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/hr. (Assume that the pitching motion originates with the arm straight back and terminates with the arm straight down. Do not ignore gravity.) (a) Find the rotational kinetic energy (in J) 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 0.604 m from the pivot in the shoulder. (Enter the rotational kinetic energy of the pitcher's arm just after the ball is released.)"(b) What force (in N) did the muscles exert to cause the arm to rotate if their effective perpendicular lever arm is 4.00 cm and the ball is 0.187 kg? (Enter the magnitude.) XN

Answer 1

To find the rotational kinetic energy of the pitcher's arm, use the formula KE = 0.5 * Moment of Inertia * Angular Velocity². To find the force exerted by the muscles to cause the arm to rotate, use the formula Force = Mass * Acceleration.

Step-by-step explanation:

(a) To find the rotational kinetic energy (KE) of the pitcher's arm, we can use the formula:

KE = 0.5 * Moment of Inertia * Angular Velocity²

Given that the moment of inertia (I) of the arm is 0.720 kg.m² and the ball leaves the hand at a distance of 0.604 m from the pivot at the shoulder, we first need to find the angular velocity (ω) of the arm. We can use the formula:

ω = v / r

where v is the linear velocity of the ball (139 km/h converted to m/s) and r is the distance from the pivot to the ball. Once we have ω, we can substitute it back into the KE formula to find the rotational kinetic energy of the pitcher's arm.

(b) To find the force exerted by the muscles to cause the arm to rotate, we can use the formula:

Force = Mass * Acceleration

Since we have the mass of the ball and the effective perpendicular lever arm, we can calculate the acceleration using the formula:

Acceleration = ω² * r

Substituting the mass of the ball and the acceleration into the force formula will give us the magnitude of the force exerted by the muscles.

Answer 2

Step-by-step explanation:

139 km /h = 38.61 m / s.

the rotational kinetic energy (in J) of the pitcher's arm

= (1/2) I ω²

Where I is moment of inertia of the arm and ω is its angular velocity

Given , I = 0.72 kg m²

ω = vR , v is linear velocity and R is length of arm ( given .604 m )

ω = 38.61 x .604 = 23.32 rad / s

(1/2) I ω²

= .5 x 0.72 x (23.32)²

= 195.77 J