Physical Principles: Components of vectors and vector addition

Student Prerequisites: For this demonstration, students need to be familiar with the following concepts - a.) triangulation, b.) two-dimensional vector addition, c.) resolution of a two-dimensional vector into components along perpendicular axes, d.) conditions of equilibrum, and e.) mass and weight.

Introduction: The mathematical description of resolution of forces demonstrates the usage of vectors in dealing with physical quantities involving more than one dimension. In this demo, we attempt to verify the vector addition of forces through a two-dimensional case. Students will apply their knowledge of trigonometry and rules of vector operations to make quantitative predictions about forces on two spring scales, and thereby to make comparison and convince themselves.

The primary objective of this interactive demonstration is to help students comprehend the vectorial nature of forces as well as basic properties of vectors as a multi-dimensional generalization of scalars in characterizing physical quantities with directions. Under this consideration, multi-dimensional vector additions can be simply reduced to compositions of handy one-dimensional scalar additions along corresponding directions, each of which can be treated separately.

Description of the Demo: Our setup consists of two spring balancing hung with a mass suspended off center between them. The whole system should be kept in a stationary state so that the condition of equilibrium can be applied:

... (1)

or specifically in component form:

, .

... (2)

There exists certain ambiguity student should be aware of in choosing a certain rectangular coordinate system to perform the resolution. For example, we may have horizontal x-axis and vertical y-axis with the origin located at the hanging point C (see picture below). As we know that components of a vector can be treated as one-dimensional scalars along respective directions, component equations (2) therefore can be embodied with consideration of geometrical relations. Students may also be asked to establish their own coordinate systems to carry out the resolution. Generally, we want to obtain such a coordinate system that calculation on it may be simplified.

We need to assume that the weight of either spring scale (and of course rope) is negligible and thus the scales and ropes and beam can be viewed as forming a perfect triangle denoted by  (see picture below). Given that

, ,

... (3)

and that the weight we use is 2kg, students then may be required to work out explicit expressions of (2) and give numerical results.

Instructions: The ropes with knots on both ends have been prepared in proper lengths to give  and  the values in (3). Ideally, these angles can also be observed from protractors. However, for a reason mentioned below, sole reliance on protractors to obtain these angles is definitely not recommended.

As always, the whole setting should be kept still before the spring scales are read.

Note to the Instructor: 1.) In practice, the protractors giving angles here unfortunately can only be used as coarse reference, because the scale disks not only block the view but also deflect ropes by their weights which make it impossible to obtain precise readings. One may consider switching the scale with the rope attached to get a better view of protractors, however if doing so, the weights of scales would cause increasing error which turns out somewhat intolerable, as has been practically observed. If students insist, this inadequacy may be partially cured by providing the actual lengths of three edges listed as following:

, , and  all in inch;

or in more detail, the length of the scale when stretched being approximately 18 inches together with the left rope 40 inches and the right rope 18 inches. One can verify that these lengths yield the angles we need. 2.) Dedicated setup time may be required.

Possible CPS Questions:

1. Are tensions equal along the two ropes tied up at point C as in the picture? [Question] [Answer]

2. Which of the following coordinate systems can in principle be used for resolution of forces? [Question] [Answer]

3. Will resolutions in different applicable coordinate systems give the same result of prediction? [Question] [Answer]

4. Based on Question 2 and 3, choose your coordinate system to predict the readings of both spring scales… [Question] [Answer]

5. What effect does changing the mass of the weight have on the readings of both scales? [Question] [Answer]