SWEET SPOTS A/Prof Rod Cross Physics Department Sydney University June 2006

What is a “sweet spot” ? Tennis racquets, baseball bats, cricket bats and golf clubs all have a sweet spot. It is the impact point that feels best. Compared with other impact points, the player feels an almost complete absence of jarring and vibration in the handle of the striking implement if the ball is struck at the sweet spot. Why does this spot exist, where is it located, how big is it and does the ball travel any faster when it is struck at the sweet spot?

Opinions Some people say that the sweet spot is located at the centre of percussion of the striking implement. Others say it is a node point where there is no vibration. Some say is the point where the ball comes off the implement at maximum speed and hence minimum effort is needed to strike the ball. Others say that the sweet spot varies in size according to how clever the manufacturer has been in designing the implement or in getting these various points to overlap.

Vibrations All objects vibrate back and forth when they are struck. In general, you can see them vibrate, you can hear them vibrate and you can feel them vibrate. A guitar string or a violin string or a tennis string will vibrate rapidly when it is struck or plucked or bowed. Heavier objects like tennis racquets and baseball bats vibrate more slowly and are harder to hear unless you put your ear up close. Guitar strings vibrate back and forth around 1000 times a second. We say that the vibration frequency is about 1000 Hz. Bats and racquets vibrate at around 100 Hz to 200 Hz.

High frequency modes Even though bats and racquets vibrate at around 100 to 200 Hz, they can also vibrate simultaneously at higher frequencies, as can a guitar string. A guitar string has many such “overtones” which is why it sounds different to say a trumpet vibrating at the same basic or “fundamental” frequency. The mix of overtones is different in all musical instruments. In practice, the only vibration mode of a tennis racquet of any importance is the fundamental mode. The ball sits on the strings too long (about seconds) to allow any high frequency modes to vibrate. The effect is similar to putting your finger on a guitar string to stop the string vibrating.

Node points When an object vibrates, some parts of the object vibrate more than others and some parts don’t vibrate at all. Those parts that don’t vibrate are called node points. A guitar string has a node point at each end where it is tied down. The most important node points of a bat or racquet are located about 15 cm (6 in) from each end for the fundamental mode. An object that is struck at a node point will not vibrate at all. If you strike a ball at such a node point you will feel no vibrations at all in the handle. This point is a good candidate for the sweet spot of a bat or a racquet or a club.

CENTRE OF PERCUSSION String Impact at tip Impact at throat Impact at COP Handle doesn’t move Handle moves to left Handle moves to right

SIMPLE BEAM Before experimenting with bats or racquets it helps (a lot) to understand the behaviour of objects with a simple shape such as a rectangular beam or a cylindrical rod. A beam or rod of length L has a node point located a distance 0.22L from each end. The COP is located a distance L/3 from the striking end. If a ball impacts at the COP then the beam will rotate about an axis at the end of the handle. Handle end Node COP L

WOOD BEAM EXPERIMENT Impact at node feels good Impact at tip feels worse COP Node

LISTEN TO VIBRATIONS Hold at node in handle, with handle close to your ear. If you hold the far end of the handle (an antinode) the vibrations are strongly damped. Impact strings or racquet frame with a ball or with the soft end of your finger (not with finger-nail). 200 Hz (high pitch) = very stiff (and light) racquet 100 Hz (low pitch) = very flexible (and heavy) racquet This way you can locate the handle and frame nodes as well as the node near the middle of the strings.

SOME EXPERIMENTS TO DO Find YOUR sweet spot on a wood beam, a bat and a racquet. (Mine is at the node point, not the COP). The node & COP points are much closer together in a baseball bat. Bats vibrate at two frequencies simultaneously. What differences do you find when using thick or short (stiff) beams & thin or long (flexible) beams and when using a stiff ball (eg golf or baseball) vs a soft tennis ball. Does the sweet spot zone get any bigger? Stiff racquets are harder to bend so they vibrate less (and faster or at a higher pitch). Does the node or the COP point shift when you hold the handle with your hand? Does the vibration frequency (pitch) change? Try adding a heavy weight at the handle end. Does the ball bounce any faster off the sweet spot? Try bouncing a ball from a fixed height off a racquet held with the strings horizontal. The bounce height is proportional to the rebound speed squared.