# Study on the Influence of Rock Shape on Rolling Distance
## 4 Theoretical Analysis  
### Model Derivation  
Analysis of the experimental results regarding the rolling distance of the test blocks clearly indicates that shape is the most significant influencing factor. This is primarily associated with the two-dimensional cross-sectional shape of the rolling test block. The ease of test block rolling can be characterized by the previously introduced rolling resistance coefficient \(RF\). However, \(RF\) alone cannot quantify the effect of stone shape on the rolling stopping distance. In order to quantify the impact of stone shape on the rolling stopping distance, an analysis from a statics perspective is performed, to investigate the connection  
Figure 19: Rolling friction coefficient between spherical ball and wooden board. Figure 20: Rolling friction coefficient between spherical ball and ceramic tile back. Figure 21: Rolling friction coefficient between spherical ball and artificial turf. between the test block's shape and rolling resistance. Furthermore, two concepts related to rolling, namely rolling resistance and rolling resistance arm, are introduced. When a stone departs from the slope with a specific initial velocity and rolls on flat ground with varying properties until its speed diminishes to zero, it is assumed that no collisions or ruptures take place during the rolling. The resistance encountered by the stone during the rolling process arises from both its own shape and the deformation that transpires during its interaction with the rolling surface. In the course of the rolling process, the torque that impedes rolling, stemming from the stone's shape, is termed rolling resistance. The lever arm responsible for creating the rolling resistance torque is denoted as the rolling resistance arm. In the case of polygonal test blocks in contact with the ground, the length of their rolling resistance arm \(\delta\) is contingent upon the contact configuration between the test block and the ground. A shorter length of contact edges with the ground translates to a smaller rolling resistance that must be overcome. As illustrated in Figure 22, for a pentagonal test block with an edge length of \(a\) in contact with the ground, the maximum rolling resistance arm \(\delta\) equals \(a/2\). As the contact length between the test block and the ground diminishes, approaching a circular configuration, the rolling resistance arm \(\delta\) diminishes to zero, resulting in zero rolling resistance for the test block. Based on the preceding analysis, it becomes apparent that the magnitude of rolling resistance encountered by a stone is governed by the fundamental conditions of its interaction with the ground, particularly the rolling resistance arm \(\delta\). Hence, when assessing the influence of stone shape on its rolling distance, it is imperative to take into account the stone's cross-sectional shape and to choose parameters that can effectively quantify its impact on rolling. Among these parameters, for a more precise selection of characteristic parameters defining the stone's shape, it is recommended to conduct a comprehensive analysis utilizing three categories of scales conventionally employed for characterizing block morphology [49], as illustrated in Figure 23. The areas of the minimum circumscribed circle and the two-dimensional cross-section perpendicular to the ground capture both the local and global aspects of falling rock shapes, offering a robust representation of the stone's shape characteristics. Figure 22: Relationship between the number of edges in the two-dimensional shape of the test block and rolling resistance. Therefore, in this paper, the difference between the area of the two-dimensional cross-section of the stone perpendicular to the ground during its rolling and the area of its minimum circumscribed circle is divided by the area of the minimum circumscribed circle.