Bat Sweet Spot Location Calculation

[schaafde]

If I have a bat speed of 74mph, how do I figure out how much angular speed that has? I cannot find a conversion or an equation anywhere.

 

[issacnewton]

To find the angular velocity, you need to have a reference point so that you can measure rate of change of the angular displacement . Can you type the question as originally specified ?

 

[schaafde]

It isn't from a question. I am conducting research on the sweet spot of a baseball bat. I am trying to determine angular momentum for the bat. I have taken the average bat speed of a college baseball player but I need to translate that into angular velocity.

 

[issacnewton]

ok, if the bat is rotating about the axis passing through its one end, then the relation between the angular velocity and the tangential velocity of any point on the bat is given by

$$v=r\omega$$

where r is the distance of the point on the bat from your hand. once you know the angular velocity, you can find the angular momentum from

$$L=I\omega$$

where I is the moment of inertia about the axis passing through its end (where you are holding the bat). The bat can be modeled as a thin rod of length x and mass m, then I is given by

$$I=\frac{1}{3}mx^2$$

I hope this helps

 

[schaafde]

In that equation for angular velocity, would the angular velocity be measured in radians per second or RPM or some other unit?

 

[Hootenanny]

In that equation for angular velocity, would the angular velocity be measured in radians per second or RPM or some other unit?

The units of angular frequency will depend on which units you use for your other variables. The SI units of angular frequency are radians per second. So if you use SI units for all your other quantities, your angular frequency will be in radians per second.

 

[AndyNewton]

Defining a "sweet spot" may have various possible meanings, given that the idea involves some aspects of what "feels good". Bearing that in mind, there is one mechanical definition that can be dealt with fairly easily by mathematics and also by a simple experiment.

The simple mechanical definition I will describe below considers the bat as a rigid mass, with one point where the ball impacts the bat, and one point where the player's hands are holding the bat. Somewhere between those two points will be centre of gravity of the bat. Impact with the ball causes two superimposed movements i) rotation about the bat's centre of gravity and ii) linear translation of the entire bat.

Movement i) causes the handle of the bat to jerk in the opposite direction to the original velocity of the ball. Movement ii) causes the handle of the bat to jerk in the same direction as the original velocity of the ball. The idea in this theory is that if these two jerking movements cancel out then the player will feel that the bat is "sweet" since there will be little shock transmitted into the players hands when striking the ball. This "sweet spot" is actually a combination of both impact spot and hand grip position.

This idea of a sweet spost can be calculated as below, or found by a neat pendulum experiment. It makes no difference how fast the bat is swung in this definition of a sweet spot. The behaviour remains the same for any impact speed.

Calculating the sweet spot:

Let impact point be distance x from the bat's centre of gravity.
Let the hand grip position be distance y from the bat's centre of gravity.
Impact point ---- centre of gravity ----- hand grip, are all in a straight line with the CofG between the other two.

During impact with ball, at any instant, there is some force F acting on the bat at the impact point.

Linear acceleration of the entire bat = F / mass of bat
The entire bat is subject to this acceleration, including the part of the bat at the handle position.

Angular acceleration of the entire bat = F.x / I
where F.x is the turning moment of the force = Force times distance from CofG
and I is the moment of inertia of the bat.

The angular acceleration of the bat gives rise to a linear acceleration at the handle position = Angular acceleration . y

If the two accelerations at the handle position are equal and opposite they can cancel out.
For that to occur we must have,

F.x.y/I = F/m

Which simplifies to give

I = m.x.y

where I is the moment of inertia of the bat
m is the mass of the bat
x is the distance from CofG to impact point
y is the distance from CofG to hand grip position.

To use the equation I = m.x.y to find sweet spot combinations of x and y you need to know m and I.
m is easy enough - just weigh the bat.
I is more difficult. You need to find a way to measure or calculate the moment of inertia of the bat.



Alternative method: Measuring sweet spot by experiment:

The equation I = m.x.y has another interesting implication.
If the bat is suspended as a pendulum, pivoting about point x, or inverted and pivoting about point y, then the period of swing of the pendulum will be the same in both cases.
Many years ago, when I encountered this theory in a dynamics textbook, I recall that the two points at x and y were called "conjugate points" of a pendulum. Google searching "conjugate points" today shows many other current meanings.

This gives a simple experimental method of determining sweet spot.
Suspend the bat betweet two pin points pushing into the hand grip area so it can swing as a pendulum, and count the number of swings of the pendulum in one minute say.
Then invert the bat, placing the pin pivot in the striking area of the bat and time the swings again. Move the pivot point up and down the bat until the swing period is equal to the original measurement made when pivoted at the hand grip... and that is the theoretical sweet spot.

 

[schaafde]

So, I would just take those calculations to where F/m is the greatest? I'm not seeing exactly how the sweet spot came into calculation.

 

[AndyNewton]

The calculation in message #7 above assumes an objective for the idea of a "sweet spot" as being any combination of the ball impact point and hand grip location such that the impact between the ball and the bat causes zero instantaneous acceleration to be felt at the hand grip.

The mathematical part, based on that assumption, starts by considering some unknown force value F, at the ball/bat contact point. When the mathematical steps have been worked through we find that the value of F cancels-out leaving the required condition as simply I = m.x.y

Therefore to achieve the desired aim - with, say two pieces of tape stuck onto the bat marked "hold bat exactly here" and "try to make the ball hit the bat exactly here" - those two pieces of tape must be at distances x and y from the centre of gravity of the bat where x and y satisfy the equation I = m.x.y

Finding values to satisfy I = m.x.y ...

m is a fixed property of the bat - and is easy to measure.

I is another fixed property of the bat - difficult to measure but can be calculated by some approximations for simple bat shapes. (Combinations of cylinders, cones, etc)

y (distance from CofG to handgrip) can be whatever you want if you can move the handgrip point. Or it can be assumed to be another fixed property of the bat if, say, you want to grip the bat with the hands touching a ridge at the very end of the handle section. In either case you need to locate the CofG to find the value of y.

... leaving only the variable "x" (distance from CofG to ball impact point). Variable x is calculated by x = I / (m.y), and the location is marked on the bat at distance x from the CofG.

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I collected these bits of theory (as message #7) years ago when I was making sticks for playing my hammer-dulcimer. A hammer-dulcimer stick is similar in overall shape to a baseball bat, but only a small fraction of the size. A curved surface at the striking end is equivalent to the ball impact point. Finger grooves can be carved in the hand end to define the grip position.

By using the theory calculation, and the pendulum timing experimental method, I found significant improvements in "sweetness" when the sticks were used.
For some sticks I simply moved my grip point a little way nearer to the centre of the stick to achieve the sweet-spot combination of striking point and handgrip point. For other sticks a small lead weight embedded inside the handle end gave the same dynamic behaviour without leaving such a long tail-end projecting backwards into my hand area.

For a hammer-dulcimer stick it is not possible to hold the stick at the extreme end because the sweet spot impact point then moves very close to the stick CofG, leaving an excessive length of stick extending beyond the string contact point. That extra length would tangle with the next set of strings on the dulcimer. In the case of a baseball bat that string tangle problem does not apply so you should be able to select any handgrip location, including the extreme end of the bat, and then find a usable matching ball-contact sweetspot.