resistance encountered when a spherical object slides on a surface?????
Answers
Explanation:
When an object is rolling on a plane without slipping, the point of contact of the object with the plane does not move. A rolling object's velocity v is directly related to its angular velocity ω , and is mathematically expressed as v=ωR v = ω R , where R is the object's radius and v is its linear velocity.
Introduction
The dynamical behaviour of a body as it passes through a gas has been the subject of much study in the fields of both astronomy and mathematics. Jeans (1940) and Chapman & Cowling (1960) considered the behaviour of the individual gas molecules in their works on the kinetic theory of gases. In astronomy the movement of objects through a rarefied gas has received much theoretical and practical attention. Applications range from the study of the passage of hydrogen dust by the solar wind (Hoyle 1960) to the interactions between the Earth and dust from the zodiacal cloud (Kortenkamp 1998).
Many small objects, with radii of the order of ∼10–0.1 μm, inhabiting the Solar System attain hypervelocity speeds. These include ionized dust particles accelerated by the Solar wind and debris ejected by disintegrating comets, which have been found with velocities of 70 km s−1 or greater. Experimental work done on the impact of hypervelocity dust particles and ions on to targets has shown that they liberate many particles from the surface of the target in a process known as sputtering. The increased resistance of the particle generated by the back impulse of the particles sputtered from its surface has been found to be around ten times the momentum of the initial impacting particle (Wallis 1986).
In this paper we consider the motion of a hypervelocity spherical object and the resistance it encounters on its passage through a gas. The radius of the object considered is supposed to be less than the mean free path of the gas through which it passes. A quite general expression for the resistance of motion arising from particle sputtering for a hypervelocity sphere moving through a gas is obtained. This is then applied to the problem of the motion of a micrometer incident on the Earth's upper atmosphere. The velocity profiles of motion of a meteor slowed by sputtering through the atmosphere will be compared with those of a hypervelocity meteor slowed down by specular refection of gas particles from its surface.
2 Review of earlier work
The majority of work carried out on the problem of the resistance by a gas on a body moving through the gas is based on Stokes' law. This states that the resistive force on a particle moving through a gas is directly proportional to the product of the radius of the body and its velocity relative to the gas. The derivation of Stokes' law assumes a continuous, incompressible, viscous medium of infinite extent. The particles forming the medium are taken to be rigid spheres.
Problems occur in applying Stokes' law to the case when the mean free path between gas molecules is greater by an order of magnitude, or more, than the radius of the body passing through the gas. Under this condition the gas does not act as a continuous medium and if the number of collisions between the gas molecules and the body is sufficiently low, so that there are no molecules back-scattered on to the body, the gas can be considered non-viscous.
The mean free path of a gas, λ, is related to the radius, r, of a spherical body in contact with the gas by the Knudsen number, Kn: