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where the numerator μ may be positive (repulsive) or negative (attractive). If such an inverse-cube force is introduced, Newton's theorem says that the corresponding solutions have a shape called Cotes's spirals [clarification needed]. These are curves defined by the equation [20] [21]
The force may be either attractive or repulsive. The problem is to find the position or speed of the two bodies over time given their masses , positions , and velocities . Using classical mechanics, the solution can be expressed as a Kepler orbit using six orbital elements .
where A is the Hamaker coefficient, which is a constant (~10 −19 − 10 −20 J) that depends on the material properties (it can be positive or negative in sign depending on the intervening medium), and z is the center-to-center distance; i.e., the sum of R 1, R 2, and r (the distance between the surfaces): = + +.
No exact solutions of the Kepler problem have been found, but an approximate solution has: the Schwarzschild solution. This solution pertains when the mass M of one body is overwhelmingly greater than the mass m of the other. If so, the larger mass may be taken as stationary and the sole contributor to the gravitational field.
The general mathematical form of such inverse-square central forces is = = for a constant , which is negative for an attractive force and positive for a repulsive one. This special case of the classical central-force problem is called the Kepler problem.
Calculating the attractive or repulsive force between two magnets is, in the general case, a very complex operation, as it depends on the shape, magnetization, orientation and separation of the magnets. The magnetic pole model does depend on some knowledge of how the ‘magnetic charge’ is distributed over the magnetic poles.
The force is negative, indicating that the force is attractive: by moving the two plates closer together, the energy is lowered. The presence of ħ shows that the Casimir force per unit area F c / A is very small, and that furthermore, the force is inherently of quantum-mechanical origin.
The force required to separate two colloid particles can be measured using optical tweezers. This method uses a focused laser beam to apply an attractive or repulsive force on dielectric micro and nanoparticles. This technique is used with dispersion particles by applying a force which resists depletion forces. The displacement of the particles ...