It is important to know the location of the sweet spot on your pickleball paddle so that you can learn to consistently hit the ball at this location to obtain the highest degree of accuracy, velocity, and power. However, locating the sweet spot is not easy. It is complicated by several different factors, including the mass distribution of the paddle, how the player grips the paddle, and other factors, which may not have a direct effect on sweet spot location, such as paddle shape, thickness, and materials.
In a previous article we answered the questions, “What is the Sweet Spot?” and “Why is the Sweet Spot Important?” In this article, we will discuss the mathematics for calculating the location of the sweet spot and demonstrate use of these mathematics to answer the question, “Where is the Sweet Spot?” of an actual paddle.
The Sweet Spot Equation
In more exact scientific terms, the sweet spot is called the “center of percussion” (cp). We will use this terminology throughout this article, because in a future article, we will introduce the idea of an “effective sweet spot”, that accounts for the center of pressure, center of gravity, geometric center, and dynamic center of the racquet. For those of you who are “mathematically-challenged”, please bear with me, as I bring out some interesting facts and observations along the way.
In this example, I use a Vulcan V530 Power paddle, but this technique will work for any paddle. In full disclosure, some of the embedded links in our website are affiliate links, meaning that at no cost to you, Pickleball Science will earn an affiliate commission if you click through the link and finalize a purchase. Purchase of merchandise through these affiliate links will help support the website so that we can continue to provide meaningful content to our readers.
The dimensions of the Vulcan V530 paddle are shown in Figure 1. This drawing was generated using the SketchUp program to simplify some of the area calculations and to keep track of dimensions. You can obtain a limited trial version of SketchUp for free or you can calculate the areas and keep track of the dimensions by hand.
Based on the field of rigid body dynamics, we can derive an expression to calculate the distance from the center of gravity (cg) to the center of percussion (cp):
Lcg-cp = Icg / (Wtotal * Lpp-cg) Equation 1
Where:
Lcg-cp is the distance from the center of gravity (cg) to the center of percussion (cp)
Icg is the mass moment of inertia about the paddle cg
Wtotal is the total weight of the paddle
Lpp-cg is the distance from the pivot point (pp) to the center of gravity (cg)
Notice that there is something very interesting about the above equation: the location of the sweet spot is not directly dependent on the size, shape, or thickness of the paddle, but rather on the weight and inertia of the paddle!
According to Newton’s First Law of Motion (conservation of momentum), inertia is simply the ability of an object to resist a change in motion when subjected to external forces. That is, if the paddle is moving along an intended direction at a certain velocity, it will resist changes in direction or velocity when it encounters another body, such as a pickleball. Inertia is a property of the weight distribution of the paddle about an axis, for which in our case is the pivot point somewhere along the longitudinal axis of the paddle. We calculated the location of the pivot point in our previous article, Why is the Sweet Spot Important?
Another interesting observation is that the pivot point is not a property of the paddle, but it is dependent on the biomechanics of the player’s wrist (i.e., size and movement) and the location along the handle where the player grips the racquet. Therefore, the same paddle can have a different sweet spot location depending on the player using it!
While the paddle face geometry or shape may have an indirect effect on the weight distribution, paddle geometry is not an explicit variable in Equation 1. We will examine the connection between paddle face geometry, paddle materials, and the sweet spot in a subsequent article. For the time being, you should be wary of any manufacturer’s claim that the shape of their paddle faces affects the size or location of the paddle sweet spot. The shape of the paddle face may affect other factors, such as your reach and its response to shock and vibration loads caused by impact with the pickleball, which we will discuss in a future article.
Determination of Paddle Weight and cg Location
We can readily measure the total weight of the racket (Wtotal) by placing it on a kitchen scale (Figure 2):
Here, it shows that Wtotal is about 8.2 oz. The units that we use in our analysis for weight and length do not matter, just as long as we are consistent. We can also readily determine the location of the center of gravity (cg) of the paddle. It must lie on the longitudinal centerline of the paddle (a line drawn down the center of the handle), where the location along the longitudinal centerline is determined by balancing the paddle on a sharp edge, such as the edge of a triangular ruler (Figure 3).
This shows that the balance point, or cg location (shown by the green dot) is along a line that intersects the inside vertex of the “V” in the Vulcan logo, which is about 6.50” from the top edge of the racquet or 9.25” from the bottom of the handle. We calculated the pivot point location at 3.00″ from the end of the handle in our previous article, Why is the Sweet Spot Important? Knowing the cg location (9.25”) and the pivot point location (3.00”), we can readily determine the distance from the pivot point to the cg (Lpp-cg):
Lpp-cg = 9.25 – 3.00 = 6.25”
Calculation of the Mass Moment of Inertia
Now, we only need to calculate Icg, in Equation 1, which is the mass moment of inertia of the paddle about the cg or principal axis. Let’s first subdivide the weight of the paddle into three distinct regions: (1) WA, which is the weight of the paddle face above the principal axis, (2) WB, which is the weight of the paddle face below the principal axis, and (3) WH, which is the weight of the handle (Figure 4).
