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Chapter 9: Problem 6

A curve ball is a type of pitch in which the baseball spins on its axis as it heads for home plate. If a curve ball is thrown at \(35.8 \mathrm{~m} / \mathrm{s}\) (80 mph) with a spin rate of 30 rev \(/\) s, how many revolutions does it complete before reaching home plate? Assume that home plate is \(18.3 \mathrm{~m}(60 \mathrm{ft})\) from the pitching mound and that the baseball travels at a constant velocity.

Short Answer

Expert verified
The baseball completes about 15.3 revolutions before reaching home plate.

Step by step solution

01

Determine the time of flight

First, we find the time it takes for the baseball to reach home plate. Use the formula for time, which is equal to the distance divided by speed. Given that the distance to home plate is \(18.3 \, \text{m}\) and the speed of the baseball is \(35.8 \, \text{m/s}\), we can calculate the time as follows:\[t = \frac{d}{v} = \frac{18.3}{35.8} \]Solving this gives the time in seconds that the baseball takes to reach home plate.
02

Calculate the number of revolutions

Now, we calculate how many revolutions the baseball makes in the time computed in Step 1. The baseball has a spin rate of 30 revolutions per second. Thus, the number of revolutions is equal to the spin rate multiplied by the time:\[\text{Number of revolutions} = \text{Spin rate} \times t = 30 \times t\]This will yield the total number of revolutions during the flight to home plate.

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Key Concepts

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

Projectile Motion
When a baseball is pitched, it follows a curvilinear path known as projectile motion. Projectile motion occurs when an object is thrown into the air and influenced only by gravity and its initial velocity.
For a curveball, despite not being primarily about parabolic paths, understanding projectile principles helps us appreciate the overall movement of the ball.
  • The horizontal motion is constant because it moves at a constant speed.
  • The vertical motion, while negligible for short distances like a pitch, typically involves downward acceleration due to gravity.
Baseball pitches, such as the curveball, demonstrate projectile principles, emphasizing that even spins involve these foundational physics concepts.
Rotational Motion
Rotational motion refers to the spinning of an object around an axis. When considering a baseball pitch like a curveball, the rotation significantly affects its path. Each section of a spinning ball has different speeds and accelerates around the axis.
This rotational motion influences the ball's flight since:
  • Spinning creates an aerodynamic force known as the Magnus effect.
  • This effect can cause the ball to curve, dip, or rise unexpectedly.
Thus, rotational motion is crucial in understanding why pitches like curveballs can deceive batters. Recognizing these motions helps players and analysts better predict pitch behaviors.
Angular Velocity
Angular velocity measures how fast an object spins around an axis. Unlike linear velocity, which refers to movement in a straight line, angular velocity relates to circular motion.
In the context of baseball physics, a spinning ball's angular velocity can determine its effectiveness in pitching strategies.
  • It is represented in revolutions per second (rev/s) or radians per second (rad/s).
  • In our example, the curveball's angular velocity is 30 rev/s.
  • Understanding the angular velocity can help pitchers optimize spin to manipulate the ball's path.
The precision in spin control, determined by angular velocity, can make significant differences in pitch outcomes.
Baseball Physics
Baseball physics combines various principles to explain the movement and effectiveness of pitches. From projectile and rotational motions to angular velocity, the simple act of throwing a baseball requires a deep understanding of physics. Players must master these concepts to enhance their skills and gain a competitive edge.
  • The constant velocity and rotational spin define pitch types.
  • Pitchers rely on precise calculations to control a ball's path to the plate.
  • For example, calculating revolutions gives insight into how much the ball may curve.
By merging these physics principles, players can significantly improve their understanding and execution of effective baseball pitches.

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Most popular questions from this chapter

In redesigning a piece of equipment, you need to replace a solid spherical steel part with a similar steel part that has half the radius. How does the moment of inertia of the new part compare to that of the old? Express your answer as a ratio \(I_{\text {new }} / I_{\text {old }}\)

A wheel rotates with a constant angular velocity of \(6.00 \mathrm{rad} / \mathrm{s}\) (a) Compute the radial acceleration of a point \(0.500 \mathrm{~m}\) from the axis, using the relation \(a_{\mathrm{rad}}=\omega^{2} r .(\mathrm{b})\) Find the tangential speed of the point, and compute its radial acceleration from the relation \(a_{\mathrm{rad}}=v^{2} / r\)

A solid uniform marble and a block of ice, each with the same mass, start from rest at the same height \(H\) above the bottom of a hill and move down it. The marble rolls without slipping, but the ice slides without friction. (a) Find the speed of each of these objects when it reaches the bottom of the hill. (b) Which object is moving faster at the bottom, the ice or the marble? (c) Which object has more kinetic energy at the bottom, the ice or the marble?

What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface? (a) a uniform solid cylinder; (b) a uniform sphere; (c) a thin-walled, hollow sphere; (d) a hollow cylinder with outer radius \(R\) and inner radius \(R / 2\).

A solid copper disk has a radius of \(0.2 \mathrm{~m},\) a thickness of \(0.015 \mathrm{~m}\), and a mass of \(17 \mathrm{~kg}\). (a) What is the moment of inertia of the disk about a perpendicular axis through its center? (b) If the copper disk were melted down and re-formed into a solid sphere, what would its moment of inertia be?
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