How Does Angular Momentum Affect the Velocity of a Disc on a Frictionless Table?

[spaghetti3451]

Homework Statement

A disc of mass M, which may be considered to be a point mass, is placed on a frictionless horizontal table. A massless string is fastened to the disc and is passed through a small hole at the centre of the table. The lower end of the string is tied to the end of a flexible rope of mass m per unit length which lies on the floor just under the hole in the table. Initially the lower end of the massless string is held firmly just at floor level and the disc is made to move in a circle, radius r₀ about the hole, with a velocity of constant magnitude v₀.

At a subsequent time, the end of the massless string is released from the floor, allowing the radius r of the circular motion of the disc to vary and the rope to be lifted off the floor. You may assume that the contact point between the rope and the floor remains vertically below the hole in the table and may neglect the effects of any horizontal motion of the rope along the floor.

The subsequent velocity of the disc has a radial component v_r and an azimuthal component v_∅. Show that the azimuthal component of the velocity is given by v_∅ = v₀r₀/r.

Homework Equations

The Attempt at a Solution

Need a quantity that is conserved under the change of situation. Is it angular momentum?

 

[Mechdude]

Homework Statement

...

Need a quantity that is conserved under the change of situation. Is it angular momentum?

remember the rope with mass gains P.E. while moving upwards, what form of energy does the mass on the table have? some of that is converted to P.E., and so the (insert answer to my question here) should remain constant.

 

[spaghetti3451]

The rope with mass gains P.E. while moving upwards; the mass on the table has kinetic energy. Some of that is converted to P.E., and so the total energy should remain constant.

Thank you so much!

But I'm not sure how the fact that the energy is conserved leads to the formula. Any hints?

 

[Mechdude]

But I'm not sure how the fact that the energy is conserved leads to the formula. Any hints?

i doubt it will lead to that, but yes, angular momentum is still conserved to answer your first question. To get the formula conservation of angular momentum would still work, though it would seem like a one step (so short it seems like a trick) solution imho.

 

[spaghetti3451]

L = r X p. Initally, the disc has only no radial component, so the angular momentum is mass*v₀*r₀. Later, the disc has a radial component and an azimuthal component, but the cross product of radial component with the the position gives zero, so that the angular momentum is mass*v_∅*r. Equating the two gives the above formula. What do you think?

 

[spaghetti3451]

Would anyone care to say yes or no?