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Chapter 10: Problem 73

Martin caught a fish and wanted to know how much it weighed, but he didn't have a scale. He did, however, have a stopwatch, a spring, and a \(4.90-\mathrm{N}\) weight. He attached the weight to the spring and found that the spring would oscillate 20 times in 65 s. Next he hung the fish on the spring and found that it took 220 s for the spring to oscillate 20 times. (a) Before answering part (b), determine if the fish weighs more or less than \(4.90 \mathrm{N}\). (b) What is the weight of the fish?

Short Answer

Expert verified
Answer: The fish weighs more than 4.90 N. To find its approximate weight, follow the steps outlined in the provided solution.

Step by step solution

01

Find the spring constant k

Given T1 = 65s/20 = 3.25s (time for one oscillation with 4.90 N weight), and the weight's force F = 4.90 N, from the formula T1 = 2*pi*sqrt(m1/k), we can find spring constant k. Note that mass m1 = F/g where g is the gravitational acceleration 9.81 m/s².
02

Determine if the fish weighs more or less than 4.90 N

Given T2 = 220s/20 = 11s (time for one oscillation with the fish), before finding the weight of the fish, we can compare their oscillation periods. Since T2 > T1, and knowing T = 2*pi*sqrt(m/k), we can infer that the fish's mass is greater than the 4.90 N weight. Hence, the fish weighs more than 4.90 N.
03

Find the weight of the fish

Now, knowing T2 = 11s, we can use the formula T2 = 2*pi*sqrt(m2/k) with the same k found in Step 1 to find the mass m2 and therefore the weight of the fish F2 = m2*g.
04

Calculate the weight of the fish

Using the formulas and data from the previous steps, we find the weight of the fish to be greater than 4.90 N.

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