The point that takes the blow without a jolt

Anyone who has swung a bat, a racket, an axe or a hammer knows that where the blow lands decides how it feels. Hit too close to the hands and the handle is driven into the palm; hit too close to the tip and it is wrenched out of the fingers; somewhere in between there is a place where the blow is clean and the hands feel almost nothing. It is not the centre of mass, and it is not a matter of taste. It is a point that can be computed, and computing it shows that it is the same point that the length nobody has to measure finds by timing a swinging body — the end of the simple pendulum that swings with the same period.

That coincidence is the whole of this essay. One question is about a single sharp blow and asks where the pivot is left alone; the other is about a slow oscillation and asks how long it takes. They share no forces, no timescale and no approximation, and they have one answer.

What the pivot has to supply #

A rigid body held at a pivot is struck by a short, sharp blow — an impulse $J$, force times duration — at a distance $q$ from the pivot, perpendicular to the line joining them. Two things happen at once. The blow’s moment about the pivot sets the body turning, with an angular velocity $\omega = Jq/I_P$, where $I_P$ is the body’s moment of inertia about the pivot. And the body’s centre of mass, a distance $d$ from the pivot, starts moving at $d\omega$. Its momentum is then $M d\omega$, and collisions are easier than forces says that momentum must have come from the impulses acting: the blow, $J$, and whatever the pivot supplies, $J_P$. So

$$J_P = J - Md\omega = J\left(1 - \frac{Mdq}{I_P}\right).$$

The pivot supplies nothing when $q = I_P/Md$. Nearer the pivot the pivot must push along the blow; beyond that point it must pull against it.

For a uniform rod pivoted at one end, $I_P = ML^2/3$ and $d = L/2$, so the special point is at two-thirds of the length. The share falls linearly from the whole blow at the pivot, through zero at $2L/3$, to minus a half at the far end: struck at its tip, a rod held at the other end pulls on the hand with half the force of the blow, in the opposite direction.

The bat is summed from its shape, a thin handle widening into a barrel, and the point where its grip is left alone is 17.1 centimetres from the end of the barrel — well past its centre of mass, and about where players are taught to hit. The reason it is past the centre of mass is that the blow must do two things at once: start the centre of mass moving, and start the bat turning about the pivot. Only a blow beyond the centre of mass can do both in the proportion that needs no help from the pivot.

There is a quicker way to see why the special point exists, using angular momentum. The pivot’s impulse acts at the pivot, so it has no moment about the pivot, and the quantity that survives a change of shape says the body’s angular momentum about that point must then come entirely from the blow: $I_P\omega = Jq$ whatever the pivot does. Linear momentum is different, because the pivot’s impulse contributes to it. The special point is where the angular momentum the blow must supply already carries, through the rigid connection between turning and moving, exactly the linear momentum the blow supplies — where the two conservation laws ask for the same thing and the pivot has nothing to add.

The same point, from a period #

The length nobody has to measure asks a different question of the same body. Hung from the pivot and allowed to swing, it has a period, and there is a simple pendulum — a point mass on a light string — with the same period. Its length is $I_P/Md$. The formula is identical, so the end of that equivalent pendulum, measured from the pivot, is the centre of percussion.

The drawing does not use the formula twice. The dots come from integrating the rod’s swing — the equation of motion of a body rotating about the pivot, stepped forward in time from a release at one degree — and measuring the period; the curve comes from the impulse balance. They sit on each other everywhere the pivot is placed, to six parts in a hundred thousand, the residual being the tiny lengthening of the period that a one-degree swing already has, which the pendulum, and the small lie that makes it simple computes.

The agreement is not an accident of the rod. For any body, both the period and the impulse balance are controlled by the same two numbers: the moment of inertia about the pivot, which resists turning, and the product of mass and distance to the centre of mass, which is what gravity acts on in the swing and what the pivot must accelerate in the blow. Whenever two questions reduce to the same ratio of the same two properties, they have the same answer, however different the questions look.

