# The Science of Tearing Paper-bag Handles

My mother came back from a market which bags the products in paper-bags with handles, and asked me to move the bags from the trunk of the car to the house. Being the lazy human I am, I hung a few bags on each arm so I could cut the number of trips back and forth. As I was walking to the front door, the handles of a bag tore, the bag plummeting to the concrete ground. A glass jar of peppers had been smashed in to a zillion little pieces. As you might expect, my mother was furious. "You're so lazy! If you hadn't hung so many on your arm, the peppers and their jar would still be intact!"

I disagree, here's why...

Scenario Lazy

Scenario Peppers

Conclusion

Note that Bag A will have $N_A$ on it regardless of Bag B's existence. Sure, my arm had $N_A + N_B$ ($> N_A$), but it wasn't the thing that broke. So, I conclude, that the tearing of the bag was inevitable, and that the peppers' fates were written by someone other than me (e.g.: manufacturer didn't put enough glue to handle expected weight, cashier put more weight than permitted, etc).

Is my reasoning correct? Or am I missing something that proves that I'm guilty?

• For spherical bags in a vacuum that might be correct...
– Aron
Mar 16, 2017 at 8:09
• I think this is not a physics problem, but a psychological one. When you placed multiple bags on one arm, two things happened. You were overly focused on you cleverness at stacking the bags, and you were less aware of what was happening to the individual bags, including the one that tore. Consequently, you were not able to respond quickly enough and save the peppers! BTDT.
– Jim
Mar 16, 2017 at 20:02
• @Jim It is an engineering problem. Where simplistic models comes face to face with the real world and comes away with a black eye.
– Aron
Mar 17, 2017 at 1:30
• Agree with @Jim here. Carrying only one bag at a time, or perhaps one bag in each hand, would have given you the chance to feel that handle ripping, taking appropriate action (like, quickly but non-destructively lowering the bag to the ground). Then again, your mom could've left a large enough margin of error instead of loading the bags to (over?) capacity. And as a bottom line, the secret to a harmonic together is to not insist on guilty / non-guilty, but to allow for errors and making amends... Mar 17, 2017 at 13:41
• Just don't use the handles on paper bags. Carry them as though they had no handles, by supporting them from underneath. Whoever designed these stupid bags should apologize to your mother. Mar 17, 2017 at 15:23

I think you are guilty. The shop assistant (or your mother) was able to load the bags into the car without causing the handles to break. They probably did not try to carry many bags at the same time.

If you hang the bags from a rod with sufficient spacing between them so that each handle hangs vertically, then the handles all bear only the weight of the bag's contents.

However, I think you probably held the bags in each hand rather than hung them from your extended arm (which would require enormous effort) or from a pole (which is unlikely to have been handy, and you are too lazy to look for one). When the bags hang from the same point the tension $T$ in the handles of the outer bags is higher than the weight $W$ of the bag, because of the large angle $\theta$ which the handle makes with the vertical. The vertical force $T\cos\theta$ provided by the handle must equal the weight $W$ of the bag's contents; the horizontal force $T\sin\theta$ is balanced by contact forces $N$ between the bags.

If the filled bags are wide, the handles of the outer bags will be at a large angle $\theta$ to the vertical, requiring a large force $T=\frac{W}{\cos\theta}$. This force tends towards infinity as the handle becomes horizontal $(\theta \to 90^{\circ})$. The outermost handles are much more likely to break than the innermost handle $(\theta = 0^{\circ})$.

Edit 1

Scenario #1 in RowanC's answer can be analysed in the same way. Assuming that the upper part of the bags have a trapezoidal shape, they spread out in an arc, with the middle bag supporting some of the weight of the outer bags.

Balancing forces on all 3 bags we get $T_2+2T_1\cos\theta=3W$. Balancing forces on the outer bags we get $2T_1(1-\cos^2\alpha)=W$ since $\theta=180^{\circ}-2\alpha$. Therefore $T_2(1-\cos^2\alpha)=(4-5\cos^2\alpha)W$.

• If $2(1-\cos^2\alpha) \lt 1$ then $T_1 \gt W$ - the outer bags bear more than their own weight. This happens for $\alpha \lt 45^{\circ}$.
• If $2(4-5\cos^2\alpha) \gt 1$ then $T_2 \gt T_1$ - the middle bag bears more weight than the outer bags. This happens when $\alpha \gt 33.2^{\circ}$.

