# Solving Equation of Motion for Particle on Sphere's Surface

• delve
In summary, the equation of motion for a particle constrained to move on the surface of a sphere is given by using spherical coordinates and writing the applied force as $\mathbf{F}$=F_r\mbox{e_r}+F_\theta\mbox{e_\theta}+F_\phi\mbox{e_\phi}. However, since the particle is constrained, there must be a reaction force acting on it. This means that the total force is tangent to the surface and the equation of motion only concerns theta and phi. The external force, Fr, is not equal to zero and allows for calculation of the reaction force and potential damage to the surface.
delve
Here's a question from a book: A particle of mass m is constrained to move on the surface of a sphere of radius R by an applied force $$\mathbf{F}(\thetha, \phi)$$. Write the equation of motion.

Now here is the answer, but there is something I don't understand about it:

Using spherical coordinates, we can write the force applied to the particle as $$\mathbf{F}=F_r\mbox{e_r}+F_\theta\mbox{e_\theta}+F_\phi\mbox{e_\phi}$$ But since the particle is constrained to move on the surface of a sphere, there must exist a reaction force that acts on the particle.

Why isn't the force just $$\mathbf{F}=F_\theta\e_\theta+F_phi\e_\phi$$, and therefore, no reaction force?

Last edited:
delve said:
Why isn't the force just $$\mathbf{F}=F_\theta\e_\theta+F_phi\e_\phi$$, and therefore, no reaction force?

That is right. The radial part of the external force is compensated by the reaction force, so in total the force is tangent to the surface. The equation of motion should concern only theta and phi.

But if they speak only of the external force, it is not equal to zero but Fr. It allows calculating the reaction force (and possible damage to the surface).

great, thank you for your help :)

## What is the equation of motion for a particle on a sphere's surface?

The equation of motion for a particle on a sphere's surface is given by the spherical coordinate form of Newton's Second Law: m(r̈ - r(θ̇2 + ϕ̇2sin2θ)) = F, where m is the mass of the particle, r is the radial distance from the center of the sphere, θ is the polar angle, ϕ is the azimuthal angle, and F is the net force acting on the particle.

## What are the initial conditions needed to solve the equation of motion for a particle on a sphere's surface?

The initial conditions needed to solve the equation of motion for a particle on a sphere's surface are the initial position r0, θ0, and ϕ0, as well as the initial velocities r, θ, and ϕ. These initial conditions determine the trajectory of the particle on the sphere's surface.

## How do you account for the curvature of the sphere when solving the equation of motion for a particle on its surface?

The curvature of the sphere is accounted for in the equation of motion through the terms involving the polar angle θ and the azimuthal angle ϕ. These terms take into account the changing direction of the net force on the particle due to the curvature of the sphere.

## What is the significance of the angular momentum in the motion of a particle on a sphere's surface?

The angular momentum of a particle on a sphere's surface is conserved, meaning it remains constant throughout the particle's motion. This is due to the symmetry of the sphere, and it allows for the prediction of the particle's motion without having to solve the equation of motion at every instant in time.

## How does the equation of motion for a particle on a sphere's surface differ from that of a particle in flat space?

The equation of motion for a particle on a sphere's surface includes additional terms involving the polar and azimuthal angles, which account for the curvature of the sphere. In flat space, these terms are equal to zero and the equation of motion simplifies to Newton's Second Law in Cartesian coordinates.

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