Final answer:
The spring constant is calculated using Hooke's Law by applying the force due to the gemstone's weight and the displacement of the spring. The calculated constant is 678.46 N/m. When a different mass of 5.2 kg is used, the spring compresses 7.5 cm.
Step-by-step explanation:
Solving for the Spring Constant:
To determine the spring constant (k), we use Hooke's Law, which states F = kx, where F is the force in newtons (N), k is the spring constant in newtons per meter (N/m), and x is the displacement of the spring in meters (m). When a gemstone with a mass of 1.8 kg compresses a spring by 2.6 cm (which is 0.026 m), the force applied is equivalent to the weight of the gemstone, which is the mass times the acceleration due to gravity (F = mg). In this case, F = 1.8 kg × 9.8 m/s^2 = 17.64 N.
Now, we can solve for k as follows:
17.64 N = k × 0.026 m
k = 17.64 N / 0.026 m
k = 678.46 N/m
The spring constant is therefore 678.46 N/m.
Compression with a Different Mass:
If a 5.2 kg mass is placed on the same spring, we find the new displacement (x) using Hooke's Law by rearranging it to x = F/k. We calculate the new force (F = 5.2 kg × 9.8 m/s^2 = 50.96 N) and divide by the previously found spring constant (k = 678.46 N/m):
x = 50.96 N / 678.46 N/m
x = 0.075 m or 7.5 cm
The spring would compress by 7.5 cm with a 5.2 kg mass.