# Particle Equilibrium Demonstrator

### Overview

Understanding the concept of particle equilibrium is critical for success in the mechanics curriculum. The equilibrium demonstrator described here helps students to grasp this concept as well and provide a means of visualizing vector components in two or three dimensions

### Principle

The primary principle relevant in this example is Newton’s first law of motion, which states that if the vector sum of all forces acting on an object is zero then the object must either be at rest or moving with constant velocity. Since in this case we have specified that the object will be specifically at rest, we can look at the problem from the other direction and say that since the object is static, the vector sum of all the forces acting upon the object must be zero. This is shown using the following equation. Since we are talking specifically about particles in this case, moments acting on the object are ignored. $\sum \vec{F} = 0$

This equation can be further separated into its vector components as shown in the following equations.

#### In the two-dimensional case $\sum F_x = 0$ $\sum F_y = 0$

#### In the three-dimensional case $\sum F_x = 0$ $\sum F_y = 0$ $\sum F_z = 0$

### What You Need

Item Quantity Description/Clarification
Frame 1 per group The frame can be made out of any thin material such as cardboard, wood, plastic, or metal. My example is laser cut from clear acrylic using a template (downloads linked at bottom of page).
Rubber band 3 per frame Any rubber bands should work as long as they are not too long for the frame.
S-hooks 6 per frame S-hooks attach the rubber bands to a weight and to the frame.
Weight 1 per frame You can provide a weight or students can use their own objects, car keys work well. If students are using their own weights then a scale will be necessary to quantify the weight.

### How It’s Done

Students are provided with the two pieces of the frame and sufficient materials to complete the activity. Worksheets can also be supplied to guide students through the activity. See sample 2D worksheet and sample 3D worksheet. These worksheets include blanks for students to fill out with the necessary values and a pre-defined calibration for converting rubber band elongation into Newtons of force which is specific to the rubber bands selected.

To complete the activity, students must first hang their weight from holes in the frame using the rubber bands. They they measure the start and end points of the rubber band to determine the orientation of each rubber band in the XYZ coordinate system. These vectors allow the student to predict the force in each rubber band using the equilibrium equations. This is considered the theoretical force.

Next the students measure the elongation of each rubber band convert that into force using the provided calibration. This is considered the experimental force.

Finally students can calculate a “performance metric” which provides a measure of how well their experimental and theoretical forces matched as shown on the worksheets linked above.

An example of a completed worksheet is shown here.

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