91Ó°ÊÓ

radial and tangential components) just before it is released.One method of pitching a softball is called the "windmill" delivery method, in which the pitcher's arm rotates through approximately \(360^{\circ}\) in a vertical plane before the 198-gram ball is released at the lowest point of the circular motion. An experienced pitcher can throw a ball with a speed of \(98.0 \mathrm{mi} / \mathrm{h}\). Assume the angular acceleration is uniform throughout the pitching motion and take the distance between the softball and the shoulder joint to be \(74.2 \mathrm{~cm}\). (a) Determine the angular speed of the arm in rev/s at the instant of release. (b) Find the value of the angular acceleration in \(\mathrm{rev} / \mathrm{s}^{2}\) and the radial and tangential acceleration of the ball just before it is released. (c) Determine the force exerted on the ball by the pitcher's hand (both

Step by step solution

01

Convert the ball's speed

First, let's convert the speed from miles per hour (mi/h) to meters per second (m/s), the unit more commonly used in physics. 1 mile is approximately 1609 meters and 1 hour is 3600 seconds, therefore, \(98.0 \ \mathrm{mi/h} \approx 43.7 \ \mathrm{m/s}\).

02

Calculate the angular speed

Now, calculate the angular speed, which is the speed divided by the radius of the circle. In this case, the radius is the distance from the shoulder joint to the ball, which is given as 74.2 cm or 0.742 m. Hence, the angular speed \( \omega \) is \( \omega = \frac{43.7}{0.742} \ \mathrm{rad/s} \approx 59 \ \mathrm{rad/s}\). Remember that the angular speed is usually expressed in rad/s for SI consistency.

03

Calculate the angular acceleration

The angular acceleration is the change in angular speed per unit time. Since the arm starts at rest and takes a whole rotation (\(2\pi \ rad\)) to reach the final angular speed, and given that the angular acceleration is assumed to be uniform, the time of one revolution \(t\) can be expressed as \( t = \frac{2 \pi}{\omega} \). Thus, the angular acceleration \( \alpha \) is \( \alpha = \frac{\omega}{t} = \frac{59}{t} \ rad/s^2\).

04

Calculate the radial and tangential accelerations

The radial (centripetal) acceleration \(a_r\) is given by \(a_r = \omega^2 * r = {(59)^2 * 0.742} \ m/s^2\). The tangential acceleration \(a_t\) is \( \alpha * r = \frac{59}{t} * 0.742 \ m/s^2\).

05

Calculate the force exerted on the ball

The last step is to calculate the force exerted on the ball by the pitcher. This can be expressed as the sum of the radial and tangential forces (\(F = F_r + F_t\)), which are the product of the mass of the ball and the respective acceleration (remember that \(F = m * a\)). Hence, \(F = m * a_r + m * a_t\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Speed

Angular speed is a key concept in understanding how fast an object rotates around a fixed point. In the case of a softball pitcher using the "windmill" technique, the arm moves in a circular path. The angular speed, denoted by \( \omega \), is calculated by dividing the linear speed of the ball by the radius of the circular path. This provides a measure of how quickly the arm is rotating in radians per second (rad/s).

• To calculate angular speed, use the formula \( \omega = \frac{v}{r} \), where \( v \) is the linear speed and \( r \) is the radius.

Angular speed helps in predicting the velocity at which the ball will be released, crucial for knowing how fast it will travel once it leaves the pitcher's hand.

Angular Acceleration

Angular acceleration, represented by \( \alpha \), is the rate at which the angular speed changes with respect to time. In our exercise, the pitcher's arm starts from rest, meaning the initial angular speed is zero. As the arm completes the circular path, it reaches its final angular speed at the point of release.

• To find angular acceleration, you need to know the final angular speed \( \omega_f \) and the time taken to reach that speed \( t \): \( \alpha = \frac{\omega_f}{t} \).
• The time \( t \) for one full revolution can be calculated by \( t = \frac{2\pi}{\omega_f} \), given uniform angular acceleration.

Understanding angular acceleration is essential for determining how the pitcher's arm increases its speed uniformly as it approaches the release point.

Radial and Tangential Acceleration

In rotational motion, like the pitcher's arm, we examine two types of accelerations: radial and tangential. These accelerations provide a complete picture of the forces at play just before the ball is released.

• Radial acceleration \( a_r \) is also known as centripetal acceleration, aimed at keeping the object in a circular path. It's given by \( a_r = \omega^2 \times r \), where \( \omega \) is the angular speed and \( r \) is the radius.
• Tangential acceleration \( a_t \) is responsible for the change in speed along the path and is calculated with \( a_t = \alpha \times r \), where \( \alpha \) is the angular acceleration.

Both types are crucial for determining how the ball achieves the necessary speed and direction when it leaves the pitcher's hand.

Centripetal Force

Centripetal force is vital for maintaining an object in a circular path and is directly related to radial acceleration. For a softball undergoing a windmill pitch, centripetal force ensures that it travels in the intended circular motion before release.

• Calculate centripetal force \( F_c \) using the formula \( F_c = m \times a_r \), where \( m \) is the mass of the ball and \( a_r \) is the radial acceleration.

Centripetal force is exerted by the pitcher's hand on the ball, allowing it to travel smoothly in a circle. Once the ball is released, this force ceases to act, and the ball follows a linear path determined by its velocity at the point of release.