Question

In softball, the pitcher throws with the arm fully extended (straight at the elbow). In a fast pitch the ball leaves the hand with a speed of 139 km/h.

(a) Find the rotational kinetic energy of the pitcher's arm given its moment of inertia is 0.720 kg ⋅ m² and the ball leaves the hand at a distance of 0.600 m from the pivot at the shoulder.
(b) What force did the muscles exert to cause the arm to rotate if their effective perpendicular lever arm is 4.00 cm and the ball is 0.156 kg?

Answer 1

To find the rotational kinetic energy of the pitcher's arm and ball together, calculate the angular velocity using the given linear velocity and distance. To find the force exerted by the muscles to cause the arm to rotate, calculate the angular acceleration and use it to find the torque.

Step-by-step explanation:

In order to calculate the rotational kinetic energy of the pitcher's arm and ball together, we can use the formula: Rotational Kinetic Energy = 0.5 * moment of inertia * angular velocity^2.

Given that the moment of inertia is 0.720 kg⋅m² and the ball leaves the hand at a distance of 0.600 m from the pivot at the shoulder, we need to find the angular velocity to calculate the rotational kinetic energy. The angular velocity can be found using the formula: angular velocity = linear velocity / radius. The radius in this case is the distance from the pivot to the point where the ball leaves the hand.

To find the force exerted by the muscles to cause the arm to rotate, we can use the equation: torque = force * lever arm. The torque is equal to the moment of inertia times the angular acceleration. The angular acceleration can be found using the formula: angular acceleration = linear acceleration / radius. Finally, we can find the force by rearranging the previous equation: force = torque / lever arm.