Combined static and sliding friction on an inclined plane

The friction force on a circular object rolling without sliding down an inclined plane is due to static friction. If the object commences in a sliding mode at the top of the incline, and if the incline angle is small, then a small additional sliding friction force arises to reduce the sliding speed of the contact point to zero if the incline is sufficiently long. While the object is sliding, the total friction force is equal to the static friction force plus the sliding friction force. The effect is described theoretically and is supported by measurements of the linear and angular velocities of a ball that commences to slide down a smooth incline.

needed to avoid slipping increases and so does the linear and angular acceleration. If the incline angle exceeds a critical angle, the object slips since the static friction force is no longer large enough to prevent slipping. At high incline angles the object slides down the incline with v > Rω, accelerating at a rate depending on the coefficient of sliding friction and the incline angle. In the latter case, the mass distribution and radius also affects the angular acceleration but the linear acceleration is independent of the mass distribution and radius.
Many experiments have been reported on sliding and rolling down an inclined plane, where the linear or angular acceleration of the object is measured as a function of the incline angle. Most of those experiments are consistent with simple theoretical models, but two recent experiments with rolling cylinders found unusual behavior of the friction coefficients near the critical angle where rolling ceases and sliding commences [5,6]. To investigate further, the author set up a simple experiment to film the motion of a ball on an inclined plane so that v and ω could both be measured as functions of time. It was assumed (incorrectly) that it would easy to determine the point at which rolling stopped and sliding commenced since v = Rω if the object rolls without sliding, otherwise v > Rω.
The motion of a circular object on an inclined plane is usually described by assuming that the object is released at the top of the incline with v = 0 and ω = 0 and that rolling friction is negligible. In practice, those assumptions may not be valid. For example, if the object is released with v > Rω or v < Rω on a low-angle incline, either deliberately or unintentionally, it might slide all the way to the bottom of the incline. That was a problem encountered in the present experiment. A theoretical model of the process is described in the following section, based on previous models [4,8], indicating that static, rolling and sliding friction can all contribute simultaneously to the total friction force. The model is compared with experimental results in section 5, highlighting another problem. That is, the coefficient of sliding friction can decrease as the sliding speed decreases, as it does at small incline angles during the transition from sliding to rolling without sliding. Figure 1 shows a solid sphere of radius R and mass M moving down an incline at speed v and angular velocity ω. The incline angle is θ, q = N Mg cos is the normal reaction force on the sphere, the component of Mg down the incline is q Mg sin , and a friction force F acts in a direction up the incline. It is assumed that the line of action of N acts through a point located at a distance D ahead of the center of the sphere, since a finite value of D is needed to account for rolling friction [4]. The equations of motion are

