center of mass of the system over the beam surface. Now use the pole to walk across the beam to show that walking is easier. Questions: Why does using the pole make walking easier? (A: It increases the moment of inertia.) What happens to the center of mass of the system when the pole is held lower than the person’s center of mass?
    OS0

  • B1-22: CENTER OF GRAVITY USING TEETER TOTTER

    B1-22
    Determine the center of gravity of the human body
    The fulcrum of the teeter totter is used to locate the vertical plane through the pivot line of the board. If desired, a piece of tape can be used to indicate this plane when the teeter totter is nearly balanced. Planes can be determined with the subject lying on the board, standing on the board facing forward, and standing on the board facing to the side. The intersection of these three planes is the center of mass of the body. It is not easy to obtain an exactly balanced position; usually the best that you can do is to oscillate back and forth about the equilibrium point by slight shifts in your position. Questions: Would the board remain in an equilibrium position if masses were placed on each end? If so, what are the requirements?

  • B1-24: CENTER OF MASS - CARTS ON BALANCE BOARD

    B1-24
    Show that the center of mass may remain at rest during motion within the system.
    The board is balanced on its fulcrum with the carts touching one another at the center of the board. Releasing the spring causes the carts to push against each other and separate, but the board remains balanced. Questions: What would happen if the two carts had different masses? Would the board become lopsided?

    The experiment can be made more complex by putting an extra weight into one cart, so that the masses of the carts are unequal.

  • B2-01: SUM OF FORCES IN A LINE

    B2-01
    Demonstrate that forces in one dimension add algebraically
    This demonstration consists of two strings connected at one end of a brass spring that is suspended at the other end from one side of a support frame. The strings run over pulleys attached to each side of the frame, and are then connected to weights. Different masses can be hung on the two sides, creating differing forces in each direction. Using the scale provided, the displacement of the spring can be measured, and the forces can be seen to add algebraically.
    B2, FS1

  • B2-02: SUM OF FORCES - SPRING SCALES

    B2-02
    Show that the sum of forces exerted on the mass by the scales is constant
    A weight is set on a platform scale, and the upper spring scale hooked to the weight. As an upward force is applied, the sum of the readings of the two spring scales remains constant, equal to the initial weight on the platform. In these photos, the sum of the forces on the 1kg mass remains slightly less than 10N as more of the lifting is transferred to the upper scale.
    B2, ME1

     

  • B2-03: EQUILIBRIUM OF FORCES - INCLINED PLANE

    B2-03
    Demonstrate equilibrium of forces

    Place the box on the inclined plane with a mass in it, and let the students see it slide down the slope. Challenge them to predict the mass that is needed to hang off the end in order to keep the box stationary.

    Given an angle of inclination a of the inclined plane, this apparatus demonstrates that the mass m required to hold a cart of mass M at equilibrium on the inclined plane is given by:

    mg = Mg sin a
    B2, ME1