Particle on the outer surface of a sphere

Understanding Particle on the Outer Surface of a Sphere

  • A particle on the outer surface of a sphere moves in a circular path.
  • It is subject to gravitational force (mg), where m is the mass and g is the acceleration due to gravity.
  • It is also subject to a normal reaction force from the sphere’s surface.
  • Understanding the equilibrium between these forces will help you analyze the motion of the particle.

Normal Reaction and Gravitational Force

  • The gravitational force is always directed towards the center of the sphere, along the line that passes through the center of the sphere and intersects the point where the particle is.
  • The normal reaction is always perpendicular to the sphere’s surface and acts along the radial direction.
  • The equation of motion of the particle is governed by resolving these forces into radial and tangential components.

Decoding the Motion of the Particle

  • The motion of a particle in a vertical circle is synonymous with the motion of a pendulum bob or a roller coaster moving in a loop.
  • The angular speed and angle of inclination govern the tension in the string or the normal reaction on the roller coaster.
  • If the particle is projected with a speed less than root(5gr), it will not reach the top point of the sphere. Here, g is the acceleration due to gravity and r is the radius of the spherical path.
  • If the projection speed equals root(5gr), it will just reach the top, but cannot go beyond it. This speed is also called the minimum necessary speed.
  • If projected with a speed more than root(5gr), the particle covers the full circular path.

Energy Considerations

  • The loss of potential energy is exactly equal to the gain in kinetic energy for a freely falling particle. This principle, derived from the law of conservation of energy, applies to particles moving on the outer surface of a sphere too.
  • Energy considerations can provide an alternative solution to many problems involving a particle on the sphere.

Understanding Centripetal and Centrifugal Forces

  • Centripetal force is the net force acting on the particle towards the center of the circle. It is responsible for keeping the particle in its curved motion.
  • The apparent centrifugal force is felt by an observer situated on the particle looking outwards. Note that this force is a pseudo force as the frame of reference is not inertial.