Final answer:
The magnitude of the maximum force acting on a 30.0 kg mass undergoing simple harmonic motion with a position function x(t) = 8.0cos((15.0)t) is 54000 N. This is found by using the angular frequency to calculate maximum acceleration, and then applying Newton's second law to find the force.
Step-by-step explanation:
To calculate the magnitude of the maximum force acting on a mass m = 30.0 kg given the position x(t) = 8.0cos((15.0)t), we first need to determine the acceleration at the point where it is maximum, which occurs at the maximum displacement of the oscillation. The maximum displacement is the amplitude A = 8.0 m in the given equation. Since we're dealing with simple harmonic motion (SHM), the maximum acceleration amax can be found using the formula:
amax = -ω2A
Where ω (angular frequency) can be determined from the position function as ω = 15.0 s-1. Substituting the values into the formula, we get:
amax = -(15.0 s-1)2 × 8.0 m = -1800 m/s2
The negative sign indicates that the acceleration is in the opposite direction of the displacement. To find the force, we apply Newton's second law:
Fmax = m × amax
Fmax = 30.0 kg × 1800 m/s2 = 54000 N
Therefore, the magnitude of the maximum force acting on the mass during its oscillation is 54000 N.