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Chapter 4: Problem 41

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) Draw free-body diagrams of the ball during the pitch and just after it left the pitcher's hand.

Short Answer

Expert verified
The force exerted was approximately 153.41 N. The free-body diagrams show gravity throughout and added horizontal force during the pitch.

Step by step solution

01

Convert Mass to Kilograms

First, we need to convert the mass of the baseball from grams to kilograms since the standard unit of mass in physics is kilograms. The mass of the baseball is given as 145 grams. We use the conversion: \[ 1 \text{ gram} = 0.001 \text{ kilograms} \].Thus, the mass \( m \) of the baseball in kilograms is:\[ m = 145 \times 0.001 = 0.145 \text{ kg} \].
02

Use Kinematic Equation to Find Acceleration

The final velocity \( v_f \) of the baseball is given as 46 m/s and the initial velocity \( v_i \) is 0 m/s (starting from rest). The displacement \( s \) during which the force was exerted is 1.0 m. We use the kinematic equation: \[ v_f^2 = v_i^2 + 2a s \]Solving for acceleration \( a \) gives:\[ 46^2 = 0^2 + 2 \cdot a \cdot 1 \]\[ a = \frac{46^2}{2} = \frac{2116}{2} = 1058 \text{ m/s}^2 \].
03

Apply Newton's Second Law to Find Force

We now use Newton's second law, \( F = ma \), to find the force.We have the mass \( m = 0.145 \text{ kg} \) and acceleration \( a = 1058 \text{ m/s}^2 \).Therefore, the force \( F \) is:\[ F = 0.145 \times 1058 = 153.41 \text{ N} \].The pitcher exerted a force of approximately 153.41 N on the baseball.
04

Draw Free-Body Diagram During the Pitch

During the pitch, the following forces act on the baseball:- **Gravity (weight):** Acts downward with magnitude \( mg \), where \( g = 9.8 \, \text{m/s}^2 \).- **Applied Force:** Horizontal force exerted by the pitcher.The free-body diagram includes a downward arrow for gravity (weight) and a horizontal arrow in the direction of the pitch representing the applied force.
05

Draw Free-Body Diagram After the Ball Leaves

After the ball leaves the pitcher's hand, the only significant force acting on the baseball is gravity:- **Gravity (weight):** Continues to act downward with magnitude \( mg \).The free-body diagram shows only a downward arrow representing gravity, as no horizontal forces are acting after the release.

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Key Concepts

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

Kinematics
Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. In our baseball exercise, we use kinematics to determine how the speed of the ball changes over time. One of the key equations in kinematics is the equation of motion:\[ v_f^2 = v_i^2 + 2a s \]Here's a breakdown of each term:
  • **\( v_f \)** is the final velocity, which in this case is given as 46 m/s.
  • **\( v_i \)** is the initial velocity, typically starting from rest, hence 0 m/s.
  • **\( a \)** stands for acceleration, which we need to find.
  • **\( s \)** represents the displacement, equivalent to the distance over which force was applied, here 1.0 m.
By using this equation, we find the acceleration needed to reach the final speed. Understanding this concept is crucial because knowing the object's acceleration helps us determine other related physical quantities.
Force Calculation
Force calculation involves determining the amount of push or pull exerted on an object. In the context of this exercise, we calculate the force the pitcher applied to the baseball using the formula:\[ F = ma \]Here:
  • **\( F \)** denotes force, measured in newtons (N).
  • **\( m \)** is the mass of the object, which we first converted to kilograms: 0.145 kg.
  • **\( a \)** is acceleration, calculated using the kinematics principles, 1058 m/s虏 in our example.
By substituting the mass and the calculated acceleration into this formula, we determined the magnitude of the force to be approximately 153.41 N. Understanding how to calculate force is essential in many areas of physics and engineering, as it allows us to design systems and predict object behavior under various conditions.
Newton's Second Law
Newton's Second Law of Motion is pivotal in understanding how objects move. This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In equation form, it looks like:\[ F = ma \]This formula highlights:
  • The relationship between force, mass, and acceleration.
  • That force is what causes an object to accelerate.
  • The significance of mass in determining how much acceleration will result from a given force.
In our baseball scenario, applying this law allowed us to determine the force exerted by the pitcher. The bigger the force applied, the greater the acceleration, assuming mass stays constant. Recognizing the importance of this law helps in designing and understanding numerous mechanical applications, from sports to vehicle dynamics.
Free-Body Diagrams
Free-body diagrams are crucial tools in physics, used to depict the forces acting on an object. During the pitch, the baseball experiences different forces, which we can represent in two separate diagrams. This visual representation helps understand the influence of each force without distractions from other elements.**During the Pitch:**
  • The baseball feels:
    • **Gravity:** Pulls the ball downward with force magnitude of \( mg \), where \( g = 9.8 \text{ m/s}^2 \).
    • **Applied Force:** Directed horizontally, showing the pitcher's influence.
**After the Release:**
  • Only gravity continues to act.
    • There's a single downward arrow for weight, as horizontal influences cease post-pitch.
By labeling forces clearly in a free-body diagram, we develop a better understanding of each force's role, especially useful in problems involving multiple forces or when analyzing motion.

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Most popular questions from this chapter

You walk into an elevator, step onto a scale, and push the "up" button. You also recall that your normal weight is 625 \(\mathrm{N}\) . Start answering each of the following questions by drawing a free-body diagram. (a) If the elevator has an acceleration of magnitude \(2.50 \mathrm{m} / \mathrm{s}^{2},\) what does the scale read? (b) If you start holding a 3.85 -kg package by a light vertical string, what will be the tension in this string once the elevator begins accelerating?

An oil tanker's engines have broken down, and the wind is blowing the tanker straight toward a reef at a constant speed of 1.5 \(\mathrm{m} / \mathrm{s}(\) Fig. \(\mathrm{P} 4.38) .\) When the tanker is 500 \(\mathrm{m}\) from the reef, the wind dies down just as the engineer gets the engines going again. The rudder is stuck, so the only choice is to try to accelerate straight backward away from the reef. The mass of the tanker and cargo is \(3.6 \times 10^{7} \mathrm{kg}\) , and the engines produce a net horizontal force of \(8.0 \times 10^{4} \mathrm{N}\) on the tanker. Will the ship hit the reef? If it does, will the oil be sate? The hull can withstand an impact at a speed of 0.2 \(\mathrm{m} / \mathrm{s}\) or less. You can ignore the retarding force of the water on the tanker's hull.

An advertisement claims that a particular automobile can "stop on a dime." What net force would actually be necessary to stop a \(850-\) -kg automobile traveling initially at 45.0 \(\mathrm{km} / \mathrm{h}\) in a distance equal to the diameter of a dime, which is 1.8 \(\mathrm{cm} ?\)

A large box containing your new computer sits on the bed of your pickup truck. You are stopped at a red light. The light turns green and you stomp on the gas and the truck accelerates. To your horror, the box starts to slide toward the back of the truck. Draw clearly labeled free-body diagrams for the truck and for the box. Indicate pairs of forces, if any, that are third-law action- reaction pairs. (The bed of the truck is not friction less.)

A chair of mass 12.0 kg is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force \(F=40.0 \mathrm{N}\) that is directed at an angle of \(37.0^{\circ}\) below the horizontal and the chair slides along the floor. (a) Draw a clearly labeled free-body diagram for the chair. (b) Use your diagram and Newton's laws to calculate the normal force that the floor exerts on the chair.
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