Final answer:
At maximum displacement in simple harmonic motion, a point on a vibrating string has zero velocity as it changes direction. Velocity is highest at the equilibrium position and is influenced by amplitude, system stiffness, and the object's mass.
Step-by-step explanation:
When a point on a vibrating string, or any object in simple harmonic motion (SHM), reaches its maximum displacement, its velocity goes to zero. At this point, the object is changing its direction of motion, and hence its velocity must pass through zero. The velocity is indeed a maximum at the equilibrium position (x = 0), where the displacement is zero and the object is neither stretching nor compressing the system.
Velocity is also determined by three main factors: the amplitude of the motion, the stiffness of the system described by the force constant k, and the mass of the object. A larger amplitude results in a greater maximum velocity (Umax), while a stiffer system leads to a higher Umax since it exerts more force for the same displacement. However, objects with larger masses have a smaller Umax, as they accelerate more slowly for a given force.
The maximum velocity Umax could be represented by the expression Umax = Aw, where A is the amplitude and w is the angular frequency. This maximum velocity occurs as the mass passes through the equilibrium position in both the positive and negative directions. Conservation of energy principles can be used to show that when the object starts with zero velocity at maximum displacement (x = X), the total energy is stored as potential energy, which equals ½kX².