Physicists from China built a theoretical model of the game in pancakes, describing the movement of a solid disk thrown on the surface of the liquid, and tested it in the laboratory. It turned out that the condition of the body bouncing is determined by the angle of its tilt in relation to the horizon and horizontal speed, and the curvature of the trajectory of the disk is due to the gyroscopic effect and the effect of Magnus. The article is published in the journal Physics of Fluids.
Game of pancakes is known for several thousand years - on how to make pebbles jump on the surface of the reservoir, wrote in the 2nd century AD, and the first scientific explanations of this phenomenon appeared in the 18th century.
Recently, interest in the physics of this process has increased - understanding the mechanism of bouncing solids above the surface of the liquid is important for the development of aircraft and swimming vehicles, as well as military devices. For example, during World War II, British engineer Barnes Wallace developed a bouncing aerial bomb, which before the explosion several times bounced off the surface of the reservoir, approaching the target (more on this and other examples of unusual weapons can be read in the material "To chop the enemy from a gun"). However, despite numerous experiments and hydrodynamic simulations, it was not possible to describe exhaustively the behavior of the rotating stone bouncing over the water until recently.
Physicists led by Kun Zhao of the Beijing Institute of Electrical Mechanical Science theoretically analyzed the movement of a rotating body thrown at an angle to the surface of the water and tested the model in the laboratory.
Like a stone, the authors examined a solid disk of constant radius and thickness, which moves progressively near the surface of the water and rotates around its axis. To build and solve motion equations (i.e. to determine the trajectory of the body), the scientists have identified the main forces acting on the disk - gravity, as well as lifting (perpendicular to the speed of the disk and directed upwards) and braking (directed strictly against the speed) hydrodynamic forces from the water. From the air, the disk also experiences aerodynamic lifting and braking effects, but physicists have neglected them, as in previous studies the contribution of these forces has been insignificant.
In addition, the authors took into account the Magnus effect, an additional deflecting horizontal force that arises from the difference in the direction of vortex flows on opposite sides of the rotating disk when it glides over the surface of the water.
To describe the trajectory of the disk, physicists used five coordinates: three spatial (setting the position of the center of the mass) and two angulars (defining tilt of the disk relative to the horizontal plane). By solving Lagrange equations, the scientists calculated the body's trajectory under different initial conditions - the initial height of the disk above the surface of the liquid, its tilt and angular speed of rotation, as well as the speed of the throw.
Visualization of a theoretically predicted trajectory for a disk with a diameter of 5 centimeters and a thickness of 2 millimeters, thrown at an angle of 15 degrees to the surface of the water from a height of 10 centimeters, at an initial rotation rate of 80 revolutions per second and an initial speed along the axis x
In addition to theoretical calculations, the researchers also made experimental measurements. As a stone to play pancakes, they used aluminum discs with a diameter of 5-8 centimeters and a thickness of 2-5 millimeters, to which the accelerometer and gyroscope were attached. When tossing discs were positioned at a slope of 0-60 degrees at an altitude of 0-30 centimeters above the surface of the liquid and unwinded with an electric motor up to 0-50 revolutions per second. The initial speed of the throw (several meters per second) was reported to the discs using an air compressor. In addition to the sensors on the disk itself, its movement was recorded by high-speed cameras.
As a result, physicists have determined that the disk bounces when the lifting hydrodynamic force tells it to accelerate above the critical - about four times more acceleration of free fall. In turn, the magnitude of this force is proportional to the square of body speed and the sine of the angle between the plane of the disk and the horizontal, which makes these parameters key to creating "pancakes." In addition, it was found that the deviation of the disk velocity from the original direction is determined by the combination of the Magnus effect and the gyroscopic effect, with the first mechanism dominating the slow rotation of the disk (less than 18 revolutions per second), and the second - at a rapid.
The authors note that the theoretical calculations almost coincided with experimental measurements, which makes the developed model potentially suitable for further use, such as aerospace, marine, or military engineering.