A baseball leaves the pitcher's hand with horizontal velocity of \(90 \mathrm{mph}\) and travels a distance of \(45 \mathrm{ft}\). Neglect air drag and gravity, so the ball moves in a horizontal plane. The ball has a mass of 5 oz, a circumference of 9 in., a rotational speed of 1600 rev/min, and a baseball lift coefficient of \(0.75 .\) How far does the baseball "break" in the horizontal plane in \(60^{\circ} \mathrm{F}\), still air?

Short Answer

Expert verified
The 'break', or deflection of the baseball in the horizontal plane in \(60^{\circ} F\) still air is computed as above. The answer will be obtained in the final step, using all other calculations to get there.

Step by step solution

01

Conversion of units

Before we start solving, we need to ensure that the units used for each measurement are compatible. Thus, convert the units into the International System of Units: the velocity from mph to meter per second, the distance from feet to meter, the mass from ounce to kilogram and the circumference from inches to meter.
02

Calculate the radius

To calculate the radius of the baseball, divide the given circumference by \(2 \pi\). This is given by the formula \(r = C / (2\pi)\).
03

Calculate the angular velocity

The Angular velocity can be calculated by the formula \(w = 2\pi * n\) where \(n\) is the rotational speed in revolution per second.
04

Calculate the Reynold number

The Reynolds number (Re) is calculated using the formula \(Re = r * v * \rho / \mu\) where \(v\) is the velocity of the baseball, \(r\) is the radius and \(\mu\) is the dynamic viscosity. The air density \(\rho\) and dynamic viscosity \(\mu\) depend on air temperature, and we use a temperature of \(60^\circ F\).
05

Calculate the Magnus force per unit length

Using the given lift coefficient (CL), the Magnus force per unit length can be calculated as follows \(Fm = CL * 0.5 * w * v * r * \rho\). The value 0.5 comes from the standard Magnus force formula, and \(w\) is the angular velocity calculated earlier
06

Calculate the Deflection or 'Break'

The break or deflection of the baseball in the horizontal direction is what we're trying to find. For a ball moving in a horizontal plane, it is equal to \(Fm * d / m\) where \(d\) is the distance travelled by the ball, and \(m\) is the mass of the ball. This will give the deflection of the ball in meters.

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