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Chapter 9: Problem 120

For many years, hitters have claimed that some baseball pitchers have the ability to actually throw a rising fastball. Assuming that a top major leaguer pitcher can throw a 95 -mph pitch and impart an 1800 -rpm spin to the ball, is it possible for the ball to actually rise? Assume the baseball diameter is 2.9 in. and its weight is 5.25 oz.

Short Answer

Expert verified
No, it is not possible for a top major leaguer pitcher to throw a rising fastball with a speed of 95 mph and a spin of 1800 rpm given the Magnus force is less than the gravitational force.

Step by step solution

01

Identify the relevant quantities

The ball's weight is 5.25 oz, which is equivalent to 0.1485 kg. Its diameter is 2.9 inches, corresponding to a radius of approximately 0.0367 m. The speed of the ball is 95 mph, or approximately 42.47 m/s in SI units, and its spin rate is 1800 rpm.
02

Calculate the gravitational force

The gravitational force acting on the ball can be calculated using the formula \( F = mg \), where \( m \) is the mass of the ball and \( g \) is the force of gravity. Substituting the known values, \( F = (0.1485 kg)(9.8 m/s^2) \) yields \( F = 1.4563 N \).
03

Calculate the Magnus force

The Magnus force, which is the lift force created by the ball's rotation in the air, is described by the formula \( F = 0.5 \cdot C_L \cdot \rho \cdot A \cdot v^2 \). The lift coefficient \( C_L \) for a baseball is about 0.5 in the relevant speed range. The air density \( \rho \) is approximately 1.225 kg/m^3, and the ball's cross-sectional area \( A \) is \( \pi r^2 \), where \( r \) is the radius of the ball. Calculating for \( F \) using these values and the given velocity \( v \), the result is \( F = 0.5 \cdot 0.5 \cdot 1.225 \cdot \(\pi (0.0367)^2\)\cdot (42.47)^2 \) which gives \( F = 0.055 N \).
04

Compare the forces

Comparing the two forces, it is clear that the Magnus force (0.055 N) is much less than the gravitational force (1.4563 N). Therefore, the ball will not rise. Hence, it is not possible for a major leaguer to throw a rising fastball with the given speed and spin.

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