Final answer:
The spring constant is 18 N/m. The speed of the ball halfway to its equilibrium point can be calculated using the equation v = (1.5 m/s)cos((2π / T)(T/4) + 90°). At the halfway point, half the energy has been converted from elastic potential energy to kinetic energy.
Step-by-step explanation:
(a) Determine the spring constant: Using the formula for the maximum speed of an object undergoing simple harmonic motion (vmax = Aω), we can rearrange the formula to solve for the angular frequency (ω = 2πf). The angular frequency can then be used to find the spring constant (k) using the formula k = mω^2, where m is the mass of the object. Substituting the given values, we have k = (0.50 kg)(1.5 m/s)^2 / (0.25 m)^2 = 18 N/m.
(b) Calculate the speed of the ball halfway to its equilibrium point: Since the ball undergoes simple harmonic motion, its speed at any point is given by v = Aωcos(ωt + φ), where t is the time and φ is the phase constant. At the halfway point, the displacement is half the amplitude (A/2). Substituting the given values, we have v = (1.5 m/s)cos((2π/T)(T/4) + φ), where T is the period. Since the ball is symmetric, cos(π/2 + φ) = sin(φ), and sin(φ) can be found using the maximum speed and amplitude. Solving for φ = arcsin(vmax/A), we find that φ = arcsin(1.5 m/s / 1.5 m) = 90°. Therefore, v = (1.5 m/s)cos((2π / T)(T/4) + 90°).
(c) Determine the fraction of energy converted from elastic potential energy to kinetic energy: At the halfway point, the ball has reached its maximum potential energy and half the maximum speed, which means that half the energy has been converted from elastic potential energy to kinetic energy.