Rocking oscillatory motion and a technique for its measurement

Abstract

Equations are derived giving the period of rocking oscillatory motion of various bodies when placed on a flat horizontal surface, displaced from equilibrium, and released. The bodies include a sphere and a circular disk with a flat surface, a circular disk with a circular hole, and elliptical disks with circular and elliptical holes. A technique that can be used in an undergraduate mechanics laboratory is described for determining the period of rocking oscillatory motion. The technique consists of measuring the frequency with which electrical continuity is completed when a metallic body instantaneously makes contact with two conducting plates as it undergoes rocking oscillatory motion. The timing is accomplished with digital electronics.

Circular arc pendulum: experiment and motion’s analysis

“…In this type of oscillation, the oscillating object is not hanged by a rod or a rope, but it rather oscillates around its stationary position. This situation is verified in many cases, for instance, (a) a body on a ship or a moving vehicle and (b) a construction or an object exposed to an earthquake [1,[8][9][10][11][12][13]. One of the most important benefits of studying rocking oscillation is the isolation of seismic waves to minimize the risk of buildings and bridges collapse [9,12,13].…”

“…rest position), the vertex of this arc contacts the flat surface at the point (O) while the arc's centre of mass (C) is located at a distance (d) above the flat surface. The arc's centre of curvature (or geometrical centre (M)) is vertically collinear with both O and C. If this stable arc suffers a non-sliding motion due to a certain little impulse, it would oscillate around the axis of contact exhibiting a rocking oscillatory motion [10].…”

“…By analyzing the motion of the oscillating arc, it is obvious that the given energy transfers the arc's vertex to be in contact with the flat surface at another point ( Ó), see figure 2(ii). In this case, the rolled horizontal distance (x) can be given by equation (10).…”

“…x = R 1 .θ (10) where, θ is the geometrical central angle enclosing the distance x as shown in figure 2.…”

“…There is another type of oscillations which can be frequently observed in nature. When a rigid body is located on a ground and begins to oscillate due to either a disturbance in the ground or by applying a force on the body itself, the oscillation in this case is called a rocking oscillation [1,9,10]. In this type of oscillation, the oscillating object is not hanged by a rod or a rope, but it rather oscillates around its stationary position.…”