Problem 66 - University Physics with Modern Physics

Question

A baseball pitcher can throw a 5.00 -oz baseball with a speed measured by a radar gun to be 90.0 mph. Assuming that the force exerted by the pitcher on the ball acts over a distance of two arm lengths, each 28.0 in, what is the average force exerted by the pitcher on the ball?

Step-by-Step Solution

Answer

Answer: The average force exerted by the pitcher on the ball is approximately 80.23 N.

Step 1: Convert units

First, let's convert the given values into the standard metric units: weight into mass (kilograms, kg) and speed into meters per second (m/s).

Step 2: Weight to Mass Conversion

To convert the weight of the baseball (5.00 ounces) into mass (kg), use the conversion factor 1 ounce = 0.0283495 kg. Mass of the baseball (m) = 5.00 oz * 0.0283495 kg/oz = 0.141748 kg

Step 3: Speed Conversion

To convert the speed of the baseball (90.0 mph) into meters per second (m/s), use the conversion factor 1 mph = 0.44704 m/s. Speed of the baseball (v) = 90.0 mph * 0.44704 m/s/mph = 40.2336 m/s

Step 4: Distance Conversion

To convert the distance over which the force is exerted (28.0 inches per arm length) into meters (m), use the conversion factor 1 inch = 0.0254 m. Distance (d_arm) = 28.0 inches * 0.0254 m/inch = 0.7112 m Total distance = 2 * d_arm = 2 * 0.7112 m = 1.4224 m

Step 2: Find initial and final kinetic energies

Before the pitcher throws the ball, its initial kinetic energy (KE_initial) is 0, since it's initially at rest. After the pitcher applies force to the ball, the final kinetic energy (KE_final) is given by the formula KE = 0.5 * m * v^2. KE_final = 0.5 * m * v^2 = 0.5 * 0.141748 kg * (40.2336 m/s)^2 ≈ 114.105 J

Step 3: Calculate the work done

Now, we will use the work-energy principle. The equation is Work = KE_final - KE_initial. Since KE_initial is 0, we have: Work = KE_final ≈ 114.105 J

Step 4: Find the average force exerted by the pitcher

Finally, we will find the average force (F_avg) exerted by the pitcher. To do this, we will use the equation: Work = F_avg * Distance F_avg = Work / Distance ≈ 114.105 J / 1.4224 m ≈ 80.23 N The average force exerted by the pitcher on the ball is approximately 80.23 N.