Final answer:
A particle's acceleration proportional to the negative of its displacement indicates simple harmonic motion (SHM). The motion is periodic and oscillatory, oscillating about the equilibrium position with a position function dependent on time and characterized by the angular frequency.
Step-by-step explanation:
When a particle's acceleration a is proportional to and in the opposite direction of its displacement x, such as a = -b⋅x, it undergoes a type of motion known as simple harmonic motion (SHM). The equilibrium position in this context refers to the point where the net force on the particle is zero, typically designated as x = 0. The force acting on the particle can be described by Hooke's Law, which states that the force, Fx, is equal to -kx (where k is the force constant).
In the context of SHM, the particle oscillates between turning points with an amplitude A. The position of the particle as a function of time can be represented by x(t) = A cos (ω t + δ), where angular frequency (ω) is given by the formula ω = √(k/m) and depends on the mass m of the particle and the force constant k. This motion describes a periodic path in an oscillatory manner back and forth across the equilibrium position, which in the case of a one-dimensional motion, is along the x-axis.