Final answer:
In geometry, the circumference and arc length of a circle are both directly proportional to its radius, meaning as the radius decreases, these dimensions also decrease. The area of the circle shrinks as well, yet it still remains greater than half the area of a square enclosing it, approximately three-quarters of it. In contrast, centripetal acceleration inversely varies with the radius.
Step-by-step explanation:
The question relates to the concept of how certain properties of a circle change as its dimensions are altered. Specifically, it involves the circumference and area relative to the radius of the circle. It is a fundamental concept in geometry which falls under the mathematics discipline.
Arc length is directly proportional to the radius of the circular path, which means if the circumference is shrinking, the radius is also decreasing. If R is the radius of a circle, then the circumference is given by the formula 2πR. In terms of area, a circle's area, represented by πR², is smaller compared to the area of a square enclosing it, which has an area of 4R². However, as the radius shrinks (just as the circumference decreases), the area of the circle (around three-quarters of the square's area) would also reduce.
Furthermore, arc length is also directly proportional to the angle of rotation, which means with a constant angle, as the radius decreases, the arc length would decrease too. This can be observed in a circle as its radius becomes smaller, the length of any arc segment reduces as well. Meanwhile, centripetal acceleration is inversely proportional to the radius of the curvature, implying that as the radius decreases, the centripetal acceleration increases.