Final answer:
The period of the motion is 0.67/2π seconds. The first times the mass is at the position x = 0 are t = 0 and t = T/2.
Step-by-step explanation:
The period of the motion can be determined by looking at the coefficient of t in the argument of the cosine function. In this case, the argument is 2πt/0.67s. The coefficient of t is 2π/0.67, and the period T is the reciprocal of this coefficient. Therefore, T = 0.67/2π seconds.
To find the first time the mass is at the position x = 0, we need to solve the equation x = 0. Substituting x = 0 into the expression x = (3.6 cm)cos[2πt/(0.67s)], we get 0 = (3.6 cm)cos[2πt/(0.67s)]. Solving for t, we find that t = 0 and t = T/2 are the times when the mass is at the position x = 0. Using the period T calculated earlier, we can find t = 0 and t = T/2 as the first times the mass is at the position x = 0.