Particle moving on the inside of a cone

Motion of a Particle Inside a Cone

A particle moving on the inside of a cone follows a circular path. This creates a motion similar to that of an object moving in a circle, but with an added dimension.
The motion of the particle is affected by gravitational force, normal reaction force from the cone, and frictional force if any.

Gravity acts vertically downwards on the particle with a force of mg, where m is the mass and g is acceleration due to gravity.
The particle experiences a normal reaction force (R) perpendicular to the surface of the cone.
If the cone is rough, there will be a frictional force at the point of contact between the particle and the cone.

Centripetal force is the net force causing the particle to move in a circular path.
For a particle inside a cone, this force is provided by the horizontal component of the reaction force.
The vertical component of the reaction force must balance the gravitational force acting on the particle.

The forces acting on the particle inside the cone will give rise to two equations of motion: one for the direction parallel to the surface and one for the direction perpendicular to the surface.
For motion perpendicular to the surface: R cos(θ) = mg, where R is the normal reaction, θ is the angle of the cone, m is the mass of the particle and g is acceleration due to gravity.
For motion parallel to the surface: R sin(θ) = mω²r, where R is the normal reaction, θ is the angle, m is the mass of the particle, ω is the angular speed and r is the radius of the circular path.

Conical pendulum is another common application of a particle moving inside a cone, where the particle is tied to a string and made to move in a horizontal circular path.
The tension in the string acts as the normal reaction force in this case.
All the concepts and equations discussed above apply to such scenarios as well.