We know that WA must equal one-half of the paddle weight, or 4.1 oz since we balanced it on a sharp edge. We do not know the weights of WB or WH, since the paddle face has a different density than the handle. We can estimate the density of the paddle face (ρ) by dividing the weight above the cg axis (WA) by the area (AA) of the face above the cg axis. The area (AA) was calculated by SketchUp.
ρ = WA / AA = 4.1 / 50 = 0.082 oz/in2
The density of the bottom portion of the paddle face must be equal to the density of the top portion of the paddle face, so we can calculate the weight of the bottom portion of the paddle face (WB) by multiplying the density (ρ) and the area of the bottom portion of the paddle face AB:
WB = ρ AB = 0.082 oz/in2 * 25 in2 = 2.05 oz
The remaining weight must be attributed to the handle (WH):
WTotal = WA + WB + WH
WH = WTotal – (WA + WB) = 2.05 oz
Conveniently, we see that the weight of the handle is equal to the weight of the paddle face below the cg. Each of the above weights act at the center of gravity of their respective regions (xA, xB, & xH). For areas associated with the racquet faces (WA & WB), the centers of gravity are simply the geometric centers (or centroids) of the respective areas. Using Sketchup to calculate the centroids, xA = 3.125” above the cg axis and xB = 1.58” below the cg axis.
We now need to determine the cg location of the handle (xH), which we can calculate by balancing the torques (or moments) about the cg axis. Figure 5 shows a free body diagram of the paddle balancing on a fulcrum at the paddle cg location.
Summing the torques (or moments) about the fulcrum location we have:
xH = [(WA * xA) – (WB * xB)] / WH
xH = [4.1*3.125) – (2.05*1.58)] / 2.05
xH = 4.67” below principal axis
The weights and dimensions of the three regions of the paddle are summarized in Figure 6 below.
We can now calculate the mass moment of inertia of the racquet about the cg location (Icg) by treating WA, WB, and WH as concentrated masses acting at distances xA, xB, and xH from the cg:
Icg = (WA * xA2) + (WB * xB2) + WH * xH2) = 89.75 oz-in2
Determination of the Sweet Spot Location
We now know everything in Equation 1 (above) and can now proceed to calculate the distance from the paddle cg to the paddle cp:
Lcg-cp = Icg / (Wtotal * Lpp-cg) = 89.75 / (8.2 * 6.25) = 1.75”
Since the cg is 9.25” above the bottom of the handle, the center of percussion of the Vulcan V530 Power paddle is:
Lcp = Lcg + Lcg-cp = 9.25 + 1.75
Lcp = 11.00” from the bottom of the handle along the longitudinal centerline.
Determination of Paddle Face Geometric Center
Another important characteristic of a pickleball paddle is the geometric center of the paddle face. When you think of the “sweet spot” you automatically think that it should be located at the geometric center of the paddle face, where you should hit most of your shots. However, nothing in our analysis above involves the geometry of the paddle – the location of the center of percussion depends only on the location of the pivot point, and the weight and inertia of the paddle. There is no guarantee that the geometric center of the paddle face will be at the center of percussion.
Out of curiosity, I calculated the geometric center of the paddle face (cf) using the SketchUp program. You can also estimate the geometric center of the paddle face by drawing diagonals (shown in green) across the corners of the paddle as shown in Figure 7. Here, the center of gravity (cg) is indicated by the green dot and the center of pressure (cp) is indicated by the yellow dot.
These results are somewhat surprising. The geometric center (cf) is located exactly at the center of pressure (cp) of the paddle! Wow! This cannot be an accident, so the paddle designers at Vulcan did an excellent job by placing the center of percussion at the geometric center of the paddle face. By hitting the ball at the geometric center of the Vulcan 530 Power paddle, you are also hitting the ball at the center of percussion, which is ideal.
Further Questions
Now that we know the answers to the questions, “What is the Sweet Spot?”, “Why is the Sweet Spot Important?“, and “Where is the Sweet Spot?”, we can explore other aspects of the sweet spot and how it affects racquet design, selection, and usage. In future articles, we will interpret these analysis results and address several questions such as:
- How do the structural dynamic (vibrational) characteristics of the paddle affect the sweet spot?
- How large is the sweet spot?
- Will the sweet spot be larger on a lighter or heavier paddle?
- How does the shape of the paddle face (such as an elongated face or an oval face) affect the size and location of the sweet spot?
- How does the location of the sweet spot change by changing your grip (and therefore the pivot point) along the handle?
- How will the use of weighted tape (or other weighting devices) on the edge of the paddle face or on the handle affect the location of the sweet spot?
- Where are the optimum locations to place additional weight depending on the size and location of the sweet spot?
