Yes, except why assume the phase constants are zero?GeoStudy said:Homework Statement
I just need a hint. So we are given:
F = -kr
We are asked:
Show that:
(a) The orbit is an ellipse with the force center at the center of the ellipse.
Homework Equations
I guess we break it up into its components:
The Attempt at a Solution
m d2x/dt2 = -kx => x = Acos(ωt + βx)
m d2y/dt2 = -ky => y= Bsin(ωt + βy)
where βx = βy = 0
A central force is a type of force that acts on an object in a way that is always directed towards a fixed point, known as the center. Examples of central forces include gravitational force and electric force.
The equation for the motion of a particle in a central force field is given by Newton's second law of motion: F=ma, where F is the net force acting on the particle, m is the mass of the particle, and a is the acceleration. This equation can be rewritten as F=mv²/r, where v is the velocity of the particle and r is the distance from the center of the force.
The direction of a particle's motion is always tangential to its circular path, while the central force acts along the radius of the circle, towards the center. This means that the central force is always perpendicular to the direction of the particle's motion.
The magnitude of the central force determines the speed at which the particle travels along its circular path. The greater the magnitude of the force, the faster the particle will move. However, the direction of the force also plays a crucial role in determining the path of the particle.
Angular momentum is a conserved quantity in a central force field, meaning it remains constant throughout the particle's motion. This is because the central force only acts along the radius of the circle, causing the particle to move in a circular path and maintaining a constant distance from the center. The magnitude of the angular momentum depends on the mass, velocity, and distance from the center of the particle.