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 6.04 cm. Through what arc length s does the 25.0-cm mark on the stick move?
To find the arc length covered by the 25.0-cm mark on the meter stick, we need to determine the angle covered by this mark when the stick is rotated.
Since the meter stick is being rotated about the end labeled 0.00 cm, the arc length covered by the 25.0-cm mark will be proportional to the angle it covers. Therefore, we can use a proportion to find the arc length:
Arc length / Angle covered = Total arc length / Total angle covered
Let's represent the arc length covered by the 25.0-cm mark as s and the total arc length as 6.04 cm. The total angle covered by the meter stick can be considered as 360° (since one complete rotation covers 360°).
Plugging in the values, we get:
s / Angle = 6.04 cm / 360°
Now, we need to determine the angle covered by the 25.0-cm mark. From the end labeled 0.00 cm to the 25.0-cm mark is a distance of 25.0 cm. Since the total length of the meter stick is 100.0 cm, the proportion of the distance covered by the 25.0-cm mark is:
Distance / Total length = 25.0 cm / 100.0 cm
Simplifying the above proportion:
Distance / 100 = 25.0 cm / 100.0 cm
Distance = (25.0 cm / 100.0 cm) * 100
Distance = 25.0 cm
Therefore, the 25.0-cm mark on the stick moves an arc length of 25.0 cm.
To find the arc length through which the 25.0 cm mark on the stick moves, we need to determine the angle through which the meter stick is rotated.
Let's assume that the meter stick is rotated by angle θ radians. Since the rotation is about the end labeled 0.00 cm, the arc length of 6.04 cm represents the distance traveled along the circumference of the rotation.
We can use the formula for arc length to relate the angle and the arc length:
s = rθ
In this case, the radius r is the distance from the center of rotation (labeled 0.00 cm) to the 25.0 cm mark. As the stick is a meter long, the radius is 25.0 cm.
Therefore, we can rearrange the formula to solve for the angle θ:
θ = s / r
Substituting the given values:
θ = 6.04 cm / 25.0 cm
θ ≈ 0.2416 radians
Now that we have the angle, we can find the arc length s through which the 25.0 cm mark moves by using the same formula:
s = rθ
Substituting the values:
s = 25.0 cm * 0.2416 radians
s ≈ 6.04 cm
Therefore, the 25.0 cm mark on the stick moves through an arc length of approximately 6.04 cm.