Answer with details plz: A uniformly dense sphere of radius R and mass m rolls d

Question

Answer with details plz:

A uniformly dense sphere of radius R and mass m rolls down an inclined track of vertical height h, along a short horizontal length, and through a vertical circular loop of radius C. Neglect dissipative effects, and assume C >> R. You may take the acceleration due to gravity to be g. Show that the minimum height of the incline for this to happen is h = 27/10C Notice that this implies h >> R. When the sphere is released from the height found in (a), what are the linear and angular speeds of the sphere as it moves along the short horizontal length? When the sphere is released from the height found in (a), what are the linear and angular speeds of the sphere at the top of the loop?

Explanation / Answer

moment of inertia of the sphere=0.4 mr^2

also let the angular velocity be w.

let the linear velocity be v.so,

v=w*r

now balancing forces at the top of the loop,

mu^2/r=mg

or u=(gr)^0.5

so conserving energy,

0.5mv^2+0.5*0.4*mv^2+mgh=2mgC+0.5mu^2+0.5*0.4mu^2

we know that initial velocity v=0.so,

mgh=2mgC+0.5mgh+0.5*0.4mgh

or h=2.7 C

=27/10 C

b)again conserving energy,

mgh=0.5mv^2+0.5Iw^2

or mgh=0.5mv^2+0.5*0.4*mv^2

or (10gh/7)^0.5=v

c)linear speed= (gr)^0.5 (found above)

angular speed=v/r

=(g/r)^0.5

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