To find the period of the object's motion given by x=(5.4 cm)sin(7.6πt), we identify the angular frequency from the sine function's argument and use the relationship T=1/f to calculate the period. The motion's period is approximately 0.26 seconds.
To find the period of the motion for an object connected to a spring that varies its position with time according to the expression x = (5.4 cm)sin(7.6πt), we need to analyze the argument of the sine function. The standard form for the equation of a simple harmonic motion is x = A sin(2πft) or x = A sin(ωt), where ω is the angular frequency and is related to the frequency f by ω = 2πf. The period T is the inverse of the frequency, so T = 1/f.
From the given expression, the angular frequency ω is 7.6π rad/s. Therefore, the frequency f is f = ω/(2π). Calculating this gives:
f = (7.6π rad/s) / (2π) = 3.8 s-1
So, the period T is T = 1/f, which means:
T = 1/3.8 s ≈ 0.26 s (rounded to the nearest hundredth)
The period of this motion is approximately 0.26 seconds.