Final answer:
The spring constant can be found by equating the gravitational potential energy of the rock at the cliff's height to the energy stored in the spring at maximum compression. By substituting the given values into the energy conservation equation, we calculate the spring constant to be approximately 2414 N/m.
Step-by-step explanation:
To find the spring constant k, we can use the conservation of energy principle. The potential energy of the rock at the top of the cliff is equal to the energy stored in the spring when the rock comes to a stop. Since we can ignore air resistance and other non-conservative forces, the total mechanical energy is conserved.
The gravitational potential energy (GPE) at the cliff's height can be calculated using the formula GPE = mgh, where m is the mass of the rock, g is the acceleration due to gravity (9.81 m/s2), and h is the height of the cliff. The energy stored in the spring at maximum compression is given by the formula Espring = 1/2 kx2, where k is the spring constant, and x is the compression of the spring.
Setting the gravitational potential energy equal to the spring's energy, we get:
mgh = 1/2 kx2
Plugging in the known values (m = 2.5 kg, h = 32 m, g = 9.81 m/s2, and x = 0.57 m), we can solve for k.
(2.5 kg)(9.81 m/s2)(32 m) = 1/2 k(0.57 m)2
k = (2.5 kg)(9.81 m/s2)(32 m) / (0.5)(0.57 m)2
k ≈ 2414 N/m
Therefore, the spring constant is approximately 2414 N/m, rounded to two significant figures.