Question

A meter stick is rotated about the end labeled 0.00 cm, so that the other end of the stick moves through an arc length of 8.85 cm. Through what arc length s does the 25.0-cm mark on the stick move?

Answer 1

The arc length that the 25.0-cm mark on the meter stick moves through is calculated using proportional lengths, resulting in an arc length of 2.21 cm.

Step-by-step explanation:

The student's question involves the concept of arc length in a rotational motion scenario. When a meter stick is rotated about one end, different points on the stick trace out different arc lengths, depending on their distance from the pivot point. To find the arc length s that the 25.0-cm mark moves through, we can use the principle of similar triangles or proportional lengths, since both the 100-cm mark and the 25-cm mark are moving through arcs centered at the same pivot point.

The complete meter stick (100 cm in length) moves through an arc length of 8.85 cm. This gives us a ratio of the arc length to the radius of rotation. Since the arc lengths are directly proportional to their respective radii (distances from the pivot point), we can say: Arc length of the 25-cm mark / 25 cm = Arc length of the 100-cm mark / 100 cm.

Thus, the arc length moving through the 25-cm mark is (8.85 cm * 25 cm) / 100 cm, which equals 2.21 cm.

Answer 2

2.215 cm

Step-by-step explanation:

length of stick (L) = 1 m

length of arc by the 1 m stick (La) = 8.85 cm = 0.885 m

Through what arc length does the 25.0-cm (0.25 m) mark on the stick move?

from the attached diagram:

OA = length of the stick = 1 m

OC = mark on stick = 0.25 m

AB = length of arc by the full stick length = 0.885 m

CD = length of arc by the 25 cm mark =?

length of an arc = radius x θ (θ is the angle and radius is the length of the stick in this case)

therefore

AB = OA x θ

θ = AB / OA = 0.0885 / 1 = 0.885 rad

also

CD = OC X θ

CD = 0.25 X 0.0885 =0.02215 m = 2.215 cm