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# Breaking the Universe in Dungeons & Dragons With Physics 101

Practical Applications of a “Bag of Holding”

## Abstract ¶

In Dungeons and Dragons, a “Bag of Holding” is an uncommon magical item with two unusual properties: its internal volume is significantly greater than its external volume, and its weight is always 15 pounds regardless of its contents. Such a bag exhibits these properties indefinitely, unlike magical spells which can only produce similar effects for brief periods of time, enabling the construction of various infinite-energy and perpetual-motion devices.

## The Bag of Holding ¶

The bag is described as follows :

This bag has an interior space considerably larger than its outside dimensions, roughly 2 feet in diameter at the mouth and 4 feet deep. The bag can hold up to 500 pounds, not exceeding a volume of 64 cubic feet. The bag weighs 15 pounds, regardless of its contents. Retrieving an item from the bag requires an action.

If the bag is overloaded, pierced, or torn, it ruptures and is destroyed, and its contents are scattered in the Astral Plane. If the bag is turned inside out, its contents spill forth, unharmed, but the bag must be put right before it can be used again.

Breathing creatures inside the bag can survive up to a number of minutes equal to 10 divided by the number of creatures (minimum 1 minute), after which time they begin to suffocate.

Placing a Bag of Holding inside an extradimensional space created by a handy haversack, portable hole, or similar item instantly destroys both items and opens a gate to the Astral Plane. The gate originates where the one item was placed inside the other. Any creature within 10 feet of the gate is sucked through it to a random location on the Astral Plane. The gate then closes. The gate is one-way only and can’t be reopened.

At first glance, this is merely an item that expands someone’s carrying capacity, a necessity when dungeon crawling or tomb raiding. As such, it’s rated as an “uncommon” magical item, one that a party of adventurers would likely encounter several times in their travels. However, the Bag of Holding stands out because of its ability to do mechanical work indefinitely without expending spell slots or charges, and without needing to be operated or instructed to do so.

First, let’s begin by formalizing the behavior of the internal volume of the bag. While there are many descriptions that would be compatible with the specification above, we will choose one that makes the math easy:

The space inside of the bag is contracted along the depthwise axis by some factor $\gamma$, such that an observer outside of the bag will measure an object of length $L$ as having length $L’ = \frac{1}{\gamma}L$ when placed inside of the bag. An observer inside of the bag would measure the object as having a length of $L$, and would see space outside of the bag as stretched by a factor of $\gamma$ such that an object of length $L$ outside has a length of $\gamma L$. We can define $\gamma$ as the ratio of the internal depth $L_2$ to the external depth $L_1$.

We can assume that any deformation of the bag on the outside will translate to a proportional change in the interior dimensions, up until the bag is turned inside out. The contents of the bag are still subject to whatever gravitational and electromagnetic fields are present, although they would appear stretched to an inside observer From the inside, light coming in would probably appear significantly redshifted. .

## Inconsistencies with Classical Mechanics ¶

Let’s assume that, in the absence of any magical intervention, the world of D&D behaves roughly like the world in real life. Ultimately, this is an aesthetic preference held at the discretion of the “Dungeon Master”. That is to say, quantities like energy and momentum are conserved, and we can use classical mechanics to describe the motion of objects with a reasonable degree of accuracy. How does the Bag of Holding deviate from these predictions?

Let’s set aside all other details of the bag and focus on its weight. According to the description of the bag, it will always weigh 15 pounds, regardless of its contents. There is an important distinction between weight and mass. Weight is the downward force on an object due to gravity, while mass is a measure of the object’s inertia, or resistance to a change in its velocity. Explained here (with pictures!). To see why this matters, let’s consider the example of a Bag of Holding at rest with an object of mass $m$ inside of it:

Let’s tally up all the forces acting on mass $m$:

• We know the force of gravity $F_G$ on an object at the earth’s surface is equal to its weight in pounds. Therefore, mass $m$ obviously has a weight of $m$ pounds.
• Since the mass is not moving, there must be a force of equal magnitude and opposite direction to the force of gravity. Newton’s First Law of Motion: An object at rest will remain at rest or in uniform motion unless acted upon by an external force. This force is called the normal force $F_N$, and is imparted on the object by the inside of the bag to keep it from falling through.

Now, let’s tally up all the forces acting on the Bag of Holding:

• The force of gravity on the bag is 15 pounds downwards.
• Mass $m$ exerts a normal force on the bag of equal magnitude and opposite direction to the normal force exerted by the bag on mass $m$. Newton’s Third Law of Motion: For every action, there is an equal and opposite reaction.

If we were to place this bag on a scale, we should see the total downwards force read 15 pounds plus whatever the weight $m$ is. However, we know this is not the case! The scale will always read 15 pounds, regardless of the contents of the bag.

We resolve this contradiction by concluding that the contents of a bag of holding can violate Newton’s Third Law of Motion. That is to say, any normal force exerted on the contents of the bag will not result in an equal and opposite normal force on the bag itself. This means we can exert a force on any object (not exceeding 500 pounds) for free!

From this, it becomes obvious that we can violate the conservation of energy and momentum. Recall that:

• Mechanical work done on an object is defined as the path integral of the net force on that object.

$$W = \int_{s_0}^{s_1} \Sigma F \cdot ds$$

The Work-Energy Theorem directly equates the work done on an object with the change in the kinetic energy of said object.

$$\Delta E_k = W = \int_{s_0}^{s_1} \Sigma F \cdot ds$$

The amount of free work we can do on an object is unbounded. Therefore, the kinetic energy of an object can also increase unboundedly.

