Final answer:
The amplitude of the ball's motion in SHM is 0.350 m, and its frequency is 2.39 Hz. The mass of the ball is found to be 1.54 kg. The total mechanical energy and ball's maximum speed cannot be precisely provided to match the options given without rounding, which may be inaccurate.
Step-by-step explanation:
The student's question concerns the properties of simple harmonic motion (SHM) involving a ball oscillating on a spring.
A) The amplitude of the ball's motion
The amplitude is the maximum extent of the oscillation, measured from the equilibrium (rest) position. According to the given position function x = (0.350 m) cos(15.0t), the amplitude is 0.350 m, which is the coefficient in front of the cosine function.
B) The frequency of the ball's motion
Frequency is found by dividing the angular frequency ω (15.0 rad/s, as shown in the cosine function) by 2π. This results in a frequency of 2.39 Hz.
C) The value of mass m
To determine the mass, we use the formula ω = sqrt(K/M), where K is the spring constant and ω is the angular frequency. Rearranging for M gives us M = K / ω2, which leads to a mass of 1.54 kg.
D) The total mechanical energy of the oscillator
The total mechanical energy in SHM is given by E = 1/2 K A2, where A is the amplitude. Calculating this with the given values, the energy is 12.25 J. However, this exact value is not listed in the options provided. Without rounding to one of the option choices, which may not be accurate, we can't provide a clear answer.
E) The ball's maximum speed
The maximum speed (v_max) in SHM occurs at the equilibrium position and is given by v_max = A ω. Using the amplitude and angular frequency, the maximum speed is 5.25 m/s. Again, since this precise value is not listed among the choices, we cannot provide an exact match from the options.