# Bridge builder

In this short problem we investigate the tensions and compressions
in a framework made from springs and ropes.

## Problem

A bridge, which balances on two supports and bears a central weight, is to be made in a triangular pattern, as in the diagram. Each segment of the bridge will be either a rope, which must be under tension to be stable, or a spring, which must be under compression to be stable.The pin joints are light and move freely, but will break if subjected to any net force.

Image

In this problem we investigate which parts of the bridge must be ropes and which parts must be springs

Part 1: Imagine that the strut X feels a tension of some unknown magnitude. By considering the vector directions of the forces at the left hand support prove that Y must be in compression if there is to be no net force at the support. Extend this idea to determine which struts in the bridge must be in tension and which struts in the bridge must be in compression if there is to be no net force at any of the pin joints.

Part 2: Is it possible to use ropes and springs to build a stable bridge of this shape with no net force at any of the pin joints if X is a spring?

For an extension of this problem, why not try the problem More Bridge Building ?

## Getting Started

Think about why at least one of the joints necessarily feels a net force for the following diagram:

Image

Once you have understood the principles involved with this piece of the framework you can try gradually to fill in the entire framework with compressions and tensions. Can you do this consistently? If so, what can you say about the net forces at each of the pin joints?

## Student Solutions

Andre from Tudor Vianu National College, Bucharest, Romania produced a good solution to this question. Can you build on these ideas to answer the extension parts of the question?

The net force on the horizontal axis must be zero at each pin joint, so the projections of the forces on X and Y on the horizontal axis must have opposite directions. So, even if we do not know their magnitude we can still determine whether they are in compression or in tension.

If X is under tension then we must have:

Image

The forces must be symmetrical in respect to a vertical axis which passes through the central weight. Starting from the left toward the centre, I obtained the following type of forces throughout the framework:

Image

For the second part, a similar analysis shows that the structure must be as follows:

Image

There would be a net downward force at P, so the equilibrium would not be stable.

## Teachers' Resources

### Why use this problem

A great deal of insight into Latticework problems can be
achieved without the need for any numerical computation at all: we
rely on vector methods and the simple principles of tension and
force to determine the character of the tensions and compressions
in the bridge. The use of ropes and springs will make students
think about the difference between an internal tension and an
internal compression.

This problem will help students to realise that numerical and
algebraic manipulation are not always necessary as the first step
in a mechanics problem; it also helps to train students' self
checking mechanisms.

This problem also reinforces basic ideas concerning vectors
and components.

### Possible Approach

Students will probably be tempted to try to calculate the
magnitude of all of the forces algebraically but that is
unnecessary in this question. To help reinforce this idea, you
could discuss the problem as a group, beginning with the question
of how vectors will help us in the problem.

Although students should be encouraged to solve the problem
without algebra, they may wish to perform an algebraic calculation
to convince themselves that their deductions are indeed correct.
Hopefully after some thought the principle will become clear and
'obvious' to the students.

The approach used in this question might seem unusual to many
students and could form the basis for a discussion about
mathematics. Although the question will not require any numerical
calculation, a good answer will need to be clearly and accurately
explained. Students could explain their answers to the group. Does
the group feel that the answer is robust? Are there any weaknesses
in the argument that might need clarification? Conversely, are
there any particularly clear aspects of the argument? How do
students feel about not using numbers? Are they aware that the lack
of algebra does not in any way reduce the mathematical importance
or significance of a task? Would an algebraic calculation involving
the mass of the weight make the result any more believable?

### Key Questions

- What are the differences between a spring and a string?
- In order for a pin joint to experience no net forces what must be satisfied?
- How can vectors help us in this problem?

### Possible Extension

Once the concept of resolution of vector components is
understood, there are several avenues of further investigation for
this problem, such as the following

- The bridge in this question is made from 7 triangular segments. What happens if the weight is hanging from the centre of a bridge with more triangular segments?
- What happens if the bridge is made from isosceles-triangular segments instead of equilateral-triangular segments?
- What would happen if an extra row of triangles were jointed onto the top of the bridge?
- Which springs/ropes would experience the most/least internal forces?

In addition, there is a follow-up extension problem to this
task: more bridge building
.