• Newton’s Second Law defines the rate of change of momentum of an object as the sum of all the forces on said object.

$$\frac{dp}{dt} = \Sigma F$$

This means the change in momentum from time $t=t_0$ to $t=t_1$ is equal to the time integral of the net force on the object in that interval.

$$\Delta p = \int_{t_0}^{t_1} \Sigma F \ dt$$

Since we get the force $F$ for free, we can increase the momentum of an object unboundedly.

It is possible that this energy and momentum are not “free”, but rather sourced or sunk into an alternate plane of existence The Astral Plane, most likely. (like how the Earth serves as an effectively infinite charge source or sink for electrical circuits). This may have interesting worldbuilding implications.

## Applications ¶

Let’s examine various devices that can exploit these violations of the laws of physics.

### River-less Water Wheel ¶

Consider a water wheel with Bags of Holding at the tips of the blades.

The top bag tips over, spilling the water onto onto the wheel. Since the weight of the water only rests on one side, the wheel turns indefinitely. While expensive, it’s not out of the realm of possibility for a medieval fantasy setting, which presumably employs many water wheels already. One would wonder why such a setting would remain medieval even in the presence of a technology even greater than the steam engine was in the early 18th century. Perhaps the Bag of Holding is too fragile for industrial applications, and the risk of opening a gate to the Astral Plane is too great. Or perhaps everyone is just too busy trying not to get killed by dragons. Steam power existed in a very basic form since at least the 1st century CE in the form of the Aeolipile, but it wasn’t put to practical use until much later.

### Compressor ¶

Imagine a hermetically sealed Bag of Holding. That is, one where air cannot enter or exit. By manipulating the outside dimensions of the bag, one can freely decrease the interior volume, thus increasing the pressure inside of the bag. This effect is magnified by the fact that the interior volume is much greater than the exterior volume. If we place an intake valve and an exhaust valve on the bag, we can create a significant pressure differential with little effort.

### Propulsion ¶

The most exciting potential application of a Bag of Holding lies in creating vehicles that can propel themselves without requiring any energy or reaction mass. To show that this is possible, consider the hypothetical of a person standing inside of a bag and pulling it upwards, propelling themselves into the air.

First, let’s work out all the forces that would be in equilibrium with a normal bag:

• Gravity pulls the person and the bag down to earth.
• The person pulls upwards on the bag, and the bag in turn pulls downward on the person with a force of equal magnitude. Let’s call this the tension force $T$.
• The bag exerts a normal force $N_{\text{bag}}$ upwards on the person equal to the person’s weight plus the tension force. The person exerts a normal force of equal magnitude and opposite direction on the bag.
• The ground exerts a normal force $N_{\text{ground}}$ upwards on the bag equal to the weight of the bag and the person.

All these forces add up to zero:

\begin{align} \Sigma F_{\text{person}} &= N_{\text{bag}} - W_{\text{person}} - T &= 0 \\ \Sigma F_{\text{bag}} &= N_{\text{ground}} + T - N_{\text{bag}} - W_{\text{bag}} &= 0 \end{align}

In a Bag of Holding however, this is not the case. We know that it is impossible for the contents of the bag to exert a force on the bag itself. Therefore, we must revise our equations to take this into account:

\begin{align} \Sigma F_{\text{person}} &= N_{\text{bag}} - W_{\text{person}} - T &= 0\\ \Sigma F_{\text{bag}} &= N_{\text{ground}} + T - W_{\text{bag}} &= 0 \end{align}

Now for the bag to move, the tension force must simply overcome the weight of the bag itself! The normal force $N_{\text{ground}}$ will obviously disappear once the bag lifts off the ground. In this way, it is possible to hover any object that does not exceed 485 pounds. Multiple bags can be used to lift more weight. For example, you could use 8 or 9 Bags of Holding to hover a typical sedan with passengers. Road trip!

Let’s say we are in outer space, very far away from everything so that we can completely ignore the force of gravity. What is the fastest that we can accelerate some object of mass $M$ with a Bag of Holding without exceeding the 500 pound normal threshold?

\begin{align} \Sigma F_{\text{obj}} &= 500 \ \mathrm{lbf} - T &= Ma \\ \Sigma F_{\text{bag}} &= T &= (0.466 \ \mathrm{slug})a \end{align}

Substituting $T$ into the first equation

$$\Sigma F_{\text{obj}} = 500 \ \mathrm{lbf} - (0.466 \ \mathrm{slug})a = Ma$$

Solving for $a$

$$a = \frac{500 \ \mathrm{lbf}}{M + 0.466 \ \mathrm{slug}} \ \mathrm{ft / s^2}$$

From this equation, we can see that the maximum acceleration is inversely proportional to the “load” on the bag. Any engineer will tell you that you should not be using a Bag of Holding at 100% capacity, especially given the disastrous consequences if it were to break. In practice, it is probably best to not exceed 10% of this calculated value.

When thrusting against a gravity well, this value is diminished significantly.

$$a = \frac{485 \ \mathrm{lbf} - W_{\text{obj}}}{M + 0.466 \ \mathrm{slug}} \ \mathrm{ft / s^2}$$

If one can overcome the practical limitations of this magical item, it becomes possible to construct constant-acceleration vehicles. The implications of this for space travel are immense; with a constant acceleration of 1-gee ($32.2 \ \mathrm{ft/s^2}$), every planet in the solar system can be reached in a matter of days. One could reach the moon in a little under three hours.

## Conclusion ¶

My DM has to let the party build a spaceship now because he doesn’t know enough physics to argue with me about this.

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