A softball player swings a bat, accelerating it from rest to 2.4 rev/s in a time of 0.20 s. Approximate the bat as a 0.90-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.

Verified step by step guidance

1. Step 1: Identify the given values and the quantities to calculate. The angular velocity \( \omega_f \) is 2.4 rev/s (convert to radians per second: \( \omega_f = 2.4 \times 2\pi \, \text{rad/s} \)), the initial angular velocity \( \omega_i \) is 0 rad/s, the time \( t \) is 0.20 s, the mass \( m \) of the bat is 0.90 kg, and the length \( L \) of the bat is 0.95 m. We need to calculate the torque \( \tau \).
2. Step 2: Calculate the angular acceleration \( \alpha \) using the kinematic equation for rotational motion: \( \alpha = \frac{\omega_f - \omega_i}{t} \). Substitute the known values of \( \omega_f \), \( \omega_i \), and \( t \) to find \( \alpha \).
3. Step 3: Determine the moment of inertia \( I \) of the bat. Since the bat is approximated as a uniform rod rotating about one end, the moment of inertia is given by \( I = \frac{1}{3} m L^2 \). Substitute the values of \( m \) and \( L \) to calculate \( I \).
4. Step 4: Use the rotational form of Newton's second law, \( \tau = I \alpha \), to calculate the torque. Substitute the values of \( I \) and \( \alpha \) into this equation to find \( \tau \).
5. Step 5: Ensure the units are consistent throughout the calculations (e.g., radians for angular velocity and acceleration) and verify the result for correctness.

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (lever arm). It is expressed in Newton-meters (Nm) and determines how effectively a force can cause an object to rotate about an axis. In this scenario, the torque applied by the player to the bat is crucial for understanding how the bat accelerates.

Moment of Inertia

The moment of inertia is a property of a body that quantifies its resistance to rotational motion about an axis. For a uniform rod, it is calculated using the formula I = (1/3)ml², where m is the mass and l is the length of the rod. This concept is essential for determining how much torque is needed to achieve a certain angular acceleration, as it directly influences the relationship between torque, angular acceleration, and moment of inertia.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically measured in radians per second squared (rad/s²). It indicates how quickly an object is speeding up or slowing down its rotation. In this question, calculating the angular acceleration of the bat is necessary to find the torque applied, as it relates to the moment of inertia and the net torque through the equation τ = Iα, where τ is torque, I is moment of inertia, and α is angular acceleration.