Problem 42 - College Physics

Question

Human biomechanics. The fastest pitched baseball was measured at 46 \(\mathrm{m} / \mathrm{s}\) . Typically, a baseball has a mass of 145 \(\mathrm{g}\) . If the pitcher exerted his force (assumed to be horizontal and constant) over a distance of \(1.0 \mathrm{m},(\mathrm{a})\) what force did he produce on the ball during this record-setting pitch? (b) Make free-body diagrams of the ball during the pitch and just after it has left the pitcher's hand.

Step-by-Step Solution

Answer

The force produced was 153.41 N. During the pitch, the ball experienced a horizontal force and gravity; after the pitch, only gravity acts.

Step 1: Convert the mass into kilograms

The mass of the baseball needs to be in kilograms for standard metric calculations. Given the mass is \(145 \text{ g}\), we convert it into kilograms by using the conversion \(1 \text{ kg} = 1000 \text{ g}\). Thus, the mass in kilograms is \(\frac{145}{1000} = 0.145 \text{ kg}\).

Step 2: Use the work-energy principle

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. The initial kinetic energy is zero since the ball starts from rest, and the final kinetic energy is \(\frac{1}{2} mv^2\), where \(m\) is the mass and \(v\) is the velocity of the ball. Thus, the work done on the ball is \( W = \frac{1}{2} \times 0.145 \times (46)^2 \). Calculate this to find the work done.

Step 3: Calculate the work done on the ball

Work, \(W\), is calculated as \[ W = \frac{1}{2} \times 0.145 \times (46)^2 = \frac{1}{2} \times 0.145 \times 2116 = 153.41 \text{ J} \].

Step 4: Calculate the force using the work formula

The work done is also equal to the force multiplied by the distance over which it is applied (\(W = F \cdot d\)). We can rearrange this to find the force \(F = \frac{W}{d}\). Since the distance \(d\) is \(1.0 \text{ m}\), \( F = \frac{153.41}{1.0} = 153.41 \text{ N}\).

Step 5: Draw the free-body diagrams

During the pitch, the ball experiences a horizontal force from the pitcher and gravitational force downward. After it leaves the pitcher's hand, only gravity acts on the ball. 1. **During the pitch:** - Horizontal force \(F\) towards the pitching target. - Gravitational force \(mg\) acting downward.2. **After leaving the hand:** - Only gravitational force \(mg\) acting downward.

Key Concepts

Understanding the Work-Energy Principle

The work-energy principle is a fundamental concept in physics, especially useful in human biomechanics to understand how forces interact with motion. It states that the work done on an object is equal to the change in its kinetic energy. This principle helps us calculate how much force is needed to bring an object to a certain speed over a specific distance.

In this scenario, when the baseball is thrown, it initially starts from rest. The initial kinetic energy is zero since its velocity is zero. The pitcher applies a force, increasing the ball's velocity to reach the measured speed of 46 m/s. Thus, the change in kinetic energy is the energy the ball has after reaching this speed, given by the formula \[ KE = \frac{1}{2} mv^2 \]where \( m \) is the mass of the ball and \( v \) is the velocity.

• Initial kinetic energy: 0 J (ball at rest)
• Final kinetic energy: 153.41 J (calculated after pitch)

Calculating this allows engineers and biomechanists to understand the energetic demands of pitches and improves techniques in sports and rehabilitation.

The Role of Free-Body Diagrams

Free-body diagrams are crucial for visualizing forces acting on an object. In biomechanics, they help us understand how different forces influence motion. When drawing a free-body diagram, you include all forces acting on the object, which helps in analyzing the motion or equilibrium of the object.

For the pitched baseball:

• **During the pitch:** Include the horizontal force from the pitcher's hand and the downward gravitational force.
• **After it leaves the hand:** Only gravitational force acts, as other forces cease.

These diagrams simplify the complex interactions into manageable visualizations, crucial for problem-solving in mechanics. By translating a physical situation into a diagram, it becomes easier to apply physics principles like Newton's laws to predict motion or calculate forces involved.

Exploring Kinetic Energy

Kinetic energy is the energy of motion. In the context of human biomechanics, understanding kinetic energy helps in analyzing how movement is produced and how it can be optimized. For an object like a baseball, kinetic energy is given by the formula \[ KE = \frac{1}{2} mv^2 \]where \( m \) is the object's mass and \( v \) is its velocity.

In this exercise, once the pitch is made, the kinetic energy represents how much energy the ball carries as it moves through the air. Increasing the velocity or the mass of the object results in a significant increase in kinetic energy, showing why pitchers focus on both strength and speed. This concept not only explains how hard a baseball can be thrown but also has broader applications in improving athletic performance across sports.

Calculating Force in Human Biomechanics

Force calculation is a core part of biomechanics, showing how different forces affect the motion of a body or object. Using the work-energy principle, we see that force is related to work done and distance, as expressed in the formula
\[ F = \frac{W}{d} \]where \( W \) is the work done and \( d \) is the distance over which the force is applied.

• In this exercise, the work done on the baseball by the pitcher is determined to be 153.41 J.
• Since the distance is 1.0 m, applying the formula gives a force of 153.41 N.

Calculating force using these principles demonstrates the mechanics behind pitching. This not only aids budding athletes in understanding the physical requirements of their sport but also assists in designing training regimens that optimize force application, enhancing athletic performance while minimizing injury risk.