Final answer:
The frequency of the oscillations is 0.367 Hz. The maximum speed of the mass is 1.45 m/s. The maximum acceleration of the mass is 1.33 m/s².
Step-by-step explanation:
To find the frequency of the oscillations, we can use the formula:
f = 1 / T
where f is the frequency and T is the period. The period can be found using the formula:
T = 2π√(m/k)
where m is the mass and k is the spring constant. Plugging in the given values, we get:
T = 2π√(1.5 kg / 20.0 N/m) = 2.72 s
So the frequency is:
f = 1 / 2.72 s = 0.367 Hz
To find the maximum speed of the mass, we can use the formula:
v_max = A√(k/m)
where v_max is the maximum speed, A is the amplitude, k is the spring constant, and m is the mass. Plugging in the given values, we get:
v_max = 0.10 m √(20.0 N/m / 1.5 kg) = 1.45 m/s
The maximum speed occurs at the equilibrium position, where the displacement is zero. (c) To find the maximum acceleration of the mass, we can use the formula:
a_max = A(k/m)
where a_max is the maximum acceleration, A is the amplitude, k is the spring constant, and m is the mass. Plugging in the given values, we get:
a_max = 0.10 m (20.0 N/m / 1.5 kg) = 1.33 m/s²
The maximum acceleration occurs at the extremes of the motion, where the displacement is maximum or minimum. (e) To determine the total energy of the oscillating system, we can use the formula:
E_total = (1/2) k A²
where E_total is the total energy, k is the spring constant, and A is the amplitude. Plugging in the given values, we get:
E_total = (1/2) (20.0 N/m) (0.10 m)² = 0.10 J
(g) To express the displacement xxx as a function of time ttt, we can use the formula:
x(t) = A cos(2π f t)
where x(t) is the displacement at time t, A is the amplitude, f is the frequency, and t is the time. Plugging in the given values, we get:
x(t) = 0.10 m cos(2π (0.367 Hz) t)