Final answer:
To find the length of a loop (arc length), multiply the radius by the angle of rotation in radians. The complete circumference is 2πr for a 360-degree or 2π radian rotation. The arc length for part of a revolution is the fraction of the circumference corresponding to the rotation angle.
Step-by-step explanation:
To find the length of the loop of a curve, we need to consider the relationship between arc length, radius of curvature, and angle of rotation.The arc length (Δs) along a curve is equal to the angle of rotation (Δθ) in radians multiplied by the radius (r) of the curve: Δs = r * Δθ. If we know the angle of rotation and the radius, we can find the arc lengthFor a complete revolution, the arc length is the same as the circumference of a circle, which is given by the formula 2πr. If the angle of rotation is less than a full revolution (ΰ degrees), then the arc length is a fraction of the circumference corresponding to that angle.
This fraction is ΰ/360 for degrees or ΰ/2π for radians since there are 360 degrees or 2π radians in a full circleThe arc length of a curve is the distance traveled along a circular path. It can be calculated using the formula:arc length = radius of curvature x angle of rotatioIn this case, the options A, B, C, and D represent different fractions of the arc length. Since the arc length is equal to the circumference of a circle for one complete revolution, which is 2πr, the correct answer would be option B, twice the arc length.