# Study on the Influence of Rock Shape on Rolling Distance
## 3 Analysis of Experimental Results
### Calculation of Rolling Friction Coefficients for Different Slope Materials  
When a ball of mass \(m\) rolls uniformly on various rolling materials, the magnitude of its rolling resistance depends on the level of deformation upon contact with the rolling surface, as depicted in Figure 18a. In the ideal scenario, with no deformation occurring in either the ball or the rolling surface, the ball's \(RF\) coefficient is 0. When a horizontal rightward force \(F\) is applied to the ball, it will continue rolling uniformly until it exits the rolling surface unless an additional external force is exerted.  
Nonetheless, for the balls depicted in Figure 18b,c, their rolling resistance is influenced by the properties of the rolling surface materials.  
Figure 17: “Steps” formed on different material boundaries by the test blocks: (**a**) the test block rolls from low hardness to high hardness; (**b**) the test block rolls from high hardness to low hardness.  
Figure 18: Deformation of spherical test blocks rolling on different surfaces: (**a**) no deformation on the test block and rolling surface; (**b**) the angle at which the deviation occurs constantly changes; (**c**) the rolling surface is flat but has slight undulations.  
When the ball rolls uniformly under the influence of an external force \(F\), it adheres to the following equilibrium formula:  
\[Fr\cos\alpha-Gd=0 \tag{2}\]  
where  
\(d\)--the distance from the instant center to the center of mass of the ball when it rolls on the rolling surface;  
\(r\)--the radius of the ball;  
\(G\)--the mass of the ball;  
\(\alpha\)--the angle formed by the vertical line through the center of mass of the ball and the line from the instant center to the center of mass of the ball.  
When the ball rolls uniformly on the rolling surface, the small angle \(\alpha\) between the vertical line through the ball's center of mass and the line from the instant center to the center of mass results in cos\(\alpha\) equating to 1. Consequently, Formula (2) simplifies to:  
\[F=\frac{d}{r}mg \tag{3}\]  
In this scenario, the force \(F\) acting on the ball aligns with the rolling resistance \(F_{f}\):  
\[F_{f}=\frac{d}{r}mg \tag{4}\]  
Consequently, the work performed by the frictional force arising from the rolling material's characteristics during the ball's descent on the sloped surface can be computed as:  
\[W_{f}=F_{f}L=\frac{d}{r}mg\cos\theta L \tag{5}\]  
where  
\(L\)--the rolling distance of the ball on the slope surface;  
\(\theta\)--the slope.  
In accordance with the law of energy conservation, it is apparent that the overall energy of the ball rolling on the sloped surface is dictated by the initial release height, specifically the gravitational potential energy \(E_{h}\). Consequently, when the ball descends on the sloped surface, Equation (5) can be derived as:  
\[C_{f}=\frac{W_{f}}{E_{h}}=\frac{E_{h}-\frac{1}{2}mv^{2}}{E_{h}}=\frac{d}{r}\cot\theta \tag{6}\]  
where  
\(C_{f}\)--linearly related to the slope's tangent, with the slope-dependent rolling friction coefficient \(\mu\);  
\(v\)--the rolling velocity of the ball on the slope surface (m/s).  
The data collected during the rolling of spherical balls on various slopes were systematically analyzed. Linear regression was employed to determine the relationship between \(C_{f}\) and cot\(\theta\), as depicted in Figures 19-21. This analysis yielded the rolling friction coefficients \(\mu\) for balls of different masses and sizes on different materials. The resulting rolling friction coefficients \(\mu_{1}\), \(\mu_{2}\), and \(\mu_{3}\) for the balls rolling on wood, ceramic tile, and artificial turf were found to be 0.07, 0.09, and 0.13, respectively.