A particle slides on the surface of a fixed smooth sphere starting from the topmost point. Find

A particle slides on the surface of a fixed smooth sphere...

Key Concepts

Dynamics of a Particle on a Curved Surface

This involves analyzing the forces acting on the particle as it moves along a curved path, combining Newton's laws of motion with the geometrical constraints imposed by the curve. This framework is essential for accurately modeling the motion and determining key points such as departure from the path.

Normal Force and Loss of Contact

The normal force is the perpendicular reaction force from the surface that supports the particle. In problems of this type, the point at which the normal force becomes zero marks the moment the particle loses contact. This condition is essential to determine the critical angle or point of departure.

Centripetal Acceleration

Centripetal acceleration is the acceleration that keeps an object moving along a curved path. In the context of a particle sliding on a sphere, the component of gravitational force that acts toward the center provides this acceleration, and changes in this force are critical in determining when the particle can no longer maintain contact with the sphere.

Conservation of Energy

This concept explains how the energy of the particle is transformed from gravitational potential energy to kinetic energy as it moves along the curved surface. It is used to relate the height descended by the particle to the increase in speed, laying the groundwork for analyzing its motion.

Transcript

00:01 So we are trying to find out this angle theta here through which the particle rotates on the surface of the sphere by the time it leaves the surface of the sphere.

00:08 So our first step is to say that we know that the normal contact force at this point is equal to zero because it's leaving the surface of the sphere and therefore is no longer in contact with the surface of the sphere.

00:19 This means that the only force acting on it is the centripetal force which acts in this direction.

00:25 And we know that that must be equal to the component.

00:30 Of the gravitational force, so the mass of the object's induced force, which is equal to the mg multiplied by that theta to obtain it and resolve it in that direction.

00:43 So what we can start by doing is equating these, so saying the centripetal force equal to mv squared over r in this case is equal to the mass of the object multiplied by the gravitational constant.

01:07 And then resolved in this direction.

01:09 So what we can see is that as these are two parallel lines, this angle must also be equal to theta.

01:16 So resolving this, we know that the component in this direction is mg cosine theta.

01:24 So what we can do now is neaten up this equation by cancelling out these ms and taking this r up to this side.

01:31 So the expression we're left with here is equal to v squared, equals rg cosine theta.

01:46 So our next step is to say that we know that energy must always be conserved.

01:51 So this means that whatever change in kinetic and potential energy occurs between point a and b must be equal.

01:58 So we can say that delta k -e is equal to delta p -e...

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