Mass wraps around rod in circular motion

[unscientific]

Homework Statement

A string attached to the center of a rod with radius R has a mass attached at the other end, moving with speed v₀. Because of acceleration due to gravity the mass moves down while undergoing circular motion, causing the string to wrap around the rod. Find an expression for the time taken for the string to completely wrap around the rod.

Homework Equations

v = rw

The Attempt at a Solution

-constant tangential velocity since acceleration due to gravity acts only downwards
-w = v/r, so as the string wraps around the rod, the angular velocity increases

w_(t) = v₀/r_(t)

-w_(t) is the same along all lengths of the string

-tried to figure out dr/dt, but to no avail

 

[supratim1]

use energy conservation, where x is the distance moved downwards by the mass, so

mgx = 1/2 mv^2 + 1/2 Iw^2

and use, w = v/r

 

[unscientific]

use energy conservation, where x is the distance moved downwards by the mass, so

mgx = 1/2 mv^2 + 1/2 Iw^2

and use, w = v/r

hmm, i don't really understand how the term '1/2 Iw^2' comes about - the tangential velocity stays the same, all its doing is accelerating downwards

Loss in GPE = Gain in KE
mgx = 1/2 mv_y^2 (in the y-direction)

I'm interested in finding out the time taken for a given string of length l to finish wrapping a rod of radius R entirely.

-at every point in time the mass is undergoing circular motion
-fixed tangential velocity, but increasing angular velocity
-in order to find the time taken to wrap around the rod, i must find an expression for dl/dt, where l is the length of the string. I imagine it to be some function of (r,v₀ and t)

 

[unscientific]

it's okay, i solved it! :)

 

[rwisz]

I would approach this problem by first defining the variables and parameters involved. The radius of the rod, R, and the initial velocity of the mass, v0, are given. The mass itself can be represented by its position on the string, r(t), which changes as the string wraps around the rod.

Next, I would use Newton's laws of motion to analyze the forces acting on the mass. The only force acting on the mass is gravity, which causes it to accelerate downwards. This acceleration also causes a change in the angular velocity of the mass as it moves along the string.

To find the time taken for the string to completely wrap around the rod, I would use the equation v = rw, where v is the tangential velocity, r is the radius, and w is the angular velocity. Since the tangential velocity remains constant, we can set the initial tangential velocity, v0, equal to the tangential velocity at any point along the string, v(t).

Using this equation, we can solve for the angular velocity at any point along the string, w(t). Then, by integrating w(t) with respect to time, we can find the time taken for the string to completely wrap around the rod.

In conclusion, the expression for the time taken for the string to completely wrap around the rod would involve the initial velocity, radius of the rod, and the acceleration due to gravity. It would also involve integrating the angular velocity with respect to time.