# Physics 26 How Strong Is a Paper Bridge?

We met force as a push or a pull. When we balanced forces, we found we could set two forces against each other. The new force was found by adding those forces up.

When an engineer builds a building or a bridge, forces are very important or the structure will fall down. How can an engineer move forces around to keep a structure standing?

Question: How strong is a sheet of paper bridge?

Materials:

Several sheets of paper

Tape

Plastic yoghurt cup or plastic cup of similar or slightly larger size

Marbles or other small weights – enough to fill the cup, over 3 pounds worth

String

Scissors

Scale

2 Chairs

Small blanket or several bath towels

Procedure:

Put the two chairs back to back with a gap between them two thirds as big as the paper is long

Place the two chairs back to back so they are parallel. The towels are to catch the marbles or rocks when they fall.

Place the blanket or folded towels on the floor between the chairs

Place a sheet of paper across the gap

A sheet of paper suspended over the gap between the chairs sags down in the center.

Carefully set the cup on the sheet of paper

If the paper stays put, add a marble

The sheet of paper seemed to fall even before the cup was set on it.

Continue adding marbles until the bridge falls down

Mass the cup and any marbles in it [You may have to pick these up.]

The sheet of paper couldn’t hold even the 11.7 g cup.

Take the sheet of paper and fold it lengthwise so the fold is 1.5 cm

Now make a second fold the other way 1.5 cm

The folds are made lengthwise to span the gap. If the folds went across the paper, would they change how the sheet of paper acted? Probably not.

Repeat this until the paper is accordion folded [This is called concertina.]

Put the folded piece of paper across the gap between the chairs

The concertina bridge is straight across the gap between the chairs. The cup sits up on the folds.

Carefully set the cup on the sheet of paper

If the paper stays put, add marbles until it falls down

Mass the cup and marbles

Put four small holes [big enough for the string.] equal distances apart around the rim of the cup

If the four strings are the same length, the cup should be close to level when suspended. This lets the weights distribute evenly and not tip out.

Cut 4 pieces of string about 45 cm long

Tie knots in one end of the strings

Put the untied ends through the holes, one piece in each hole

Tie the ends together [You might want to tape it together too so the knot doesn’t untie.]

Take another piece of paper and roll it up lengthwise so the roll is 10 cm in diameter

Note: The diameters don’t have to be exact. You need a large, medium and small tube.

The tube sizes can vary but one is large, one small and one in the middle. A couple of short pieces of tape keep the tubes from unrolling.

Tape the roll so it won’t unroll

Roll another sheet of paper into a 7 cm diameter tube

Roll another sheet of paper into a tube with a 1 cm diameter

Put the 4 cm roll through the string loop of the cup and suspend the cup between the chairs

The cup is suspended below the middle of the bridge. An interesting comparison could be done putting the cup at different places along the tube. Each attempt would use a new tube of the same size.

Add marbles one by one until the bridge fails

Using a wide top cup made it easy to add the rocks. Another advantage was having my hand inside the strings so I caught the cup as it fell before the rocks were scattered over the towels.

Mass the cup and marbles

The large tube bridge held 478.3 g, an increase of 75.4 g over the concertina bridge.

Repeat this with the 2 cm and 1 cm tubes

Observations:

Describe how the sheet of paper looks suspended between the chairs

Mass of cup and marbles the sheet of paper held up

Describe how the concertina or folded paper looks suspended between the chairs

A single additional rock caused the folds to flatten under the cup. The concertina bridge did hold several more rocks before crashing down to the floor.

Describe what happens to the concertina as you add marbles to the cup

Mass of cup and marbles the concertina held up

Describe how the 10 cm tube acts before and as you add marbles to the cup

Mass of cup and marbles the tube holds up

Describe how the 7 cm tube acts before and as you add marbles to the cup

Mass of cup and marbles the tube holds up

The medium tube bridge held up 699.7 g which was 221.4 g more than the large tube.

Describe how the 4 cm tube acts before and as you add marbles to the cup

Mass of cup and marbles the tube holds up

Conclusions:

Compare how the plain sheet of paper and the concertina looked suspended.

Where was all the force from the cup focused on the sheet of paper?

Where was all the force from the cup focused on the concertina bridge?

Note: Think about how the tops and bottoms of the folds act.

How is the force from the cup focused with the tubes?

Adding rocks to the cup below the large tube bridge caused the tube to flatten.

What happens to the tube bridges to make them fail?

What do you think would happen if you could make an even smaller diameter tube?

How do you think the forces would focus if the tube were a solid cylinder?

Why is a hollow tube stronger than a solid cylinder?

What I Found Out

I didn’t have enough marbles so I went out to the creek and gathered pieces of gravel about the size of marbles.

The sheet of paper barely stayed up suspended between the chairs. It bowed down in the middle. The 11.7 g cup never really sat on it before the paper fell to the floor.

After the sheet of paper was folded into the concertina, it went straight across between the chairs. It did not sag. The cup sat on it as though it was on a table.

The only difference between the sheet of paper and concertina bridges were the folds yet the weight capacity increased a lot.

I started adding rocks. Finally the folds buckled under the weight. A few more rocks and the concertina fell. It held up 402.9 g of cup and rocks.

All the force of the cup was in the middle of the sheet of paper and it couldn’t hold it up. The folds of the concertina bridge let the force push and pull between the top and bottom of the folds. The folds carried a lot of the force away from them to the chairs. This let the concertina carry weight until the folds finally broke. Then the force was more in the center and it fell down.

The tube bridges acted much the same but the small tube took longer to change. They were straight across the gap. As the rocks were added, the tubes began to flatten. As the tube bridges failed, the tubes crushed.

The force of the weight is spread around the tube bridge and passed on to the chairs. As the tube collapses, more of the force is concentrated on the middle where the strings are. When the tube fails, all the force has moved to where the strings are crushing the tube and making it bend.

I did have some more rocks but couldn’t get them to stay on the pile. The tube had started to flatten so the small tube bridge was approaching its weight capacity.

The smallest tube had not failed when the cup was filled to overflowing with rocks. Perhaps I could have added some small lead wheel weights at the beginning so the tube would fail.

An even smaller diameter tube should hold more weight as long as the center is hollow. This lets the force of the weight move away from where the strings are hanging and go into the chairs. The forces on a solid tube would stay mostly where the strings are hanging putting the weight on the bar there until the bar would break.