Final answer:
In simple harmonic motion given by the equation X = (4.0 m) cos(ωt), the amplitude is 4.0 meters. The velocity and acceleration can be calculated using the time derivatives of the position function, and the maximum speed and acceleration are determined by the maximum values of the sine and cosine functions respectively.
Step-by-step explanation:
Simple harmonic motion (SHM) is characterized by the equation X = (4.0 m) cos(ωt), where X is the maximum displacement from the equilibrium position, also known as the amplitude. To determine the amplitude, frequency, and period of this motion:
- The amplitude is the coefficient of the cosine function, which is 4.0 meters.
- The frequency (ƒ) is the number of oscillations per unit time, typically found using the angular frequency ω, where ω = 2πƒ.
- The period (T) of the motion is the time taken for one complete cycle of the motion, given by T = 1/ƒ.
The velocity (v) and acceleration (a) at any time t can be found using derivatives of the position function x(t), with velocity given by v(t) = -ωX sin(ωt) and acceleration by a(t) = -ω²X cos(ωt).
To find the maximum speed and acceleration, we look at the conditions when the sine and cosine functions reach their maxima. The maximum speed occurs when sin(ωt) = ±1, yielding Umax = ωX, and the maximum acceleration occurs when cos(ωt) = ±1, yielding Amax = ω²X.
To calculate the displacement between t = 0 and t = 1 second, we need to evaluate x(t) at these times and subtract the initial position from the final position. At t = 0, x(0) = X, and at t = 1, we need to evaluate x(1) = 4.0 m cos(ω).
The phase of the motion at t = 2 seconds is ωt, which represents the argument of the cosine function in the equation of motion.