Five hoops are each pivoted at a point on the rim and allowed to swing as physical pendulums. The masses and radii are
Hoop 1: M = 150g and R = 50 cm
Hoop 2: M = 200g and R = 40 cm
Hoop 3: M = 250g and R = 30 cm
Hoop 4: M = 300g and R = 20 cm
Hoop 5: M = 350g and R = 10 cm
Order the hoops according to the periods of their motions, smallest to largest.
A) 1, 2, 3, 4, 5
B) 5, 4, 3, 2, 1
C) 1, 2, 3, 5, 4
D) 1, 2, 5, 4, 3
E) 5, 4, 1, 2, 3

Question

Quizplus · Accepted Answer

The period of a physical pendulum is influenced by its moment of inertia and mass distribution. For hoops pivoted at the rim, the moment of inertia is $I = MR^2$, and the period $T$ is proportional to $\sqrt{I/M}$, which simplifies to $\sqrt{R^2}$ or just $R$ for a constant gravitational field. Thus, the hoop with the smallest radius will have the smallest period, and the hoop with the largest radius will have the largest period. Therefore, the order from smallest to largest period is from hoop 5 to hoop 1.

Five Hoops Are Each Pivoted at a Point on the Rim