Final answer:
The period of an oscillating mass described by the function x(t) = (2.5 cm) cos 13t is determined by using the angular frequency ω provided in the cosine function and calculating the frequency. The period is the inverse of the frequency and is approximately 0.48 seconds.
Step-by-step explanation:
The student is asking about the period of an oscillating mass, a concept studied in physics involving simple harmonic motion. To determine the period, we examine the cosine function that describes the position x(t) of the mass. The general form of the position function for simple harmonic motion is x(t) = Acos(2πft + φ), where A is the amplitude, f is the frequency, φ is the phase shift, and t is time. From the given function x(t) = (2.5 cm) cos 13t, we can infer that 13 represents the angular frequency ω (omega), where ω = 2πf. Hence, we can solve for the frequency f.
To obtain the frequency f, we use the relation f = ω / (2π), substituting the given ω = 13 rad/s. Calculating further, f = 13 / (2π) ≈ 2.07 Hz. The period T is the inverse of the frequency, so T = 1/f. Therefore, the period T ≈ 1/2.07 ≈ 0.48 seconds or 480 milliseconds. To answer the student's question, the period of the oscillating mass is approximately 0.48 seconds.
By understanding the relationship between angular frequency and period within the context of oscillations and simple harmonic motion, we can solve for the time it takes for one complete cycle of movement (the period). This foundational knowledge is essential for students learning physics, particularly mechanics and wave motion.