A wheel is breaded down an inclined phase, starting at rest from a height h. a)

Question

A wheel is breaded down an inclined phase, starting at rest from a height h. a) If the wheel has a radius R and a mass M, what is the moment of inertia of a the wheel about the center? b) If the the sphere has a velocity v_rms at a given point, what is the expression for the translational kinetic energy? c) What is the angular velocity at that point in terms of v_rms and R? d) What is the rotational kinetic energy in terms of v_cms? e) If the translational plus rotational kinetic energy at the bottom of the incline is equal to the gravitational potential energy at the top of the ramp (mgh), what is v_cm at the bottom of the incline?

Explanation / Answer

(a) I= integration(r2 dm)

so moment of inertia= MR2

(b) the expression for translational kinetic energy=(1/2)M vcm2

(c) angular velocity w= vcm /R

(d) rotational kinetic energy = (1/2)I w2

putting the value of w , we get

rotational K.E.=(1/2)*I*vcm2 /R2

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