Answer :
Final answer:
The spring constant of the spring is 1245.45 N/m, and length of the spring when a 215 N weight is attached is 38.3 cm.
Explanation:
The question requires us to find two things: the spring constant and the length of the spring with a different weight. Given a 21.0 cm long spring, it stretches to 26.5 cm when a 7.00 kg mass is hung from its free end. The spring constant, denoted as 'k', can be calculated using Hooke's law which states that the force 'F' exerted by a spring is equal to the negative product of the spring's displacement and its spring constant, or F = -k * x. Here the displacement of the spring 'x' can be calculated as the difference between the stretched and the original length of the spring, which is (26.5 cm - 21.0 cm) = 5.5 cm or 0.055 m. Given that the weight of the object hung from the spring (in this case the force) is equal to its mass multiplied by acceleration due to gravity (F = m * g), we know that 7.00 kg weight would exert a force of about 68.6 N (with g approximated as 9.8 m/s^2). Therefore, with F = 68.6 N and x = 0.055 m, the spring constant 'k' can be found as F/x = 1245.45 N/m.
In regards to the second part of the question, when the 7.00 kg weight is replaced with a 215 N weight, the spring would stretch further. Using Hooke's law (F = k * x), we can calculate the additional length the spring stretches when the 215 N weight is attached. The displacement of the spring 'x' is found as F/k = 215 N / 1245.45 N/m = 0.173 m or 17.3 cm. This is the additional length the spring stretches. Therefore, the total length of the spring with the 215 N weight is the original length of the spring plus the stretch, which comes to be 21 cm + 17.3 cm = 38.3 cm.
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