Final answer:
The particle's motion has a frequency of 1.5 Hz, a period of 0.6667 seconds, an amplitude of 4.00 m, a phase constant of π, and at t = 0.250 s, the particle's position is -4.00 m.
Step-by-step explanation:
The position of a particle is given by the expression x = (4.00 m)cos(3πt + π), where x is in meters and t is in seconds. To determine the characteristics of the particle's motion, we can compare this expression with the standard form for simple harmonic motion, x(t) = A cos(ωt + φ).
- (a) The frequency (f) is given by the coefficient of t divided by 2π, so ω = 3π rad/s and thus f = ω / (2π) = 3π / (2π) = 1.5 Hz. The period (T) is the reciprocal of the frequency, so T = 1/f = 1/1.5 Hz = 0.6667 seconds.
- (b) The amplitude of the motion is the coefficient of the cosine function, which is 4.00 m.
- (c) The phase constant (φ) is the constant term within the cosine function, which in this case is π.
- (d) To find the position of the particle at t = 0.250 s, we can substitute t into the expression to get x = (4.00 m)cos(3π(0.250) + π) = (4.00 m)cos(1.75π) = -4.00 m.