Analysis of Particle's Movement on a Sphere in a Frictionless Environment

In this article, we will analyze the movement of a frictionless particle on a solid sphere, focusing on the specific conditions that cause the particle to lose contact with the sphere. This involves understanding the principles of conservation of energy and the forces involved in the particle's movement. We will derive an exact angle at which the particle leaves the sphere.

Problem Description and Assumptions

A particle of mass ( m ) kg is released from rest at the highest point of a solid sphere of radius ( a ) metres. The particle is to be analyzed under the assumption that there is no friction between the particle and the sphere, and the sphere is fixed to the ground. The particle starts its journey when the radius vector of the particle has turned through an angle ( theta ). At the instant of interest, the particle has moved such that its speed ( v ) can be determined from the initial potential energy and kinetic energy relationships.

Equations of Motion and Forces Involved

The particle's motion is governed by the following equation:

mg cos(theta) - N m frac{v^2}{a}

Where ( N ) is the normal reaction force. When the particle leaves the sphere, the normal reaction force vanishes, leading to:

mg cos(theta) m frac{v^2}{a}

Thus,

cos(theta) frac{v^2}{ga}

This equation shows the relationship between the angle ( theta ) and the speed ( v ) of the particle.

To find ( v ), we use the principle of conservation of energy:

mgh frac{1}{2}mv^2

Where ( h ) is the vertical distance the particle has fallen relative to the initial position. For a sphere, the vertical distance can be written as:

20 frac{1}{2}v^2 10h

Giving,

v^2 20gr cos(theta)

Substituting ( v^2 ) into the force equation gives:

cos(theta) frac{2gr cos(theta)}{ga}

Simplifying, we get:

cos(theta) frac{2 cos(theta)}{3}

Solving for ( theta ), we get:

cos(theta) frac{2}{3}

Therefore, the particle leaves the sphere when ( theta ) is approximately ( 48.2^circ ).

Conclusion

The analysis demonstrates that the key factor determining when the particle leaves the sphere is the angle ( theta ) and the speed ( v ) of the particle. The relationship derived shows that the frictionless particle will lose contact with the sphere at an angle of ( theta approx 48.2^circ ). This result is independent of the sphere's size but depends on the starting angle ( theta ).