Final answer:
The question involves finding a function for the simple harmonic motion given the displacement is zero at time t=0. The function will involve displacement, velocity, and acceleration as related to the period of the motion, spring constant, mass, and amplitude.
Step-by-step explanation:
The question is asking to find a function that models the simple harmonic motion (SHM) with the given properties. In SHM, the displacement x(t), velocity v(t), and acceleration a(t) are related to the spring constant k, mass m, amplitude X, and time t. The displacement is generally given by x(t) = X cos(\(\frac{2\pi t}{T}\)), where T is the period of the motion. Since displacement is given to be zero at t=0, this confirms that a cosine function should be used. The velocity can be modeled by v(t) = -Umax sin(\(\frac{2\pi t}{T}\)), where Umax is the maximum speed, related to the spring constant, mass, and amplitude as Umax = \(\sqrt\frac{k}{m}X\). The acceleration is the second derivative of the displacement, so a(t) provides information about the force involved and is inversely proportional to the mass m and directly proportional to kX.