Final answer:
The frequency, f, is 0.527 Hz. The speed at the equilibrium position, vmax, is 2.268 m/s. The spring constant, k, is 10.8 N/m.
Step-by-step explanation:
To find the frequency, f, we need to calculate the number of cycles per second, which is given by the formula:
f = 1 / T,
where T is the period. Since the duration for 68 cycles is given as 129s, the period can be calculated as:
T = (duration for 68 cycles) / (number of cycles) = 129s / 68 = 1.897s.
Now, substituting the period into the formula for frequency:
f = 1 / T = 1 / 1.897s = 0.527 Hz.
The speed at the equilibrium position, vmax, can be calculated using the formula:
vmax = A * 2 * π * f,
where A is the amplitude. Substituting the values:
vmax = 0.857m * 2 * 3.1416 * 0.527Hz = 2.268 m/s.
The spring constant, k, can be calculated using the formula:
k = (2 * π * f)² * mass,
where mass is the mass of the particle. Substituting the values:
k = (2 * 3.1416 * 0.527Hz)² * 2.89kg = 10.8 N/m.
The potential energy at the endpoint, umax, can be calculated using the formula:
umax = (1/2) * k * A²,
where A is the amplitude. Substituting the values:
umax = (1/2) * 10.8 N/m * (0.857m)² = 4.39 J.
The potential energy when the particle is located 47.7% of the amplitude away from the equilibrium position, u, can be calculated using the formula:
u = (1/2) * k * x²,
where x is the distance from the equilibrium position. Substituting the values:
u = (1/2) * 10.8 N/m * (0.857m * 0.477)² = 0.563 J.
The kinetic energy, k, at the same position can be calculated using the formula:
k = (1/2) * mass * v²,
where v is the speed. Substituting the values:
k = (1/2) * 2.89kg * (2.268 m/s)² = 12.36 J.
The speed, v, at the same position can be calculated using the formula:
v = 2 * π * f * x,
where x is the distance from the equilibrium position. Substituting the values:
v = 2 * 3.1416 * 0.527Hz * (0.857m * 0.477) = 1.72 m/s.