Key Concepts
Center of mass

Ever wonder how balancing toys work? Simple toys that entertain by precariously balancing were popular in the Victorian era. These seemingly gravity-defying curiosities can rock back and forth at the edge of a table but remain relatively stable.

But how do they balance? The secret lies in a heavy but visually nondescript counterweight attached to the toy. Our eyes are drawn more immediately to the large, ornamental part of the toy that depicts a person or animal rocking back and forth on a tiny pivot point. On its own, this part of the toy looks unstable, which is precisely why such toys continue to be so intriguing—they make little intuitive sense; to an onlooker, it appears that the larger element should send the entire contraption off-balance.

For this fun experiment, you won't have to go searching for any century-old antique toys. All you'll need is a hammer, a ruler and a thick rubber band.

An object's center of mass is a unique point where that object's mass is centered. For simple, symmetrical objects with equally distributed mass (such as a ruler), the center of mass lies in the object's geometric center (which would be just at the six-inch mark for a one-foot ruler).

For more irregularly shaped objects—such as a person on a bicycle—you might be more familiar with the term "center of gravity," which refers to a point on an object that's identified by averaging together all the gravitational forces acting on that object. Don't let this confuse you! For our purposes, it's fine to think of these two concepts as the same because Earth's gravity produces a relatively uniform downward pull on small objects near its surface. Think of it this way: If there's more mass in one region of an object than another (say, a person on top of a bike rather than the bike's wheels), then gravity applies more force to that region.

Let's apply this concept to two children sitting on opposite ends of a seesaw. They create a simple mechanical system consisting of a lever (the seesaw's plank), a fulcrum (the point about which the plank can tilt) and a couple of weights (the children). If the two children weigh the same, the system's center of mass is located in center of the seesaw's plank and directly above the fulcrum. Put a heavier child on one end of the seesaw, and you've shifted the system's center of mass away from the fulcrum. The greater torque (or force) that gravity applies to the more massive (and thus heavier) side of the system pulls the larger child to the ground while lifting the smaller child into the air.

• Hammer
• 12-inch ruler (plastic works, but wood is preferable)
• Thick rubber band
• String (optional)
• Tape (optional)

• First, pick up your ruler and balance it lying lengthwise on your index finger. Try using the six-inch mark on your ruler as your balancing point—this is roughly the location of the ruler's center of mass, and you'll notice it's at the middle of the ruler. This is because our ruler is an object of uniform density and is shaped in such a way that it doesn't have a heavier or lighter end.
• Take your ruler and lay it flat on a table or desk. How many inches on the ruler can you nudge over the edge of the table before it falls off? Think about our hypothetical seesaw from earlier. You should find that once you nudge the ruler's center of mass over the table's edge (the fulcrum), gravity applies more torque to the more massive side of the system than the other, dragging it over the edge and onto the floor.
• Now, try to identify the center of mass on your hammer. If you hold your hammer lengthwise, can you balance it by placing your index finger beneath the point of the hammer located halfway along its length? Probably not! In fact, you should find that the hammer's head is much heavier than its handle. We can infer that the hammer's center of mass is located somewhere closer to its head.
• The goal now is to use our materials to build a new mechanical system with a center of mass located as close to the 0-inch tip of the ruler as possible. (Hint: our hammer will act as a counterweight.) Take another look at how classic Victorian balance toys are structured. Knowing what you've learned about where the hammer's center of mass is, where do you think the hammer is going to go when we build our new system?
• Loop your rubber band over your hammer so that it hangs somewhere near the middle. (Depending on how smooth the hammer handle is, you may want to affix the rubber band to the handle with tape. If your rubber band is too stretchy, you can use a loop of string approximately three inches in diameter instead).
• Loop the rubber band over your ruler. This end of the loop should hang near the 2-inch mark on the ruler. The end of the hammer's handle should intersect and form an acute angle with the ruler at around the 8-inch mark.
• Place the end of the ruler that starts with zero near the edge of the table. Does it balance? Try altering the position of the hammer's head relative to the tip of the ruler. With some careful adjustments, you can tweak the system so that its center of mass is located at the very tip of the ruler.
• Once you've gotten your system to balance, try nudging the 0-inch end of the ruler closer and closer to edge of the table. By doing so, you're moving the system's center of mass progressively closer to the fulcrum and, if you're careful, you can produce a dramatic effect by getting your system to balance on a mere sliver of the table's edge.
Extra: If you want to create an even cooler visual effect and make your system look more like a Victorian balance toy, place a very light stuffed animal on the 12-inch end of the ruler. (A stuffed monkey that hangs onto the end of the ruler with clasped Velcro hands is a crowd favorite.)

Observations and Results
Let's refer back to our hypothetical seesaw. Imagine that more of the seesaw's plank were shifted over to one side of the fulcrum, making that side of the seesaw much longer than the other (just like our ruler hanging over the edge of the table). If you place a large enough person on the shorter end of the seesaw, that person's weight will apply more net torque to that end, suspending the lighter end of the seesaw in the air. The head of our hammer performs a very similar function, but instead of pushing down on the shorter end of the seesaw like the weight of a child would, the gravity acting on the hammer's center of mass pulls on the 0-inch tip of the ruler from below the table.

Hopefully, we managed to dissect the illusion in a scientifically revealing way. So why does it fail to make intuitive sense to us on a visual level? The hammer's handle and the ruler both appear "large," and they both hang much farther over the edge of the table than they "should." At first glance, our instinct is to think, "How can that be?" Of course, the answer lies in the fact that the hammer's head is more massive than its lighter wooden handle, which puts its center of mass closer to the head (and subsequently somewhere under the table). But this is rarely the first thing that occurs to us!

Victorian toys take advantage of the same human perceptual tendency. Think about how these toys are constructed and you'll notice that the large, ornamental part of the toy is made up of thin sheets of metal whereas the toy's counterweight, although small and inconspicuous, is heavy.

More to Explore
Center of Gravity, from
Center of Mass, from PhysicsLAB
Hammer Ruler Trick, from
Center of Gravity of Symmetrical and Asymmetrical Objects, from
Torque and Position of Center of Mass of an Object, from

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