91Ó°ÊÓ

When dealing with rotational motion, the moment of inertia plays a role similar to that of mass in linear motion. It measures how much an object resists changes to its rotational motion. For a uniform rod rotating about one of its ends, the moment of inertia is determined using the formula:- \[ I = \frac{1}{3} m L^2 \]
Where:

• \(m\) is the mass of the rod.
• \(L\) is the length of the rod.

In this problem, the bat is approximated as a uniform rod with a mass of 2.2 kg and a length of 0.95 m. This approximation allows us to plug these values into the formula. A higher moment of inertia means that more torque is needed to change its rotational speed. Calculating this important parameter is the first step to understanding the rotational dynamics of the bat.

Angular acceleration describes how quickly the rotational speed of an object is changing. It is similar to linear acceleration but applied to rotational motion. To find angular acceleration, we use the change in angular velocity over time:- \[ \alpha = \frac{\omega_f - \omega_i}{t} \]
Where:

• \(\omega_f\) is the final angular velocity.
• \(\omega_i\) is the initial angular velocity (often zero if it starts from rest).
• \(t\) is the time over which the change occurs.

In this exercise, the softball bat starts from rest and reaches a final rotational speed of 2.7 revolutions per second, which we convert to radians per second. With these values, we determine how fast the rotational velocity is increasing, which is crucial for calculating torque.

A uniform rod is a simple model used to simplify complex real-world objects like a baseball bat. By assuming the mass is distributed evenly throughout its length, calculations like the moment of inertia become straightforward. This simplification allows us to apply standard physics formulas effectively.

For example, when we model a bat as a uniform rod, we assume it has consistent properties that make it easier to analyze. Although an actual bat may not be perfectly uniform, this approximation is useful for theoretical calculations and provides a good estimate of the physical behavior.