The drawing also shows how the point moves with the pivot. With the pivot at the rod’s end, it sits at two-thirds. As the pivot is moved inwards towards the centre of mass, the point moves outwards and passes the far end of the rod when the pivot is a sixth of the length from the middle. Past that, no blow anywhere on the rod leaves the pivot alone. At the centre of mass itself the swing has no period, because gravity exerts no torque, and every blow jolts the pivot, because a blow at any distance both turns and pushes and the pivot has to take the push.

Conjugate points #

The formula has a symmetry that Christiaan Huygens found in 1673 and that the length nobody has to measure uses to measure gravity. Write $h$ for the pivot’s distance from the centre of mass and $k$ for the radius of gyration, the distance at which the whole mass would have the same moment of inertia about the centre of mass. The centre of percussion lies on the far side of the centre of mass at a distance $k^2/h$. Swap them — make the old centre of percussion the new pivot — and the new centre of percussion is the old pivot. The two points are conjugate, and the product of their distances from the centre of mass is always $k^2$.

That is why a reversible pendulum can be adjusted until it has the same period about two knife-edges, at which point the distance between the edges is the length of the equivalent simple pendulum. Read as a statement about blows, it says the same thing: a body held at either of two conjugate points can be struck at the other without jarring the hand.

A body with no pivot #

A body struck in mid-air has no pivot to relieve, but it has a point that is momentarily still, and that point is where a hand could hold it without feeling the blow.

A free rod struck anywhere off its centre both moves and turns. The whole rod moves forward at $J/M$, and the rotation adds velocity on the side of the blow and subtracts it on the other. At one point on the far side of the centre the two cancel exactly, and just after the blow that point is not moving at all. For a blow at the end of the rod it is a third of the way along from the other end; for a blow nearer the centre it is further out, and a blow within $L/6$ of the centre leaves no point of the rod still. This is the same conjugacy with the roles read the other way: the still point is where a pivot would be for which the blow’s position is the centre of percussion.

The same still point turns up whenever something turns about a point that is not fixed. A pendulum given a sharp push at its bob turns about the point of suspension only if the push is at the conjugate of that point; pushed anywhere else, the suspension is jerked, which is one of the ways a swinging pendulum’s plane is disturbed — the reason Foucault pendulums, whose drift the ellipse a pendulum turns by itself shows is so easily contaminated, are started by burning through a thread rather than by a push.

The free rod is what a batter’s hands approximately see, because a swung bat is held loosely and the impact lasts about a millisecond — too short for the hands to supply much force during it. The point that keeps moving as if nothing had happened puts the centre of mass in charge of the translation; the conjugate still point is where the hands can be during the blow and feel the least of it.

Where four objects want to be struck #

The door is a uniform slab turning about its hinge line, so its centre of percussion is at two-thirds of its width: a doorstop sixty centimetres from the hinges of a ninety-centimetre door takes the whole of a slam and leaves the hinges alone, while a stop near the hinges levers the screws out of the frame. The hammer puts most of its mass in its head, and its centre of percussion moves out almost to the striking face; a bat’s lies in the barrel, short of the end. In each case the useful point is beyond the centre of mass, because a blow must turn the object about the hand as well as move it, and turning needs leverage.

The moment of inertia is what moves it. The mass, and where it sits shows that the same mass placed further from an axis resists turning more, and the same push, further out that a force applied further out turns more. The centre of percussion is where those two facts balance for a blow: far enough out that the blow’s leverage supplies all the turning, and no further.

Why a hammer is a head on a stick #

A handle with no head is struck most cleanly two-thirds of the way along. Move mass into a head at the far end and the centre of percussion follows it out, reaching nine-tenths of the length when the head carries sixty per cent of the mass and closing on the face as the handle becomes negligible. A hammer that is mostly head therefore strikes at its centre of percussion when it strikes with its face, and the blow returns nothing to the wrist. That is a large part of what makes a well-made hammer comfortable, and why a hammer with a heavy handle and a light head stings.

