Final answer:
Using the conservation of mechanical energy in the context of simple harmonic motion, the maximum displacement of the mass from the equilibrium position is found to be approximately 1.5 feet.
Step-by-step explanation:
The student is asking about the maximum displacement of a mass suspended from a spring in a physics context, specifically involving simple harmonic motion (SHM). To determine this displacement, we must consider both the initial potential energy stored in the spring due to its initial stretch and the kinetic energy given by the initial velocity. The spring constant (k) is 9 lb/ft, and the mass (m) is 1 slug, which is initially released 1 foot above the equilibrium position with an upward velocity. Using conservation of mechanical energy, the maximum displacement (x_max) can be found when the kinetic energy is zero.
Initially, the spring has potential energy (U) due to being stretched by 1 foot (since potential energy in a spring is given by U = 1/2 k x^2) and the mass has kinetic energy (KE) due to its velocity (given by KE = 1/2 m v^2). At the point of maximum displacement from equilibrium, all this initial energy will be transformed into spring potential energy, so:
1/2 k x_initial^2 + 1/2 m v_initial^2 = 1/2 k x_max^2
Substitute the given values: 1/2 (9 lb/ft) (1 ft)^2 + 1/2 (1 slug) (3 ft/s)^2 = 1/2 (9 lb/ft) x_max^2
Solving this equation for x_max will give us the maximum displacement from equilibrium, which is approximately 1.5 feet. So the correct answer to the student's question is Option 3: 1.5 ft.