The simple set up shown below, suggested by Prof. P. Schlein, is a good illustration of the vector addition of forces. The vertical angle of the strings can be measured with a large protractor.
Various similar arrangements can be done with the Blackboard Mechanics Set [2].
This device operates on a view graph for the whole class to see easily. Angles are measured by the circular protractor; magnitudes of the forces by the extension of the springs via Hooke's Law.
Label the rings one through seven, starting with the first black ring beyond the last red ring. The force in grams weight as measured to the base of the cone forming the vector arrow is:
Ring |
Force |
1 | - |
2 | 50 gms |
3 | 85 |
4 | 120 |
5 | 155 |
6 | 190 |
7 | 225 |
Force in gms. wt. = 35 X ring # - 20
A good illustration and exercise in adding vectors is provided by a sailboat. The situations of "running with the wind", sailing crosswind, and sailing "close to the wind" are well reviewed in Epstein's Thinking Physics, 2nd Ed., pp 32 - 37. All of these examples can be demonstrated on the air track using a glider with sail and an electric fan to produce the wind. You can actually get the glider to move forward about 45 degrees into the wind.
Vector Addition
This apparatus consists of a platform that rolls along the lecture table, and some toy bulldozers or carts to roll on it. You can push the rolling platform with your hand or let a bulldozer push it. The angle between the motions is completely adjustable. The demonstration can be shown in several versions:
A. As a qualitative illustration, let the bulldozers move and sketch the resulting vectors on the blackboard.
B. To illustrate the path of a boat crossing a river, set the bulldozer rolling on the platform at slight back angle and mark a direction perpendicular to the platform motion on the lecture table with blocks. With the correct adjustments, the bulldozer moving on the platform will be seen to cross the shortest distance perpendicular to the "current".
C. The apparatus is set up on butcher paper, the initial positions of the bulldozers marked, and their final positions marked after the bulldozers have moved. Then the butcher paper can be taped up on the blackboard and the vectors sketched in.
D. In an elaborate demonstration, apparatus is set up as in the illustration above with a bulldozer pushing a blinky while a picture is taken of three blinky tracks as first one bulldozer moves, then the other, and then both. The picture is developed in ten seconds and projected on the screen so the students can see the three blinky tracks.
Magnetic arrows are available which can be stuck on the blackboard. Arrows come in various lengths and can be placed tip to tail to demonstrate vector addition. A protractor and ruler are also available to measure the angle and length of the resultant vector.
A simple demonstration of a radian. Is shows the relation between the radius and the section of circumference of a circle. A straight radius is lifted off and placed on the curve of the circumference. The angle bounded is one radian.
Links:
[1] http://ephysics.physics.ucla.edu/newkin/html/position_velocity_ship.htm
[2] https://demoweb.physics.ucla.edu/node/400