A more thorough analysis could balance the torque on each bag.

• Even when you hang one or more on your forearm, they still typically but-up against one another so that each bag isn't hanging vertically anymore, and this type of analysis is relevant.
– Dave
Mar 17, 2017 at 13:38
• Also, material fatigue. Mar 17, 2017 at 13:42
• I have made a lot of anecdotal observations that support this as well. When I have had paper bags tear, it is almost always the one with the most angle on the handle. The other thing to add to this answer is it's not just a question of higher magnitudes of force; it's a lot easier to tear a piece of paper starting at one point than it is to just pull it straight apart: If the handles are at an angle the stress is no longer evenly distributed at the handle attachment point (like the bags were designed for) or on the handle itself. It could easily open a tear, and once that starts, it's over. Mar 17, 2017 at 22:45
• Also relevant is that the outermost of each bag's two handles will carry nearly all of the weight. I would never carry paper-handled bags like this! Mar 19, 2017 at 14:48
• @JasonC Good point about the attachment point. If the handles were attached to a solid wall, the attachment would fail for a smaller force if pulled at an angle (greatest effect anti-parallel at 180 degrees) vs being pulled parallel (o degrees). However, I think with paper bags the bag would bend and keep the attachment and handle aligned. Mar 20, 2017 at 19:18

The human body is a mighty complicated system. The forces involved are not simple static forces but dynamic ones that are constantly being adjusted as you walk.

When you take a step, you use your body to cushion the blow like a shock absorber. This cushioning reduces the maximum force experienced by the bag. If you're better at cushioning, you are less likely to rip the bag.

Empirically, humans are better at maintaining this cushioning for lighter loads. As the loads get higher, we start using more drastic motions to try to keep the bags where they belong, and these motions can lead to more forces.

Most likely you took a step, landed with a fair bit of force (you are carrying a load), and failed to properly absorb the shock. You probably also unconsciously did a correction as you tried to recover, which often applies far more force than the original shock.

That being said, fault cannot be proven. If you were a martial arts expert or a dancer, you may have been taught how to move with a minimal amount of shock. If so, the bags may have felt little to no dynamic forces at all, in line with your argument that the bag was doomed from the start.

• The shock likely being oriented in a direction not in line with gravity, given the way those handles are often affixed to the bag. Mar 16, 2017 at 3:16
• As the load gets higher, we start using more drastic motions to try to keep the bags where they belong. I doubt this is true. I think the dynamic forces on each handle will be no greater when carrying multiple bags at the same time than when carrying a single bag. When I carry many bags, or heavy bags, instinctively I move slower, taking smaller steps. If the bags start swinging I stop to regain control, maybe even lowering them to the ground. I do not continue and over-compensate as you suggest. Mar 20, 2017 at 19:02

I think there are two things that happen when you carry more than one bag in each hand:

1. There is increased force (net) downwards on the inner bags, as the outer bags are forced around an arc from your hand due to collisions with the inner bags (and the force that is holding them off the center line, is transferred to the inner bags - the horizontal force is netted out(although if the bread is in the middle bag, it'll be squashed), but the vertical force on inner bags is increased).
2. In the outer bags, the handles are designed to handle force in the plane of the paper, but as the handles are held at the same angle in the hand, but the bags are 'bent away', the force is no longer in this plane, increasing the stress at the joints of the handle to the bag (especially if a crease forms near the edge of the handle).

I'd predict that an inner bag, but not the most inner bag, was the one that broke - where both of the above factors played a part.

• +1. Point #2 I especially have a lot of anecdotal experience with. Mar 17, 2017 at 22:48
• Agreed on #2. Trader Joe's bags are incredibly strong..until you apply force at awkward angles, and then they go kaput. Mar 19, 2017 at 4:18

Cort Ammon is correct. It might be added that when a human carries a load on a weak handle he minimizes by subtle countermovements the inertial forces which are exerted on the handle due to the walking movements in addition to the weight of the load. This is obviously more difficult to do for more than one load.

Thus you mother is right in scolding you!

If you carried one bag only, you would be more in tune with that bag’s status. That might include supporting the bag's weight from the bottom with the other arm. When you first noticed the handle starting to tear (which you might in this case) you could quickly push it against your body to embrace it, and stop the tearing or catch it before it drops.

Too bad Mythbusters isn’t around anymore. This sounds like something some youtuber could do, after that style.