Theoretical model
where I cm = kMR 2 is the moment of inertia of the sphere about an axis through its center of mass. For a solid, uniform sphere, k = 2/5. If v > Rω or if v < Rω then the contact point on the sphere slides on the incline at velocity v P = v − Rω, so dv P /dt = dv/dt − Rdω/dt. From equations (1) and (2) it is easy to show that q q where μ E is an effective coefficient of friction, defined as the ratio of F to N, and is easily measured from equation (1) in terms of the linear acceleration down the incline. The value of μ E increases as the incline angle increases, up to a maximum value μ k where μ k is the coefficient of sliding friction. A similar effect is observed when a wood block is at rest on an incline. As the incline angle increases, the static friction force increases, up to the point where the block starts to slide down the incline. A value for μ k cannot be calculated from equation (5) but it can be measured from the linear or angular acceleration of a ball at high incline angles, and is a factor that depends on the surface roughness of the ball and the incline, and possibly on the sliding speed. Equation (5) indicates that three different factors contribute to F and μ E , namely static, rolling and sliding friction. The first term on the right side of equation (5) arises from static friction. The second term arises from rolling friction, and the third term arises from sliding friction. If the ball rolls without sliding then the third term is zero. However, if the ball is released at the top of the incline with v > Rω or with v < Rω, then the ball will commence to slide down the incline, even at low incline angles. In that case, all three factors in equation (5) contribute to the total friction force and all three can be present at the same time. It is commonly assumed that only one of the three factors is involved or is dominant in any given situation, but that is not necessarily the case.
The third term in equation (5) arises from sliding motion of the contact point but it is not equal to the coefficient of sliding friction, μ k . At high incline angles where μ E = μ k and where μ E is independent of the incline angle, the third term in equation (5) increases as θ increases, opposing the increase in the first term as θ increases since dv P /dt is large and positive. At low incline angles where μ E < μ k , dv P /dt is small and negative if v P > 0, meaning that v P approaches zero so the ball will eventually roll without sliding if it commences to slide down the incline.
which is the first term on the right side of equation (5). In that case, F arises from static friction alone. When θ is small, μ E is smaller than μ k . Exactly the same static friction force arises if D = 0 and if the contact point slides at constant speed since then dv P /dt = 0. If v P and dv P /dt are both very small, then F is almost the same as the static friction force since the sliding friction term in equation (5) is then very small. A small increase in the friction force is then sufficient to reduce v P to zero as the object slides down the incline. The latter result can arise if a ball is projected down a low-angle incline with v > Rω at the top of the incline, or simply released at low speed with v > Rω. The friction force will then act to decrease v and increase ω until v = Rω. If a ball or cylinder rolls without slipping on a horizontal surface then θ = 0 and v P = 0, in which case equations (3)-(5) give and μ E can be identified as the coefficient of rolling friction, μ R , given by which is the second term in equation (5).

(b) Sliding friction
If a ball slides on a horizontal surface with D = 0 then a = [k/(1 + k)]dv P /dt from equation (3) and μ E = −k(dv P /dt)/[g(1 + k)], which is the third term in equation (5) when θ = 0. The most commonly observed value of μ E on a horizontal surface is μ k , in which case dv P /dt = −(1 + k)gμ k /k and a = −μ k g. The ball will start rolling without slipping when v P decreases to zero, an effect commonly observed in billiards or when putting on a golf ball.
On an inclined surface, the static friction term increase as θ increases, but μ E cannot exceed the value μ E = μ k . If μ E = μ k then dv P /dt must be positive and must increase as θ increases, to oppose the increase in the static friction term. Neither of those two terms can usefully be identified as a friction coefficient if μ E = μ k .
A more useful expression for μ E is obtained directly from equation (1), indicating that (10) allows μ E to be determined from a measurement of the linear acceleration down the incline. If the friction force is large then it is possible that a = 0 and then m q = tan E and Eur. J. Phys. 44 (2023) 055006 R Cross the object will coast down the incline at constant speed. At high incline angles where μ E = μ k , it is easy to show from equations (1) and (2)  A non-zero value of v P does not necessarily imply that μ E = μ k , despite the fact that the contact point slides up or down the incline when v P is non-zero. Equation (5), as well as the experimental results presented below, indicate that μ E can remain smaller than μ k at low incline angles if dv P /dt is small. The following experiments were conducted specifically to examine the contributions of static and sliding friction to the acceleration of a ball down an incline.

