Final answer:
The spring force constant k can be determined by using the conservation of energy principle, equating the gravitational potential energy of the dropped ball to the potential energy of the spring at maximum compression. Upon substituting the mass, gravitational acceleration, height, and compression distance into the formula, the spring constant k is calculated to be approximately 134.31 N/m.
Step-by-step explanation:
To calculate the spring force constant k of the spring, we use the conservation of energy principle. When the ball is dropped, its potential energy is converted into the spring's potential energy at maximum compression. The gravitational potential energy (PEgrav) at the height just before the ball touches the spring can be calculated using PEgrav = mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height. Once the ball compresses the spring, this energy is equal to the potential energy stored in the spring (PEspring), which is given by PEspring = (1/2)kx^2, where k is the spring constant and x is the compression distance.
Equating the gravitational potential energy to the spring potential energy:
mgh = (1/2)kx^2
142 g (0.142 kg) * 9.8 m/s^2 * 62.2 cm (0.622 m) = (1/2)k * (4.35501 cm)2 (0.0435501 m)^2
Now, solve for k:
k = (2 * mgh) / x^2
k = (2 * 0.142 kg * 9.8 m/s2 * 0.622 m) / (0.0435501 m)2
k ≈ 134.31 N/m