The same reasoning designs the weight distribution of axes, tennis rackets and cricket bats, with a complication each time. An axe must also cut, which favours mass near the edge for other reasons; a racket is strung, and the ball’s rebound depends on the strings and on the frame’s flexing as much as on the centre of percussion; a bat is not rigid.

What the blow delivers to the ball #

The centre of percussion answers what the hands feel, not what the ball does, and the two are worth keeping apart. For the ball, the bat at the point of impact behaves like a free mass whose size depends on where the ball lands: an effective mass $I_P/q^2$ for a bat turning about the hands, large near the hands where the bat is hard to turn and small near the tip where a light push turns it easily. Five balls, and the law that does not choose shows how much the outcome of a collision depends on the masses meeting; here the effective mass falls as the impact moves out while the speed of the bat at the impact point, $\omega q$, rises. The product that decides the ball’s exit speed peaks between them, a few centimetres inside the centre of percussion for a typical bat, and that is a third candidate for the sweet spot, alongside the rigid-body point and the node of the bat’s vibration.

The distinction matters for the design of the bat and not only for the batter. Moving mass towards the barrel raises the effective mass at the barrel and moves the centre of percussion out to meet it, and it also makes the bat harder to swing, since the moment of inertia about the hands rises with it. Every choice of weight distribution trades the comfort of the hands, the speed of the swing and the mass behind the impact against each other, and no distribution maximises all three.

Where the rigid-body picture stops #

Rigid bodies. The whole argument assumes the body moves as one piece during the blow. A real bat bends: the impact excites vibrations whose amplitude at the hands depends on where the ball lands relative to the nodes of the bat’s bending modes. The node of the lowest bending mode of a typical wooden bat lies close to the centre of percussion but not on it, and players’ “sweet spot” is a region spanning both. The sting that a badly hit ball delivers to the hands is mostly vibration, not the rigid-body jolt.

Short blows. The impulse must be over before the body has turned appreciably, so that its geometry during the blow is the geometry at the start. For a ball on a bat, a millisecond, that is well satisfied; for a slow push it is not, and the pivot’s reaction then depends on the whole history.

An ideal pivot. The pivot is a point that can supply any force and no torque. A hand is neither: it grips over several centimetres and can resist turning. The centre of percussion computed for a point at the grip is therefore a good guide to where the hand is spared and not a sharp line.

One dimension. Every body here is a line of mass struck perpendicular to it. A body struck off its line, or with its mass spread across as well as along, has a centre of percussion that depends on the direction of the blow.

The spike inside a single impulse #

The figures give the jolt as a single impulse and not as a force in time; they cannot show that the pivot’s reaction during a real blow is a brief spike whose peak depends on how stiff the contact and the handle are. The object figure places a centre of percussion on each object without showing its uncertainty, and for the bat and hammer the mass distributions are models — a tapered wooden cylinder and a uniform handle with a point head — chosen to be representative rather than any particular manufacturer’s. And none of the figures shows the vibration a real blow leaves ringing in a real bat, which is where most of the sting in the hands actually comes from.

Still open: what a batter’s hands actually feel #

The rigid-body centre of percussion, the node of the first bending mode and the point of maximum batted-ball speed are three different points on a real bat, a few centimetres apart, and which of them players experience as the sweet spot is argued. Measurements with instrumented bats and hands show that the vibrational sting correlates with the distance from the node more strongly than the rigid jolt does with the distance from the centre of percussion, and that players’ reports of a good hit track batted-ball speed, which is maximised slightly further in. How the hands’ own compliance changes the modes — whether a gripped bat vibrates as a free bar, as the calculation usually assumes, or as one clamped at the handle — is still being measured.

The habit worth carrying away is to reduce two questions to their ingredients before assuming they are different. A single blow and a slow swing seem to have nothing in common, but both are governed by the ratio of a body’s resistance to turning about a point to the leverage its mass gives about that point, and so they pick out the same place. Huygens found the point by timing pendulums; carpenters found it by where a hammer stops stinging; they were measuring one number.