The ability to put topspin on the ball should be an important component of every player’s game. As we have discussed in previous articles (see for example, “How Fast is a Pickleball Serve?”) you must hit serves and ground strokes on an upward trajectory to clear the top of the net, and soft enough so that the ball lands in bounds. Without the aid of topspin, you have a narrow margin of error and may hit the ball too slow or too high, allowing your opponent to easily run the ball down and/or slam it back at you. Topspin combines with gravity to push the ball towards the ground while keeping it low over the net and in bounds (see “Can Topspin Enable a Faster Serve?”). This widens your margin of error and allows you to hit the ball harder, faster, more accurately, and more consistently.
Because of the recognized importance of topspin, pickleball paddle manufacturers have attempted to differentiate their paddles from others based on their ability to apply topspin to the ball. These claims are often confusing to players, since a paddle’s spin capability is attributed to a wide variety paddle characteristics, including the paddle’s face materials, face surface roughness, face friction coefficient, core materials, stiffness, sweet spot size and location, handle length, weight, etc. While paddle face surface roughness might seem to be the obvious factor that affects pickleball spin, we will see that other factors such as the paddle’s capability to generate ball velocity are also important.
In this article, we will develop a robust method for calculating a paddle’s ability to apply topspin to a ball that is based on proven scientific principles of impulse, momentum, and friction. We then propose that this methodology be used by pickleball paddle manufacturers, reviewers, and regulatory organizations for purposes of characterizing a paddle’s spin capability.
The Importance of Friction
As we discussed in “How is Topspin Generated?”, a serve or ground stroke can be broken down into a “normal” component that is perpendicular to the paddle face, and a “tangential” component that is parallel to the paddle face (Figure 1). The normal force (Fn) propels the ball back over the net to your opponent whereas the tangential force (Ft) puts topspin on the ball.
The normal and tangential forces are related through the coefficient of friction (COF) which is commonly denoted by the Greek letter “mu” or µ:
Ft = µ Fn Equation (1)
There are two types of friction with which we are concerned: static friction and dynamic (or kinetic) friction. Our definition of µ does not differentiate between the two types of friction and may consider other factors that might not be measurable in a standard friction test. It is therefore the “effective” coefficient of friction.
The differences between the static and dynamic COF are illustrated in Figure 2. If you apply a steadily increasing force (Fa) to a block on the ground, the block will not move because it is opposed by the friction force (Ft) developed between the block and the ground. Once you reach a certain break-away force, the block will start moving and the force required to keep the block moving (Fa) is reduced. Most materials (including those that are used in pickleball paddles and balls) will have a static COF that is greater than its dynamic COF.
Since the ability of the paddle to grip the ball is key to applying topspin to the ball, the USAPA performs tests on paddles and has limited the dynamic COF of the paddle faces to be 0.1875 or less. This may be erroneous. Use of the dynamic COF implies that the ball is sliding with respect to the paddle face during contact, making it difficult to apply topspin to the ball.
Based on a review of classical theory involving the oblique impact of elastic spheres, Pickleball Science believes that sliding of the ball with respect to the paddle will occur only initially when the ball makes contact with the paddle face. As the ball and paddle deform, the sliding gives way to rolling of the ball on the paddle face, thereby creating topspin. Once the rolling is established, it continues until the ball is released from the paddle. In essence, the ball “sticks” to the paddle over the period of contact and rolls off the paddle to create topspin. The static COF (which is easier to measure than the dynamic COF) might therefore be more indicative of how much spin a paddle can impart to a ball.
Impulse and Momentum
In this section, we will combine our knowledge of friction, pickleball kinematics, and momentum to develop a single equation for predicting the amount of topspin on a pickleball by knowing the paddle face friction and ball velocity.
