Final answer:
To find the spring constant of the spring, set the maximum spring force equal to the weight of the block just before losing contact. The spring constant is calculated as 818.33 N/m by dividing the force (19.62 N) by the maximum displacement (0.024 m).
Step-by-step explanation:
To solve the mathematical problem completely and find the spring constant of the spring, we'll use the concepts of simple harmonic motion (SHM) and the conditions for maximum compression. The key condition is that at the maximum compression (which is given as 2.4 cm), the block must be just about to lose contact with the spring — this is the point where the spring's force equals the weight of the block.
Firstly, we need to find the maximum force exerted by the spring when the block is pushed down to its maximum displacement (x = 2.4 cm or 0.024 m). This force will equal the weight of the block at the point just before losing contact, which is 2.0 kg × 9.81 m/s2 (acceleration due to gravity). So, the maximum force F_max = mass (m) × acceleration due to gravity (g).
F_max = 2.0 kg × 9.81 m/s2 = 19.62 N
The spring force can also be expressed in terms of the spring constant (k) and the displacement (x), F_spring = kx. Setting these two forces equal at the point of maximum compression, we get:
k × 0.024 m = 19.62 N
To find the spring constant (k), we'll divide the force by the displacement:
k = 19.62 N / 0.024 m = 818.33 N/m
Therefore, the spring constant of the spring is approximately 818.33 N/m.