Rotational inertia of arm lifting a cup

[lyl926]

Homework Statement

Doc Holliday takes his last shot of whiskey. His forearm and hand spans 18" and weighs 2lbs. The shotglass and its intoxicating contents weighs 5ozs. (there are 16 ozs in one pound). Doc remains otherwise motionless as his elbow bends, tossing back the whiskey.
Calculate the rotational inertia of the arm + drink.

Homework Equations

τ=Iα (torque=rotational inertia x alpha)

and the equations in the file i attached

The Attempt at a Solution

Since the cup is point mass and that arm I think is rigid mass(?),
I used
I = (1/3)mL² + something
I don't know which of the equations from the file the something should be..
My friend used I=mR² but I'm not sure how that works..

 

[ideasrule]

By definition, the moment of inertia of a point mass is md^2. Every other moment of inertia you've encountered--1/3ML^2, 2/5MR^2, 2/3MR^3, whatever--is derived from I=md^2.

 

[kerimek]

I would first clarify what is meant by "rotational inertia" in this context. It seems that the question is asking for the moment of inertia, which is a measure of an object's resistance to rotational motion. The formula for moment of inertia is I = mr^2, where m is the mass and r is the distance from the axis of rotation.

In this case, the arm and cup can be treated as two separate objects with their own moments of inertia. The arm can be approximated as a thin rod with a point mass at the end (the hand holding the cup). The moment of inertia for a thin rod rotating about one end is (1/3)mr^2, so in this case, the moment of inertia for the arm would be (1/3)(2lbs)(18in)^2 = 216 in-lb-s^2.

For the cup, we can use the formula for a point mass rotating about an axis perpendicular to its motion, which is simply mr^2. In this case, the distance r would be the radius of the cup, which is half of its diameter. So the moment of inertia for the cup would be (5/16)lbs(9in)^2 = 3.52 in-lb-s^2.

To find the total moment of inertia for the arm and cup together, we can simply add the two values together: 216 in-lb-s^2 + 3.52 in-lb-s^2 = 219.52 in-lb-s^2.

It's important to note that these calculations are just approximations and may not be completely accurate, as there are many factors that can affect the moment of inertia of an object (such as the shape and distribution of mass). However, this should give us a rough estimate of the rotational inertia of Doc's arm and the shotglass.