Ohio Resource Center

# Vectors, a Weight, and a Piano

Estimated Time
One class period or more depending on how much the teacher wants to expand this model
Prerequisites
• Familiarity with vectors
Materials Needed
• spring scales
• weights
• ring stands
• graphing calculator
• string
• protractor or altimeter
Ohio Standards Alignment

### Topics

Vector forces, equalization of vectors

### Overview

Students create a physical model of vector forces using ring stands, spring scales, string, and a weight. They apply the actual forces using the vectors and angles given in the problem. They find the forces needed to lift a piano from different points on a street with wires attached to buildings on opposite sides of the street at different heights.

### The "Hook"

In the early 1900's there was no way to get a piano to upper floors of a building other than by lifting it up over the street and sliding it in a window.

Critical Question
Where should you attach the lifting ropes to make the job of hoisting the piano the easiest?
Critical Question
Does changing the elevation of the point of attachment on the buildings change the forces?
Critical Question
Does changing the placement of the piano on the street change the forces?

### The Investigation

Analyze the actual forces using vectors and angles from the example. Compare your results and explain possible differences. You would like to lift a piano weighting 1500 pounds from a street that is 30 feet wide with cables attached to buildings on either side of the street. What must be the cable strength in each situation? Create a physical model of vector forces using ring stands, spring scales, string and a weight.

[A] What is the force exerted on each cable when lifting the piano from the center of the road with the point of attachment being 15 feet high on each side of the street (Building A and Building B)? See Diagram A.

[B] What is the force exerted on each cable when lifting the piano from the center of the road with one point of attachment being 20 feet high on Building B and the opposite point of attachment being 30 feet high on Building A? See Diagram B.

[C] What is the forced exerted on each cable when lifting the piano from a point 10 feet from one side of the street (Building B) and the point of attachment is 10 feet high and 20 feet high on the opposite side of the street (Building A)? See Diagram C.

### Teaching Tips

The underlying concept that is not discussed is that the vectors operating in a horizontal direction are equal to each other due to the fact that there is no movement of the piano to one side of the street or the other.

Most students will get questions [A] and [B] but they will struggle with [C] because they equate distance to force, which is not the case for vectors. The teacher can expand with different placements as they see fit. Or the teacher could ask where the placement might be if one cable has twice the force of the other, etc.

Brief solutions are provided here.

### Citation

From the teaching files of Paul R. Lenz.