Physics and Acoustics of Baseball & Softball Bats Daniel A. Russell, Ph.D. Applied Physics, Kettering University, Flint, MI 48504-4898 The contents of this page are ©2008 Daniel A. Russell

Today is The contents of this page were last modified on September 4, 2008

# Moment-of-Inertia of a Baseball or Softball Bat

 Inertia is a measure of how difficult it is to change the velocity of an object by applying a force, and is usually expressed in terms of mass. The greater the inertia of an object (i.e., the more mass an object has), the more difficult it is to change its velocity. Mass is different from weight though the two terms are often used interchangeably. [Since I am a physics professor I feel obligated to explain the difference between the two before moving on to discuss baseball and softball bats. Mass is a property of an object; a given object will have the same mass regardless of where in the universe it happens to be located. A bowling ball has the exactly same mass on earth as it does on the moon, or somewhere out in space where it is free from the pull of gravity. Trying to stop a moving bowling ball in space is just as difficult as it is on earth. Weight is a force, specifically the force that an object experiences when placed in a gravitational field. The strength of the gravitational field depends on the mass of the object creating the field as well as on the square of the distance from that object. The gravitational field of the moon is only about 1/6 that of the earth. So, while a bowling ball has the same mass on earth as it does on the moon (and thus its velocity would be just as difficult to change in both locations) the weight of the ball on the moon would be 1/6 of its weight on earth.] Rotational inertia, or the moment-of-inertia (MOI) is a measure of how difficult it is to change the rotational velocity of an object which is rotating about a pivot point. The larger the moment-of-inertia, the more difficult it is to change the rotational speed of the object. The value of the MOI depends on the total mass of the object as well as the way in which that mass is distributed about the pivot point. Consider first a point mass m (a small ball) which is rotating parallel to the ground at a distance x from a fixed pivot point, as is shown in the image at top right. The MOI of the point mass is the product of the mass and the square of the distance from the pivot point: Single point mass rotating at a fixed distance from a pivot point. If there are several point objects all rotating about the same pivot point, but at different distances from the pivot, then the total moment-of-inertia of the system is just the sum of the individual moments-of-inertia. Multiple point masses rotating at a various distances from a pivot point. In our case the object which is rotating is a bat, with mass distributed along its entire length. In order to solve this more difficult problem, we pretend that the bat is made up of very large number of very small slices, each with its own incremental mass dm and each being located at a distance x from the pivot point. If we knew how the mass dm varied with position we could add all the individual moments-of-inertia together to obtain the total inertia of the bat. However, very few people actually have the mass distribution function dm for a given baseball or softball bat at their fingertips, so calculating the moment-of-inertia for a bat isn't very practical. It turns out that the moment-of-inertia is much easier to measure than it is to calculate, so I won't attempt to explain how we would solve for the inertia, but instead will describe how it is measured and what it means. Extended object (bat) pivoting about a fixed axis may be broken up into many small pieces with varying mass and varying distance from the pivot point.

### Weight and Balance Point

To measure the weight of the bat you could just put it on a scale and record the weight directly. To find the balance point, you could use a knife edge and find the location where the bat exactly balances. This is a bit tricky but it works pretty well. Or you could follow the procedure for measuring weight and balance point as spelled out in ASTM Standard F2398-04 "Test Method for Measuring Moment of Inertia and Center of Percussion of a Baseball of Softball Bat" which is what bat manufacturers and governing associations do. This test standard requires the use of two scales, one which supports the bat at a point 6-inches from the end of the knob, and the other which supports the barrel of the bat at a distance of 24-inches from the knob. The weight of the bat is just the sum of the two scale readings
W = W6 + W24
and the balance point is calculated from
BP = (6*W6 + 24*W24) / (W6 + W24)

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