Question

part a

a baseball bat can be rotated around many different axes of rotation. three such possibilities are shown in figure 1. rank the baseball bats moment of inertia about each of these three axes of rotation.

rank the moment of inertia from largest to smallest and overlap axes labels if the same.

Explanation:

Analyze mass distribution relative to axis C

Using the Moment of Inertia knowledge point

$$ I = \int r^2 dm $$

For axis \(C\) (the longitudinal axis running through the center of the bat), the distance \(r\) of every mass element from the axis is extremely small (on the order of the bat's radius, a few centimeters). Thus, \(I_C\) is the smallest.

Locate the center of mass

Using the Center of Mass Calculation knowledge point

$$ x_{\text{cm}} = \frac{\int x \, dm}{M} $$

Because a baseball bat is thicker and heavier at the barrel end (left side) than at the handle end (right side), the center of mass (\(\text{cm}\)) lies to the left of the geometric midpoint, closer to axis \(A\).

Compare axes A and B using parallel axis theorem

Using the Moment of Inertia and Center of Mass Calculation knowledge points

$$ I = I_{\text{cm}} + M d^2 $$

Let \(d_A\) be the distance from the center of mass to axis \(A\), and \(d_B\) be the distance from the center of mass to axis \(B\). Since the center of mass is located closer to the barrel (near axis \(A\)), we have:

$$ d_A < d_B \implies I_A < I_B $$

Rank the moments of inertia

Using the Moment of Inertia knowledge point

$$ I_B > I_A > I_C $$

Answer:

• Largest: B, Middle: A, Smallest: C