e. As the radius of the circle is doubled, the acceleration is __________________ (one-fourth, one-half, two times, four times) the original value.

Break Down the Problem

1. Identify the relationship between radius and acceleration in circular motion.
2. Analyze how doubling the radius affects the acceleration.

Relevant Concepts

1. The centripetal acceleration $a$ of an object moving at a constant speed in a circular path is given by the formula:
   $a = \frac{v^2}{r}$ where $v$ is the speed of the object and $r$ is the radius of the circle.

Analysis and Detail

1. When the radius $r$ is doubled, the new radius $r' = 2r$.
2. The centripetal acceleration with the new radius becomes:
   $a' = \frac{v^2}{r'} = \frac{v^2}{2r} = \frac{1}{2} \cdot \frac{v^2}{r} = \frac{1}{2} a$ This shows that the new acceleration $a'$ is half the original acceleration $a$.

Verify and Summarize

• Original acceleration $a = \frac{v^2}{r}$ and new acceleration $a' = \frac{v^2}{2r} = \frac{1}{2}a$. Thus, when the radius is doubled, the acceleration decreases to half of its original value.

Final Answer

The acceleration is one-half the original value.