Rod Cross has written a new article: "The Double Pendulum in Tennis" Do you swing your wrist or does the racquet swing it for you? Does the forearm speed up or slow down before impact? Is there an ideal weight relationship between racquet and forearm? Do you initiate your stroke with a limp or locked wrist? When does the wrist release? What happens if you increase the force exerted by the wrist? These and many other interesting questions are discussed. This article is a followup on his other recent article on "The Physics of the Kick Serve"

I haven't read the article yet, but I will say that something I have shown to people to try to help them is that some things will happen naturally and give you better shots without forcing anything. If you will use a towel with a know or a T-shirt or a rope/chain with something at the end and make a swing, it will NATURALLY straighten out slightly in front of the body. This is the natural contact point. Remember boys and girls, "Physics is your friend." However, "gravity is a cruel mistress..."

After years of experimenting with lead tape, I have confirmed that indeed a groundstroke can be modeled as a double pendulum, and that it is quite easy to use this information to identify the optimum weight distribution for the racquet. If the arm, pivoting from the shoulder, is considered to be the upper pendulum, and the racquet (including the hand), pivoting from the wrist, is the lower pendulum, then the weight distribution of the racquet will significantly affect the natural swing frequency of the lower pendulum while having almost negligible affect on the natural swing frequency of the upper pendulum. This allows the required racquet weight distribution to be calculated based on the simple math for the physics of a single physical pendulum. For maximum control on a groundstroke, the racquet face should remain at a constant angle through the hitting zone. For this to occur, the forward component of the velocity vector of the racquethead must have equal magnitude to the forward component of the velocity vector of the hand. In other words, if the racquethead moves forward faster (or slower) than the hand during the moment of impact, then small errors in timing will result in changes in racquetface angle, leading to less accuracy of the shot. But if the racquethead moves at the same speed as the hand, then small errors in timing are inconsequential and the ball will still go toward the target. The formula for the frequency of a physical pendulum is proportional to sqrt(MgR/I), where M is the mass of the pendulum, g is the acceleration of gravity, R is the distance from the pivot to the center of mass, and I is the moment of inertia about the pivot point. For the racquet pendulum, we can use M equals mass of the racquet in kg (because the mass of the hand contributes very little to the equation), g = 980.5 cm/s^2, R = distance from butt to balance point in cm (the wrist pivot point is approximately even with the butt end of the racquet), and I = swingweight about the butt end of the racquet. It turns out that the above condition for optimum control on a forehand (where the racquethead moves forward through the contact zone at the same speed as the hand) is met for a typical average sized male tennis player (height 5’10 to 6’1”) when MgR/I = 21.0. If 21.0 is optimum for a particular player, then a frame with MgR/I = 20.8 will be much more difficult to control because the racquethead will lag behind the hand, causing the player to compensate by applying extra moment force from the wrist in order to maintain control. Many of the best top ATP pro players such as Federer, Agassi, and Sampras, play with MgR/I = 21.0. Exceptions to the rule tend to be players like Nadal, who use extreme western grips. An extreme western grip, with an extreme uppercut swingpath and extra topspin produced, reduces the need for a perfectly balanced frame. Players who are taller (6’3” to 6’6”)tend to have longer arms that naturally swing slightly slower. For these players, the optimum MgR/I value is about 20.7. And players that are shorter (5’4” to 5’9”) such as WTA players, tend use frames with higher MgR/I values, typically about 21.3. I have plotted MgR/I values vs career high ranking for ATP pros, and found that on average, players with MgR/I values in the 20.5-21.0 range have significantly better ranking than the players with values >21.0 or <20.5. And players with MgR/I <20.0 have even worse rankings. Plotting MgR/I vs ranking for WTA pros reveals that the best players tend to have MgR/I >21.0. Adding mass to a frame near the top of the grip will increase MgR/I, while adding mass head, especially near the tip, will decrease MgR/I. Adding mass to the buttcap area will have negligible effect on the MgR/I value, but it will affect the MgR/I value for the 2hb. The MgR/I value can be simultaneously optimized for a forehand and 2hb, because the 2hb has a pivot point about 10cm up the handle. Substituting R -10 for R and substituting the swingweight about the 10cm axis for I gives the MgR/I value for the 2hb. For a typical sized male player, optimum MgR/I for the 2hb = 22.6. If MgR/I = 21 for the forehand, the 2hb MgR/I constraint of 22.6 will be met as long as the balance is about 32.0cm. I carefully adjust and measure my racquets at home to get MgR/I to 21.0, and then I always fine-tune to perfection by hitting against a wall. When I fine-tune my racquets for optimized MgR/I by hitting against a wall, I try to keep my wrist relaxed as I swing naturally through the hitting zone. If my shots (on my righthanded-forehand) tend to miss to my left, then I know that MgR/I is too high (because it means that the racquet gets ahead of my hand, causing me to pull my shot left). If I tend to miss my target wide right, then MgR/I is too low. Once MgR/I is just right, it becomes much easier to control my shots, and I am much less likely to spray my forehand errantly.

When computing MgR/I, I assume you use the strung specs, since that is what you will actually be swinging when you play?