Final answer:
The spring constant of the vertically hung spring can be calculated using Hooke's law with the displacement caused by the hanging mass and the gravitational force on the mass. For the given displacement of 4.9 cm and a 6.50 kg mass, the spring constant is approximately 1301.326 N/m.
Step-by-step explanation:
To find the spring constant (k) of a vertically hung spring, we apply Hooke's law, which states that the force (F) exerted by a spring is directly proportional to the displacement (x) from its equilibrium position: F = kx. In this case, the force exerted by the spring is equal to the gravitational force on the mass, which can be calculated using Newton's second law (F = mg), where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s2).For a spring that stretches from 23.0 cm to 27.9 cm, the displacement (x) is 27.9 cm - 23.0 cm = 4.9 cm or 0.049 m. A 6.50 kg mass would exert a force of F = 6.50 kg × 9.81 m/s2 = 63.765 N. Given this force and the displacement, we can find the spring constant using Hooke's law: k = F / x = 63.765 N / 0.049 m = 1301.326 N/m.