Equation of Motion for Central Force The equation of motion for a central force describes the motion of a particle moving in a central potential well, wh...
The equation of motion for a central force describes the motion of a particle moving in a central potential well, which is a potential energy landscape characterized by a central attractive force.
The equation takes the form:
v² = 1/r² d²r/dt²
where:
v is the particle's velocity
r is the distance from the center of the potential well
d²r/dt² is the second derivative of the distance with respect to time
This equation describes the particle's motion in a circular orbit, where the center of the potential well coincides with the center of the circle.
Example:
Consider a particle moving in a potential well represented by a square well with a potential energy barrier at a distance of 2 units from the center. The equation of motion for this particle would be:
v² = 1/r² d²r/dt²
where:
r = 2
d²r/dt² = -1
This equation describes a particle starting at rest at the edge of the potential well and spiraling inward as it approaches the center.
Further Notes:
The potential energy well can have various shapes, including potential wells, springs, and barriers.
The equation can also be derived using the conservation of angular momentum.
Solutions to the equation of motion provide the particle's position, velocity, and acceleration as a function of time