Final answer:
The question requires calculating the angular velocity of a baseball catcher's arm immediately after catching a ball using conservation of angular momentum. The arm is modeled as a rod pivot around the shoulder, and its moment of inertia is calculated. After setting up the conservation equation, we can solve for the angular velocity, yielding the answer from the given choices.
Step-by-step explanation:
The student is asking for the angular velocity of a baseball catcher's arm immediately after catching a ball. To find this, we can use the principles of conservation of angular momentum, assuming that the arm and ball system is isolated and ignoring other forces such as gravity for a short period of time. Since angular momentum before catching the ball must equal angular momentum after catching the ball, and given the arm acts as a rod of length 0.5 m about the pivot at the shoulder, we can set up the following:
Initial angular momentum (Li) = Final angular momentum (Lf)
Li = (mball * vball * rarm)
Lf = Itotal * ω
Where mball is the mass of the ball, vball is the velocity of the ball before being caught, rarm is the length of the arm, Itotal is the moment of inertia of the arm-ball system, and ω is the angular velocity we need to find.
The moment of inertia for the arm (modeled as a uniform rod) plus the ball is:
Itotal = Iarm + mball * rarm2
Iarm = (1/3) * marm * rarm2, and by substituting we get:
Lf = (ω * [(1/3) * marm * rarm2 + mball * rarm2])
By setting Li equal to Lf and solving for ω, we find the angular velocity of the arm. We must convert m/s to rad/s by considering that the linear velocity at the arm's end is equivalent to the product of the angular velocity and the radius, and this provides the angular velocity answer from the choices given.