Classical Mechanics - Rotational Motion

Seven pennies are arranged in a hexagonal, planar pattern so as to touch each neighbor, as shown in the figure. Each penny is a uniform disk of mass m and radius r. What is the moment of inertia of the system of seven pennies about an axis that passes through the center of the central penny and is normal to the plane of the pennies?

A. (7/2) mr²
B. (13/2) mr²
C. (29/2) mr²
D. (49/2) mr²
E. (55/2) mr²

(GR0177 #25)

Solution:

Parallel axis theorem: I = ml² + I_CM

Moment inertia of each penny (a uniform disk): I = ½ mr²

with l = 2r,

I = m(2r)² + ½ mr² = (9/2) mr²

I_total =  I_{6 outer pennies} + I_{1 central penny}

I_total = 6 × (9/2) mr²  + ½ mr² = (55/2) mr²

Answer: E