Experimental procedure
Experiments were conducted using a smooth, laminated wood incline tilted at various angles to the horizontal so that a ball could either roll without slipping or slide down the incline. Two different balls were examined, one being a 50.8 mm (2.00 inch) diameter billiard ball, the other being a 63.8 mm diameter Henselite ball used as the 'jack' in the game of lawn bowls. The larger ball is essentially a large billiard ball, made from the same phenolic material as billiard balls and equally hard and smooth. Each ball was released from rest at the top of the incline by holding the ball with one finger and gently lifting the finger. The balls were also projected down the incline at low speed by hand to examine the effect of changing the initial conditions. The coordinate of the central axis of each ball was digitized as a function of time using Tracker motion analysis software and curve-fitted to calculate the linear velocity and acceleration. The angular velocity and acceleration were also measured by recording the angular displacement versus time of an equator line drawn around the ball. Both accelerations were measured with an estimated accuracy of ±2%. The experimental arrangement is illustrated in supplementary video Incline.mov. The curve fit procedure needed to be treated with caution to obtain reliable estimates of acceleration from the displacement data. About 20 displacement data points were digitised from the top to the bottom of the incline. In some cases, the data could be fit equally well with quadratic or cubic functions. A quadratic fit implies that the acceleration remains constant down the incline. A cubic fit indicates that the acceleration varies with time. To determine whether the acceleration actually varied with time, two samples of 6 data points were selected, one at the top of the incline and one at the bottom. An example of the procedure is shown in figure 2, assuming that the data points are given by x = 0.05t + t 2 − 0.1t 3 . It can be seen in figure 2 that a quadratic fit is almost as good as a cubic fit in this case. In theory, the acceleration, a, decreases from 2.0 ms −2 at t = 0 to 1.7 ms −2 at t = 0.5 s. The quadratic fit in figure 2 gives a = 1.85 ms −2 , corresponding to the time-average acceleration. A quadratic fit to the first and last 6 data points gives a = 1.97 ms −2 and a = 1.73 ms −2 respectively, close to the cubic fit results, confirming that the acceleration actually decreased with time.

Experimental results
Results showing the measured linear and angular acceleration of each ball are presented in figure 3(a) at six different incline angles. In figure 3, each ball was released at nominally zero speed. Equation (10) was used to calculate the value of μ E at each angle, as shown in figure 3(b). In theory, the linear acceleration is independent of the ball radius, as given by equation (3) when D = 0, but the angular acceleration is inversely proportional to R, as given by equation (4). At low incline angles, the experimental results are consistent with rolling without slipping solutions that assume D = 0 and v P = 0, indicating that both of these terms are either zero or very small. Figure 3(b) indicates that there is a transition when θ ≈ 35°F given by equation (7) with k = 2/5. where the balls slide down the incline with μ E = μ k = 0.19 ± 0.01 for the billiard ball and μ E = μ k = 0.21 ± 0.005 for the Henselite ball.
The transition region is shown in figure 4 as graphs of v/ω versus time at selected incline angles. At low incline angles, v/ω is closely equal to R, indicating that the balls roll down the incline without slipping. However, a different picture emerges if θ > 15°. At these higher angles, v/ω > R immediately after the ball is released and then v/ω approaches R slowly if θ is less than about 35 • . In other words, the balls slide down the incline even in the region where the balls are expected to roll without slipping. At angles greater than about 35 • , v/ω remains larger than R, indicating that the balls slide down the incline the whole time.
The initial value of v/ω depends on how the ball is released. In the present experiment, the balls were released by hand, nominally with v(0) = 0 and ω(0) = 0. However, the first measurements taken soon after release showed that v was larger than Rω, indicating that the balls started in a sliding mode at t = 0. It is possible that the balls started to slide just before the finger on the ball was lifted. Even so, the experimental values of μ E were smaller than μ k , as indicated by the results in figure 3 Effects of varying the initial launch speed of the Henselite ball on a θ = 18.9°incline are shown in figure 5. In figure 5(a) the ball was released at nominally zero speed but with measured initial values v(0) = 0.05 ms −1 and v P (0) = 0.008 ms −1 . The ball started sliding down the incline with v/ω ≈ 40 mm before it commenced to roll without slipping near the bottom of the incline. The measured value of μ E remained almost constant at about 0.1 the whole time, despite the initial sliding phase. If the ball was deliberately pushed at low speed down the incline then the initial value of v/ω was much larger, as shown in figure 5(b). The initial value of μ E was then approximately equal to μ k at the top of the incline but it decreased below μ k while the ball was still sliding. As v/ω decreased, μ E decreased to about 0.1, corresponding to the rolling without sliding value shown in figures 5(a) and 3(b).
The three contributions to μ E in figure 5(b) are shown in figure 6(b). The static friction term is easily calculated from k and θ. The sliding friction term was calculated from the measured values of v, ω and v P = v − Rω shown in figure 6(a). The value of μ R was then calculated from the measured values of μ E and the other two terms in equation (5).
The main contribution to μ E in figure 6(b) is the static friction term which remains constant at 0.0978 as the ball travels down the incline. The sliding friction term adds to the static friction term since dv P /dt is negative, but the sliding friction term itself is not as large as μ k . As the ball slides down the incline, v P and dv P /dt decrease towards zero so μ E decreases. The rolling friction term, μ R , remains very small. The ball commences to roll without slipping at t ≈ 0.2 s, and then μ E approaches the static friction value μ E ≈ 0.1.

