Final answer:
The velocity of the oscillating mass is found by differentiating the position function, resulting in a sine function multiplied by the derivative of the interior function of the cosine, and the energy is given by the kinetic energy formula at maximum velocity.
Step-by-step explanation:
The question pertains to the physics concept of simple harmonic motion (SHM). It focuses on a mass oscillating on a spring and involves calculating velocity, position, and understanding energy at different points in the motion. To obtain the velocity (v) of the oscillating mass, we differentiate the position function x(t) = (2.5cm)cos(13t) concerning time (t).
The derivative of the cosine function gives us -sin(13t), multiplied by the chain rule factor from the derivative of 13t, which is 13. Hence, the velocity function is v(t) = -32.5 cm/s * sin(13t). The energy of the particle at the center of oscillation is the maximum kinetic energy when potential energy is zero, which is given by E = 1/2 * m * v^2, for a 0.2 kg mass at a speed of 5 m/s results in 2.5 J. Equation can be also obtained by considering the amplitude of the oscillation and the spring constant.