Question

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time t=0 1. amplitude 60 ft, period 0.5min

Answer 1

The function modeling the simple harmonic motion with an amplitude of 60 ft and a period of 0.5 min is x(t) = 60 cos(2π(1/30)t).

Step-by-step explanation:

The student is asking for a function to model simple harmonic motion (SHM) with a given amplitude and period. The amplitude is the maximum displacement from the equilibrium position (60 ft) and the period is the duration of one complete cycle of the motion (0.5 min or 30 seconds). Since the displacement is at its maximum at t=0, a cosine function would be the appropriate choice for the displacement function.

To derive the function, we use the general formula for the displacement in SHM:

x(t) = A cos(2πft + φ)

Where:

• A is the amplitude
• f is the frequency, which is the reciprocal of the period (f = 1/T)
• φ is the phase constant
• t is the time

For this problem, the amplitude A is 60 ft, the period T is 0.5 min, which gives us a frequency f of 2 cycles/min or 1/30 cycles/sec. The phase constant φ is 0 since the displacement is a maximum at t=0. Consequently, the function is:

x(t) = 60 cos(2π(1/30)t)