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Hoofbeat

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Please could someone check my answers to the following and advise me on any of the bits I'm stuck on. Thanks

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Q. A disk of radius a and mass m is suspended at its centre by a vertical torsion wire which exerts a couple [tex]\fontsize{5} -c\theta [/tex] on the disk when it is twisted through an angle [tex]\fontsize{5} \theta [/tex] from its equilibrium position. Show that the oscillations of the disk are simple harmonic, and obtain an expression for the period [I've done this proof so that bit's fine]

A wire ring of mass m and radius a/2 is dropped onto the disk and sticks to it. Discuss what happens to (a) the period, (b) the amplitude, (c) the energy of the oscillations for the 2cases where the ring is dropped on (i) at the end of the swing when the disk is instantaenously at rest (ii) at the midpoint of the swing when the disk is moving with is maximum angular velocty. Assume the ring and disk are concentric.

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i) At end of swing

a) The period increase as the moment of inertia has increased

b) The amplitude increases as the velocity remains same as before, but the time period has increased

c) The energy is unchanged as at the end of the swing where v=0, the object only has potential energy.

ii) At midpoint of motion

a) The period increases due to increase in moment of inertia

b) ?

c) PE=0. There is a change in energy as the extra disk increases friction so the KE is lower?

Thanks.

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Q. A disk of radius a and mass m is suspended at its centre by a vertical torsion wire which exerts a couple [tex]\fontsize{5} -c\theta [/tex] on the disk when it is twisted through an angle [tex]\fontsize{5} \theta [/tex] from its equilibrium position. Show that the oscillations of the disk are simple harmonic, and obtain an expression for the period [I've done this proof so that bit's fine]

A wire ring of mass m and radius a/2 is dropped onto the disk and sticks to it. Discuss what happens to (a) the period, (b) the amplitude, (c) the energy of the oscillations for the 2cases where the ring is dropped on (i) at the end of the swing when the disk is instantaenously at rest (ii) at the midpoint of the swing when the disk is moving with is maximum angular velocty. Assume the ring and disk are concentric.

====

i) At end of swing

a) The period increase as the moment of inertia has increased

b) The amplitude increases as the velocity remains same as before, but the time period has increased

c) The energy is unchanged as at the end of the swing where v=0, the object only has potential energy.

ii) At midpoint of motion

a) The period increases due to increase in moment of inertia

b) ?

c) PE=0. There is a change in energy as the extra disk increases friction so the KE is lower?

Thanks.

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