Final answer:
The spring constant (k) can be found by equating the force exerted by the spring, given by Hooke's law (F = -kx), to the weight of the frame (mg), when the frame is at rest. The spring constant is then k = mg/x, where x is the spring's displacement and m is the mass of the frame.
Step-by-step explanation:
To find the spring constant (k) for a spring when a frame hangs at rest, we apply Hooke's law, which states that the force (F) exerted by a spring is equal to the negative product of the spring constant (k) and the displacement (x): F = -kx. Here, x is the displacement of the spring from its equilibrium position. From the setup, when the frame is at rest, the spring is stretched by an amount (s), which corresponds to x in Hooke's law. The restoring force F is equivalent to the weight of the frame, which is the product of the mass (m) of the frame and the acceleration due to gravity (g).
In equilibrium, the force exerted by the spring equals the weight of the frame: -kx = mg. The negative sign indicates that the force exerted by the spring is opposite to the direction of displacement. Solving for the spring constant k, we get k = mg/x. To find k, we need to measure the amount the spring is stretched (s), which is the displacement x, and know the mass of the frame.
Once the spring constant is determined, this value can be used to analyze the collision between the putty and the frame, as well as the subsequent motion of the frame using principles like conservation of momentum and conservation of energy.