The constrained motion of a particle along a circular curve: Newtonian and Relativistic model
Kostas Papamichalis, Dr of Theoretical Physics
Download the theoretical analysis of the models and the included activities, by clicking here: Constrained Motions
The objective of the simulation is to study the constrained motion of a paticle (bead) along a circular curve. The study is accomplished within two different theoretical context: a) the Einstein General Theory of Relativity and b) the Newtonian Mechanics. The mass of the bead and the initial condition of the mechanical system are identical in the two models. The motion of the bead is considered with respect to a Cartesian coordinate system, with space origin O. The center of the circular orbit coincides with O; its radius is L. The system is deterrmined by one degree of freedom: the deviation angle "theta" of the bead radius-position from the negative vertical axis Ox2. The initial value of theta is controlled by the user. The initial angular velocity is zero. The bead is moving in a homogeneous vertical gravitational field, whose strength g is also controlled by the user.
The left window depicts the motion of the bead according to the relativistic model and the right one, according to the Newtonian model. The graph-windows show in real-time the change of the deviation angle and the angular velocity with time, for each model.
The deviation angle and the angular velocity of the beam versus time
The upper graph depicts the change of the deviation angle theta in real time. The lower graph depicts the angular velocity in real time. The blue curves correspond to the relativistic model and the red to the Newtonian model.
If the gravitation field is strong enough, the maximum magnitude of the beam velocity according to the Newtonian model, could exceed the light-velocity; that is not possible according to the relativistic model. The magnitude of the velocity in the relativistic picture cannot exceed the value: c*sqrt(1-2gL/c2). Hence, the observed delay of the relativistic system compared to the Newtonian could be explained. Nevertheless, the relativistic effects are most exciting when the gravitation field g approaches the crucial value 12.5 (in the sim-units).