Discussion
The theoretical calculations and experimental results described in the present paper show that a ball can slide down an incline with only a small increase in the friction force above the static friction force when v P and dv P /dt are small. That result is surprising since the sliding friction force is usually given by F = μ k N. The results were checked by rolling several different solid  and hollow cylinders down the incline, with essentially the same results. The result is therefore not peculiar to a sphere. Equally surprising is the fact that the effect has not previously been observed, either because the increase in the friction force is too small to have a noticeable effect on the acceleration down the incline or because simultaneous measurements of v and ω are not common.
The experiment in [6] shows a possibly related effect. The authors observed an unexpected increase in angular acceleration of a cylinder when it rolls down an incline at angles close to the critical transition angle. The linear acceleration was not measured in that experiment, so it was not clear whether the cylinder was rolling with or without slipping. The launch method was not quoted either. It is possible that high values of angular acceleration may have arisen in their experiment due to an increase in the sliding friction force in the region where sliding was not expected. The authors in [6] came close to this conclusion by assuming that the friction force alternates between static and sliding friction near the transition region. However, they did not suggest that the total friction force could increase slightly above the static friction force as a result of low-speed sliding of the contact region.
It is not easy to explain why the sliding friction force is very small when the contact point on a sphere or cylinder slides at low speed, but that is the result observed experimentally and predicted by equation (5). A possible clue is that rolling noise is very different from sliding noise. If a ball or cylinder is dragged slowly by hand without rotating across a smooth, horizontal surface, it makes a high-pitched scratching noise and leaves long scratch lines on the surface. A similar result is obtained if sandpaper is dragged across the surface. On the other hand, a ball or cylinder that rolls down an incline makes a different noise which is lower in frequency and smaller in amplitude. Any given contact point on a rolling ball or cylinder lifts off the surface almost as soon as it makes contact. The duration of the contact time of any given contact point might therefore account for a difference in the noise and a difference in the sliding friction force.
Alternatively, it is possible that the contact point at the bottom of the ball commences to slide when it first comes into contact with the surface and then slides to a stop before lifting off the surface. That effect would be more likely if v P is very small when the contact point first contacts the surface. In that case, the time-average friction force will be less than the sliding friction force on the ball. A similar effect is commonly observed when a ball bounces obliquely off a surface. In that case, the bottom of the ball can slide to a stop during the impact and then grip the surface before bouncing off the surface [10]. The mechanism responsible for a reduction in the coefficient of sliding friction was not resolved in the present paper.

Conclusion
If a circular object slides down an incline, it is commonly assumed that the relevant coefficient of friction is the coefficient of sliding friction. That is not the case at small incline angles where it is commonly assumed that the object rolls without slipping and the friction force is due to static friction. The experimental and theoretical results in the present paper present a different picture at small incline angles. That is, if the object starts sliding at the top of the incline, then the friction force is only slightly larger than the static friction force. In that situation, the sliding friction force is relatively small since the sliding speed of the contact point is very small. The object continues to slide down the incline until the sliding speed decreases to zero and then it rolls without sliding if the incline is long enough.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).