With rare exception,^{1–3} the force of friction on a rolling object is not usually a topic discussed in introductory physics textbooks. Although the invention of the wheel is one of the essential world achievements, rolling friction is typically ignored and the inability of students to explain or model the deceleration of a rolling rigid object on a rigid horizontal surface necessitates a mechanism for this phenomenon. (Detailed analyses of this mechanism and measurements of the coefficient of rolling friction can be found in Refs. 4–6.) Every introductory physics textbook discusses kinetic and static friction and many papers are devoted to teaching these forces,^{7} but it is well known that measuring static and kinetic friction in educational labs is troublesome.^{8,9} To make matters worse, the coefficient of rolling friction is much smaller than the coefficient of kinetic friction,^{1–5} so student measurements of rolling friction are even more troublesome. In addition, the usual measurements of time (about 1 s) and distance traveled (less than 1 m) are comparatively small, which makes surface uniformities and accurate leveling (or measuring the angle of incline) of the track critical factors that influence the uncertainty of the measurements. To counter these difficulties, a method of measuring the coefficients of rolling friction based on the oscillations of steel balls on a large concave lens was proposed in Ref. 4. This method is free from the above-named deficiencies.

In this note, a ball oscillating on a concave trackway is used to find the coefficient of rolling friction *μ _{r}* using typical laboratory equipment. This simple experiment can be carried out by students and the results are highly reliable. It is known that a variety of factors can influence friction forces, including adhesion, deformation, elastic hysteresis, abrasion, the effect of impurities, etc.

^{7}However, the nature of rolling friction is not the subject of this paper. Instead, we use a phenomenological approach that presumes the magnitude of the rolling friction force $fr$ is proportional to the normal force

*N*and has a direction opposite to the motion

^{2,10}

where $\mu r$ is assumed to be constant (independent of velocity). In this case the only dissipative force that causes the change of mechanical energy (kinetic plus potential energies) is the force of rolling friction.^{10} The assumption of Eq. (1) should be considered an initial, but commonly used, approximation; deviations of this formula have been observed.^{11}

A steel ball of diameter of 3.90 cm and mass *m* = 225 g was used as the rolling object. A manufactured wood track (a so-called *stringless pendulum*^{12}) and a plastic ruler with a central groove were utilized as concave tracks (see Fig. 1). The ball was released near the bottom of the track and its (oscillatory) position was measured using a pasco^{13} motion sensor. Figure 2 shows typical graphs of the ball's position and velocity versus time. Because the track curvature is small, the motion of the ball can be approximated as one-dimensional with a constant normal force (*N* = *mg*). Indeed, on the wood track, which has a radius of curvature of 0.57 m, the ball's largest displacement from the equilibrium position (0.15 m, see Fig. 2) results in a normal force that differs from *mg* by only 3% [$N=mg\u2009cos\u2009(0.15/0.57)=0.97\u2009mg$]. From Fig. 2, one can see that these oscillations are damped with a linearly modulated amplitude; this confirms the assumption that the force of rolling friction is constant and does not depend on the speed of the ball^{14,15} in this experiment.

Assuming the ball rolls without sliding, a rolling friction force and the force of gravity are the only forces responsible for the change in kinetic energy *K*.^{10} By choosing initial and final positions of the ball at the bottom of the concave track (perhaps after many oscillations) and applying the work-kinetic energy theorem, one can confirm that the ball's change in kinetic energy equals the work done only by the force of rolling friction.^{10} The kinetic energy of a solid sphere rolling on a flat surface is the sum of the translational $Kt$ and rotational $Kr$ (with respect of the center of mass) kinetic energies. Assuming the ball is in contact with the bottom (rather than the edges) of the track, the total kinetic energy is then^{2}

where $v$ is the speed of the ball's center of mass. Meanwhile, the work done by the force of rolling friction is

where $s$ is the total distance travelled by the ball. Equating the work done to the change in kinetic energy then allows us to find the coefficient of rolling friction as

The motion sensor data allows us to determine the ball's velocity at any point in time and to compute the total distance traveled by the ball. For example, using Fig. 2, let us choose an initial time of *t _{i} = *1 s (the ball is at the bottom of the track with

*v*0.33 m/s) and a final time of

_{i}= –*t*33 s (the ball is again at the bottom of the track with

_{f}=*v*= 0) for oscillations of the ball on the wood track. The pasco software interface software allows us to compute the (positive) area under the velocity curve, which gives the total distance traveled by the ball,

_{f}*s =*3.8 m. Substituting these values into Eq. (4) gives $\mu r=2.0\xd710\u22123$. Using the same procedure, the coefficient of rolling friction for the steel ball on the plastic ruler is found to be $\mu r=0.75\xd710\u22123$.

Alternatively, *s* can be found as the sum of the ball's oscillation distances. For linearly modulated amplitudes, the amplitudes represent an arithmetic progression and therefore $s=(2(Ai+Af)\Delta t/T)$, where *A _{i}* and

*A*are initial and final amplitudes,

_{f}*T*is the period of oscillations, and Δ

*t = t*is the time interval. For example, for ball oscillations on the wood track (Fig. 2),

_{f}– t_{i}*A*= 0.15 m,

_{i}*A*= 0,

_{f}*t*1 s,

_{i}=*t*= 33 s, and

_{f}*T*= 2.7 s, leading to

*s*= 3.6 m. This method results in a 5% difference with the computation made by calculating areas under the graph of velocity versus time.

The proposed method of measuring the coefficient of rolling friction can be utilized using a plastic ruler and standard physics equipment^{12} (the 2D collision^{16} apparatuses can also be exploited as an aluminum concave track). This method is easy to implement, requires little time, and it allows students to explore the dependence of $\mu r$ on the ball's diameter and the ball and track materials. Moreover, this lab integrates different topics in introductory physics including: kinematic relations (differential and integral) between velocity, displacement, and distance traveled; the work-kinetic energy theorem; and damped oscillations that are linearly modulated. We believe such a lab would be a good learning experience for introductory students.