# Inquiry Investigation of Factors Affecting Deflection of a Center-Loaded Beam

The heart of experimental physics involves determining power relationships between variables related to specific physical phenomena. The electrical power generated from flowing water spinning a turbine is proportional to the cube of the volume of water moved per unit time. For a fixed mass of gas under a constant temperature, absolute pressure and volume are inversely proportional. The intensity of light emitted from a point source is inversely proportional to the square of the distance from the source. The magnetic field measured along the axis of a small neodymium magnet is inversely proportional to the cube of the distance from the magnet. These are all examples of power relationships that are commonly discovered by students in the physics laboratory.

Richard Born (Northern Illinois University) has developed a great lab using the Vernier Structures & Materials Tester. In this investigation, he has outlined the methodology for students to perform inquiry activities to determine the relationships of factors affecting the deflection of a center-loaded beam supported at both ends. The equation that models these factors is:

*∆* is the beam’s elastic displacement at mid-span, *F* is the load, *L* is the span length, *E* is the modulus of elasticity, and *I* is the area moment of inertia. If we consider a solid rectangular beam of length *L*, base *b* and height *h*, then the area moment of inertia is *bh*^{3}/12, and the equation becomes

This equation suggests a perfect opportunity for the physics student to experimentally investigate power relationships. Students are likely to identify the factors that will affect this deflection, with minimal coaching. From that point, students could, without knowledge of the equation, design experiments to determine that elastic displacement is directly proportional to the load, directly proportional to the cube of the beam’s length, inversely proportional to the beam’s base, and inversely proportional to the cube of the beam’s height. The physics student would be involved in NGSS Science and Engineering Practices including:

- Asking questions and defining problems
- Developing and using models
- Planning and carrying out investigations
- Analyzing and interpreting data
- Using mathematics and computational thinking
- Constructing explanations and designing solutions
- Engaging in argument from evidence
- Obtaining, evaluating, and communicating information

A discussion of dimensional analysis would be an appropriate summary activity.

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