Work done pulling a string through a hole on a rotating particle

[matpo39]

hi, I am having a little trouble with the last part of this problem.

A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string, whose other end passes through a hole in the table, where i am holding it. Initially the particle is moving in a circle of radius R_0 with angular velocity w_0, but i now pull the string down through the hole until length R remains between the whole and the particle. (a) what's the particles angular velocity now? (b) assuming that I pull the string slowly that we can approximate the particles path by a circle of slowly shrinking radius, calculate the work i did pulling the string.

i was able to get part a by using conservation of angular momentum and i got

w= (R_0/R)^2*w_0

for part b i know that to find the work done i need to take the integral of F.dr, but i and not really sure how i would set that up.


thanks

 

[NateTG]

If you know the work energy theorem, there's an easy answer.

Otherwise:
What is the tension in the string when the mass is at radius $$r_i$$?
What force do you need to pull at to pull the mass in?

 

[matpo39]

ok I got the tension to be
(m*v^2)/r_0 = m*r_0*w_0^2 = F

so the i took the intagral of (F*dr) evaluated at r_0 to r and got (1/2)*m*w_0^2(r^2 - r_0^2)

is this right? because change in KE = work and for change in KE i get

(1/2)*m[ (r*w)^2 - (r_0 *w_0) ^2] , which is close to the formula i was expecting to get.

 

[NateTG]

Looks good to me.