Particle constrained to move on a sphere

In summary, the problem involves a particle of mass m moving on a sphere of radius R, constrained by an applied force F(theta, phi). The relevant equations are x = R sinθ cosφ, y = R sinθ sinφ, and z = R cosθ, where theta and phi are functions of time.
  • #1
dray
2
0

Homework Statement


A particle of mass m is constrained to move on a sphere of radius R by an applied force F(theta,phi). Write the equation of motion.


Homework Equations


x=vtRcos(phi)sin(theta)
y=vtsin(phi)sin(theta)


The Attempt at a Solution


F=m(del)2(x+y)
 
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  • #2
Hi dray. Welcome to PF.

Forces are vectors. How would you write the given force as a vector? What does the problem say? Is this a force in the radial direction only or does it have angular components?

What does your attempt at a solution represent? It cannot be an expression of Newton's Second law because the "del" operator has to do with spatial derivatives and the Second Law involves a second time derivative.

Your relevant equations should be

x = R sinθ cosφ
y = R sinθ sinφ
z = R cosθ

There is no explicit time dependence in these expressions, but theta and phi themselves are functions of time.
 

1. What is a particle constrained to move on a sphere?

A particle constrained to move on a sphere is a physical system where a particle is confined to move along the surface of a spherical object. This type of system is commonly used in physics and mathematics to model various phenomena, such as planetary motion and atomic structure.

2. How is the motion of a particle constrained to move on a sphere described?

The motion of a particle constrained to move on a sphere is described using spherical coordinates, which include the radius, longitude, and latitude of the particle's position. The particle's movement is also affected by the forces acting upon it, such as gravity or electromagnetic forces.

3. What are some applications of a particle constrained to move on a sphere?

Particle constrained to move on a sphere systems have numerous applications in various fields, such as astrophysics, geophysics, and molecular biology. Some examples include studying the orbits of planets, analyzing the Earth's magnetic field, and understanding the behavior of molecular structures.

4. How does the motion of a particle constrained to move on a sphere differ from a particle moving in a straight line?

A particle constrained to move on a sphere has a more complex motion compared to a particle moving in a straight line. This is because the particle's position is not only determined by its speed and direction but also by its location on the sphere's surface and the forces acting upon it. Additionally, a particle on a sphere can have a range of motion, while a particle moving in a straight line has limited motion along a single axis.

5. Are there any real-life examples of a particle constrained to move on a sphere?

Yes, there are many real-life examples of a particle constrained to move on a sphere, such as the Earth's orbit around the sun, the moon's orbit around the Earth, and the rotation of a ball on a smooth surface. Additionally, the motion of electrons around the nucleus of an atom can also be modeled as a particle constrained to move on a sphere.

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