The position of a body moving in simple harmonic motion is given by

m
t
x 





+
=
4
9
cos
3
Ï€
. For this motion, what are the (a) amplitude, period, phase
constant, frequency, and angular frequency? Determine the body’s (b) maximum
speed and maximum acceleration.

To determine the amplitude, period, phase constant, frequency, and angular frequency of a body in simple harmonic motion, we can start by analyzing the given equation:

x = A * cos(ωt + φ)

where:
- x is the position of the body
- A is the amplitude of the motion
- ω is the angular frequency (2πf)
- t is the time
- φ is the phase constant

From the given equation, we can identify the following values:

Amplitude (A): In this case, the amplitude (A) is 4/9.

Frequency (f): The frequency (f) can be determined as the reciprocal of the period(T). The equation shows that the body completes one full oscillation (2π) when t = T. So, ωT = 2π and ω = (2π)/T. From the given equation, we know that ω = 3π. Therefore, the frequency (f) is ω / (2π) = 3.

Period (T): From the above calculation, we know that ωT = 2π. Substituting the given value ω = 3π, we can solve for T. Therefore, T = (2π) / (3π) = 2/3.

Phase constant (φ): The phase constant (φ) is the initial phase of the motion. In the given equation, φ is not specified. When φ is assumed to be zero, it means the motion starts at its maximum amplitude. So, we can assume φ = 0.

Angular frequency (ω): The angular frequency (ω) is already given as 3π.

Now let's determine the body's maximum speed and maximum acceleration.

Maximum speed (v_max): The maximum speed occurs when the displacement is maximum (A) and the acceleration is zero. At maximum displacement, the velocity is zero, so we need to find the derivative of the position equation with respect to time (x). Differentiating x = A * cos(ωt + φ) with respect to t gives us v = -A * ω * sin(ωt + φ). Substitute the value A = 4/9 and ω = 3π into the equation to find v_max.

v = -(4/9) * 3π * sin(3πt)
The maximum value of sine is 1, so v_max = -(4/9) * 3π.

Maximum acceleration (a_max): The maximum acceleration occurs at the extreme points of the motion (positive and negative maximum displacement). At these points, the acceleration is maximum and the velocity is either zero or at its maximum (v_max). Differentiating v = -A * ω * sin(ωt + φ) with respect to t gives us a = -A * ω^2 * cos(ωt + φ). Substitute the value A = 4/9 and ω = 3π into the equation to find a_max.

a = -(4/9) * (3π)^2 * cos(3πt)
The maximum value of cosine is 1, so a_max = -(4/9) * (3π)^2.

Therefore, the answers to the given questions are:

(a) Amplitude: 4/9
Period: 2/3
Phase constant: 0
Frequency: 3
Angular frequency: 3π

(b) Maximum speed: -(4/9) * 3π
Maximum acceleration: -(4/9) * (3π)^2