Given: Find: A) Using the Parallel-Axis Theorem, give a formula for the moment o

Question

Given:

Find:

A) Using the Parallel-Axis Theorem, give a formula for the moment of inertia of the sphere in the figure about its axis of rotation.

B) Derive a formula for the rotational kinetic energy of the sphere, using the formula for the moment of inertia in part (a). Then substitute in the values  M = 0.200 kg,  R = 0.200 m, = 35.0 rad/s and d = 0.250 m to obtain a numerical value for the rotational kinetic energy.

C) Obtain a formula for the translational kinetic energy of the center of mass of the sphere.  In other words, calculate the (translational) kinetic energy of a point particle of mass  M that moves in a circle of radius d at angular speed .

D) Obtain a formula for the rotational kinetic energy of a uniform solid sphere of radius R and mass M if it rotates at angular speed about an axis through its center, instead of about the axis shown in the figure.

E) Add the two results of (c) and (d).  Show that this resulting formula is the same as the one you used to calculate the rotational kinetic energy in part (b), where all of the kinetic energy was considered to be rotational kinetic energy.

Any help with any of these parts would be great! I'm very lost so any help is much appreciated, thanks!

Explanation / Answer

the parallel axis theorem or Huygens-Steiner theorem can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular distance (r) between the axes. The moment of inertia about the new axis z is given by: i=i(through centre of mass)+m*(perpendicular distance between centre of mass and axis)^2 A)I=2/5(mr^2)+md^2....here I(THROUGH CENTRE OF MASS )=2/5(mr^2) b)K.E=1/2(I w^2)=1/5(mr^2)w^2+1/2(md^2)w^2 ... putting value of i from above ke for given values=9.616 J c)translational k.e=1/2(mv^2)=1/2(m(w*d)^2).........bcoz v=w*d d) if axis is trough center then I is gien as I=2/5(MR^2) K E=1/2(Iw^2)=1/5(mr^2)w^2 ...using I e)clearly adding c and d gives b

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