# Square Wheels

Square wheels roll smoothly on a surface with bumps of the right size and shape.

- About 20 cardboard toilet paper tubes (all approximately the same diameter)
- Foamboard, stiff cardboard, or matboard to serve as a base for the cardboard tubes, about 4 x 30 inches (10 x 75 cm)
- Hot-glue gun and glue sticks
- Poster board or matboard, approximately 8 x 10 inches (20 x 25 cm)
- Scissors
- Pencil or pen
- Ruler
- Pushpin
- Drinking straw
- Two bamboo skewers
- Paper clip
- String, about 12 inches (30 cm)

- Using hot glue, attach a cardboard tube across one end of the base. Continue gluing on tubes, with each tube just touching the one before it, until you reach the other end of the base (see photo below).
- Measure the diameter of three or four of the toilet-paper rolls. The diameters should be approximately 1 11/16 inches (4.3 cm). If this is the case, cut four square wheels from the poster board, with sides of 2 inches (5 cm). If the diameter of your tubes is significantly different from this measurement, then make the sides of the square wheels equal to 1.2 times that diameter. (See the Math Root below.)
- Locate the center of each square wheel by drawing two diagonals, as shown in the photo below.
- Use the pushpin to poke a small hole in the center of each square wheel, taking care not to bend or crease the wheel.
- From the poster board, cut a rectangle measuring 2 x 5 inches (5 x 12 cm).
- Cut two sections of straw, each 2 inches (5 cm) long.
- Glue the straw sections to the rectangular piece of poster board 3/8 inch (1 cm) from each end (see photo below). This assembly will be the body of a small cart.
- Cut the skewers into two 5-inch-long (12-cm) pieces, each with a point at one end. These will be axles. (If you can’t cut the skewers with the scissors, just break the skewers, or cut them with a utility knife.)
- Slide one of the square wheels onto a skewer until it’s about 3/4 inch (2 cm) from the non-pointed end. Slide the pointed end through the straw, and then slide the other square wheel onto the skewer. Adjust the wheels so they’re aligned with each other, and are fairly close to the edge of the cart. The wheel-and-axle assembly should turn freely in the straws. Assemble the other set of wheels the same way. When all the wheels are on, the cart should look like the photo below.
- Use the pushpin to poke a hole between the straw and one end of the cart body, equidistant from the edges, as shown below. Put one end of the paper clip through the hole, and adjust until it’s positioned as shown in the photo below.
- Tie a loop in the end of the string, and place it on the paper clip (see photo below).

Place the cart at one end of the cardboard-tube “road" (see photo below). Make sure the wheels on each axle are aligned with each other. Also, be sure that the wheels are reasonably perpendicular to the axles and are not excessively tilted or wobbly. If they are, use a small amount of hot glue to hold them in place on the axle.

Pull gently on the string so the cart travels along the road. Notice that the cart rolls along smoothly and the axles stay at a reasonably constant height.

A smooth ride on square wheels—amazing, no? This works because, as the cart rolls, the vertical distance from each axle to the horizontal base of the road stays about the same. Study the cart carefully as it rolls and you’ll see that the axle neither rises nor falls: High spots in the lumpy road are exactly cancelled by flat spots on the square wheel.

A special shape called a *catenary curve*, not a circle, is the curve that will give an absolutely level ride with square wheels. A road made with circles—as we’ve done in this Snack—is a reasonably close approximation, however, and is easier to build from commonly available materials.

To see a catenary curve, find a chain or heavy rope, hold one end in each hand, and let it hang upside down. Inverted, a catenary curve will also provide the greatest strength to an arch supporting only its own weight, such as the Gateway Arch in St. Louis, Missouri.

**Math Root**

*Calculating Wheel Size*

In order to travel smoothly over the array of tubes, the sides of the square wheels must be 1.2 times the diameter of the tubes. The equations below explain how this relationship is derived; the diagram shows how the math applies to the square wheels and the “road.” Note that *l* is the side of the square, and *d* is the diameter of the circle (which represents the tube). The circumference of the tube equals 2π*r*.

$$\begin{gathered}

\cos 45=\frac{A C}{A B}, \text { or } A B=\frac{A C}{\cos 45}=\frac{r}{\cos 45} \\

D B=A B-A D=\frac{r}{\cos 45}-r=r\left(\frac{1}{\cos 45}-1\right)=r\left(\frac{1}{.71}-1\right)=r(1.41-1)=0.41 r \\

l=\overparen{D F}+2 D E \\

\overparen{D F}=\frac{2 \pi r}{4} \\

D E=D B=0.41 r \\

l=\frac{2 \pi r}{4}+2 \times 0.41 r=0.5 \times 3.14 r+0.82 r=1.57 r+0.82 r=2.4 r \\

r=\frac{d}{2}, \therefore l=2.4 \times \frac{d}{2}=1.2 d

\end{gathered}$$

Click here for an image of the proof above.

Regester, Jeffrey, “A Long and Bumpy Road,” *The Physics Teacher*, April 1997. (Also reprinted in *Apparatus for Teaching Physics: A Collection of “Apparatus for Teaching Physics” Columns from The Physics Teacher*, 1987–1998, edited by Karl Mamola, American Association of Physics Teachers, 1998, pages 46-47.)