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 7.40 cm. Through what arc length s does the 25.0-cm mark on the stick move?
To determine the arc length through which the 25.0 cm mark on the stick moves, we need to first understand the relationship between the distances from the rotation point and the corresponding arc lengths.
In this case, we can assume that the rotation is around a fixed point, creating a circular arc. The distance from the rotation point to the 0.00 cm mark remains constant, as the stick rotates without translating.
Let's denote the distance of the 25.0 cm mark from the rotation point as d. Since the stick is a meter long, we can find d by subtracting the distance of the 0.00 cm mark from the total length of the stick:
d = 25.0 cm - 0.00 cm = 25.0 cm
Now, we can set up a proportion to find the arc length s corresponding to the distance d:
d / s = r / R
Where:
d is the distance from the rotation point to the 25.0 cm mark (25.0 cm),
s is the arc length we want to find,
r is the radius of the circular path created by the rotation, and
R is the radius of the stick (100.0 cm, as it is a meter stick).
We can rearrange the equation to solve for s:
s = (d * R) / r
Substituting the given values:
s = (25.0 cm * 100.0 cm) / 7.40 cm
Simplifying:
s = 2500 cm² / 7.40 cm
Calculating:
s ≈ 337.84 cm
Therefore, the 25.0 cm mark on the stick moves through an arc length of approximately 337.84 cm.