A solid uniform sphere of mass m and radius R rolls without slipping down an inc

Question

A solid uniform sphere of mass m and radius R rolls without slipping down an incline of height h.

(a) What forms of mechanical energy are associated with the sphere at any point along the incline when its angular speed is ?? Answer in words and symbolically in terms of the quantities m, g, y, I, ?, and v. (Do this on paper. Your instructor may ask you to turn in this work.)

(b) What force acting on the sphere causes it to roll rather than slip down the incline?

Score: 0.5 out of 0.5

Comment:

(c) Determine the ratio of the sphere's rotational kinetic energy to its total kinetic energy at any instant.

=

Explanation / Answer

a) Potential energy = m*g*y

linear kinetic energy = 0.5*m*v^2

rotational kinetic energy = 0.5*I*w^2

Here, I = (2/5)*m*R^2 (moment of Inertia of sphere)

w is the angular speed of the sphere.

2) As the sphere rolls down the pont which is in contact with the surface is always

instantaneously at rest.

so, The force which causes rolls with out slipping shoud be static frctional force

3) KEr = 0.5*I*w^2

= 0.5*(2/5)*m*R^2*w^2

= 0.2*m*v^2

KEt + KEr = 0.5*m*v^2 + 0.2*m*v^2

= 0.7*m*v^2

so,

KEr/(KEt + KEr) = 0.2/0.7

= 2/7

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