In our article “Paddle Weight and Momentum”, we developed relationships between the product of the paddle force multiplied by the contact time (the so-called “impulse”) and the mass multiplied by the velocity of the ball (the so-called “momentum”). The impulse-momentum relationship can be expressed for both the tangential and normal forces and velocities of the ball:
Ft t = mb vt (Equation 2)
Fn t = mb vn (Equation 3)
Solving for Ft in Equation (2) and Fn in Equation (3) and substituting into Equation (1), we can arrive at the following simplified equation:
vt = µ vn (Equation 4)
We can calculate the ball’s tangential velocity (vt) by multiplying the ball’s radius (rball) by its rotational speed, commonly denoted by the Greek letter “omega” or ω. The units for ω are usually given in radians per second, but we can convert these units to revolutions per minute (RPM) and express the tangential velocity (vt) by the following:
vt = rball * ωRPM * (2π/60) Equation (5)
Substituting Equation (5) into Equation (4) we get:
rball * ωRPM * (2π/60) = µ vn Equation (6)
Since rball and vn must be in a consistent set of units (e.g., inches or meters) and we like to express ball velocity in terms of miles per hour (MPH), we can further simplify and express Equation (6) as follows:
ωRPM = 112 * µ * vMPH Equation (7)
Where ωRPM is the rotational speed of the ball in RPM, µ is the effective COF, and vMPH is the translational velocity of the ball as it leaves the paddle face in MPH.
Results & Interpretation
According to videobloggers who evaluate the spin capability of paddles, it appears that 1800 RPM is the upper limit for the amount of spin a player and paddle can generate on a ball. In our article, “Can Topspin Enable a Faster Serve?”, we developed the equations of motion of a pickleball subjected to aerodynamic drag and topspin. When we evaluate these equations with a topspin of 1800 RPM, we find that the maximum velocity of the ball during a serve as it leaves the paddle is about 70 MPH with an average speed of 52 MPH. This serve will have a velocity of about 40 MPH when it reaches the ground at the opposite baseline. It is not physically possible to make an in-bounds serve with an initial velocity faster than 70 MPH without increasing the topspin of the ball beyond 1800 RPM.
Using Equation (7) with a ball speed (vMPH) of 70 MPH and the maximum allowable COF of 0.1875 (as prescribed by USAPA), we find that the maximum rotational speed of the pickleball (ωRPM) must be 1470 RPM. But wait a second! Didn’t the videobloggers and our kinematic analysis find that the maximum rotational velocity of the ball is 1800 RPM? Where is the discrepancy? Are the mathematical equations wrong? This is unlikely because others have independently witnessed that it is possible to achieve a spin rate of 1800 RPM with an average ball velocity of 52 MPH.
The answer lies in the fact that the ball does not slide against the paddle during contact, and the COF must be greater than 0.1875. It is therefore erroneous to use the dynamic COF when evaluating the spin capability of a paddle. Back-solving Equation (7) by knowing that the spin rate (ωRPM) is 1800 RPM and the initial ball velocity (vMPH) is 70 MPH yields an effective COF (µ) of 0.23, which may be more reasonable. The static COF (which is typically greater than the dynamic COF) might therefore be a better indicator of the spin capability of a paddle.
Recommendations
Up until now, the pickleball community (including players, paddle manufacturers, reviewers, and regulatory organizations) have characterized a paddle’s spin capability based solely on surface roughness, which we have demonstrated is only part of the problem. According to Equation (7), the ability of a paddle to generate topspin is dependent on the paddle’s friction coefficient as well as its ability to generate velocity.
For example, while one paddle may seem to be rougher or grittier than another one, it may actually produce less topspin. This might be caused by the paddle’s inability to generate sufficient velocity because the face sheets are too stiff, or the paddle swing weight may be too high for the player to generate swing velocity, or the paddle sweet spot is not in an ideal location, etc. Because of this and other factors, there is no “one-size-fits-all” paddle solution for every player, and paddles should not be selected based on surface roughness alone.
In a similar vein, videobloggers who evaluate paddles using high-speed photography by counting the ball rotations as a function of time are only telling only part of the story. The other important bit of information that we need know is the corresponding ball velocity. This could be accomplished by using high-speed photography to count ball rotations together with motion studies to calculate ball velocity as it comes off the paddle. In effect, we would re-arrange Equation (7) so that the ball rotational velocity (ωRPM) is “normalized” by the ball translational velocity (vMPH). This directly yields the effective COF (µ):
µ = ωRPM / (122 * vMPH) Equation (8)
Low spin rate paddles might have µ < 0.18, medium spin rate paddles might have µ in the 0.18 – 0.21 range, and high spin rate paddles might have µ > 0.21.
In future articles, we will attempt to analytically predict the spin capability of several paddles based on physical characteristics that can be measured or analyzed in a lab, such as the static COF, face stiffness, paddle swing weight, and sweet spot location.
