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Answer :
Final answer:
The angle subtended by a 1.50 m arc on a 2.50 m radius circle is 0.60 radians or 34.38 degrees. The radius for a 14.0 cm arc with a 128° angle is approximately 6.28 cm. The arc length intercepted by a 0.700 radian angle on a 1.50 m radius circle is 1.05 m.
Explanation:
Let's tackle each part of the question step by step:
- For an arc of length 1.50 m on a circle's circumference of 2.50 m, the angle in radians subtended is found using the formula angle in radians (θ) = arc length (s) / radius (r). Here, r = 2.50 m, so θ = 1.50 / 2.50 = 0.60 radians. To convert radians to degrees, use the conversion factor 180/π. Thus, 0.60 radians = 0.60 * (180/π) ≈ 34.38 degrees.
- The radius of a circle where an arc of 14.0 cm subtends a 128° angle is found by converting degrees to radians (since 1 radian = 180/π degrees), then using the relationship s = rθ. First, convert 128° to radians: 128 * (π/180) ≈ 2.23 radians. Thus, the radius r = s/θ = 14.0 cm / 2.23 ≈ 6.28 cm.
- For a 1.50 m radius circle and an angle of 0.700 radians, the intercepted arc length s is calculated using the relationship s = rθ. Hence, s = 1.50 m * 0.700 ≈ 1.05 m.
These calculations illustrate the direct relationship between arc length, radius, and the subtended angle in both radians and degrees for circular motion.
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