Jumping Rope by the Numbers

Last year, applied mathematician Jeffrey Aristoff and mechanical engineer Howard Stone, both of Princeton University, were at the gym waiting for a pickup game of basketball. To warm up, Stone started jumping rope. As the rope whizzed over the head of his colleague, Aristoff wondered, "Is it known how jump ropes bend in the wind?" A few literature searches later, he concluded that the answer was, "not really." Now, the two have solved the problem themselves.

How engineers have fun. A high-speed video of a woman jumping rope reveals that the rope bends slightly out of the plane; mathematical modeling shows that this reduces drag.
Credit: Jeffrey Aristoff

The problem is not the child's play that it might seem. Ever since the 1940s, physicists have described the movement of slender structures through fluids—such as a jump rope through air—as a flat plane whose speed is limited by drag. In this case, the plane is defined by the U-shaped area under the rope. But because it is farther from the axis of rotation, the middle of a jump rope travels faster than the ends, so it experiences greater drag. That should bend the rope slightly out of the plane, and that new shape changes the forces on the entire rope. Luckily, Aristoff and Stone are experts on the interactions between fluids and soft, bendy materials. (Aristoff was part of the team that cracked the mysterious interaction between thirsty cats and dishes of milk.)

The duo started by capturing the bending action of a loop of nylon string attached to a spinning motor, which they filmed with a high-speed stroboscopic camera. Sure enough, the rope does bend slightly out of the plane at speeds typical of rope-skippers (see video). The bending of a jump rope is "simply too fast for the naked eye to discern," says Aristoff, now based at Numerica Corp. in Loveland, Colorado.

Then came the hard part—capturing the phenomenon with a mathematical model. The duo boiled it down to a balance between two ratios: the length of the rope versus the distance between its ends, and the force of drag versus the inertia, or "centrifugal force," of the spinning rope. They converted those relationships into a coupled pair of nonlinear differential equations. To see how well their math predicts the shape of a jump rope, they plugged a range of realistic rope-jumping values into the equations and ran a simulation on a computer.

Not only does their model capture the bendy behavior of a typical jump rope, it also helps explain the behavior of a range of similar materials, from reeds blowing in a breeze to lobster antennae bending in a current, the researchers report in the Proceedings of the Royal Society A. Being able to bend significantly reduces the drag on such objects, they found. For a jump rope, bending translates to a 25% drag reduction, allowing it to spin faster than it would if it were stiff.

"It is an intuitive and convincing model," says Douglas Holmes, a materials scientist at Virginia Polytechnic Institute and State University in Blacksburg. "As usual, nature has known this for a long time. Biological materials, while generally far less stiff than our engineered structures, can often survive in hostile environments by deforming to reduce the forces they're exposed to." The next step, Holmes says, is to expand the model to more complex versions of bendy materials in fluids. For example, he asks, "are some structures better suited for the swirling winds of a tornado while others are fit for turbulent ocean waves?" And as for jumping ropes, "an ideal rope is short, thin, and smooth," Aristoff says, and if your goal is to set a speed record, "jumping rope at high altitude, where the air is less dense, could be advantageous."