How Will the Body Move When the Monkey Climbs the Rope?

[Petrulis]

Homework Statement

Here is the drawing which shows the situation:

http://img260.imageshack.us/my.php?image=monkeygn4.jpg

The disc is weightless. There is a rope which is rolled over the disc. A monkey is fastened at the point A. At the point B there is a body bound to rope. The mass of the monkey is equal to the mass of the body, so at the beginning the system is in balance. The monkey starts to move up the rope with speed v in respect of the rope. How will the body (which is bound at the point B) move while the monkey will be climbing up the rope?

The Attempt at a Solution

The monkey and rope are initially at rest, but then the monkey starts moving along the rope at speed V (relative to rope). It doesn't say how long it takes the monkey to reach that speed from rest, but let's call that time "t".

It also doesn't say whether the rope moves or stays stationary while this acceleration is taking place. But let's assume that the rope moves down (on the monkey's side) at speed V_r, while the monkey moves up (relative to the ground) at speed V_m. The monkey's speed V along the rope, is a combination of those two speeds: V = V_m + V_r.

So, the monkey accelerates upward at rate of V_m/t. Also, the only forces acting on the monkey are the rope's tension T, and the monkey's weight mg. So, by Newton's 2nd Law:

Fnet = m*a

T - mg = m*V_m/t

or:

T = m*V_m/t + mg

or:

T = m(V - V_r)/t + mg

Now, the object on the other side of the rope feels this same tension T at the same time. Therefore, its upward acceleration is:

A_o = (T - mg) / m

= (m(V - V_r)/t + mg - mg) / m

= (V - V_r)/t

But we also know that the rope is going up (on the object's side) at speed V_r after t seconds. So another expression for the object's upward acceleration is:

A_o = V_r/t

combining this with the previous equation gives:

V - V_r = V_r

or:

V_r = V/2

Furthermore, since V = V_m + V_r, we have:

V_m = V - V_r = V - V/2 = V/2

So this means the monkey's upward speed (V_m) relative to the ground, is V/2. And the object's upward speed (V_r) is also V/2.

Well, it is what my brain have done to solve this problem. Isn't here any mistakes? And this solution is quite long. Maybe there is a shorter way to reach the answer without using parameters like t (maybe using angular momentum conservation law)?

 

[Petrulis]

Please somebody have a look :)

 

[m2003]

Dear student,

Thank you for sharing your solution to this problem. It appears that your approach is correct and your solution is reasonable. However, there may be a more efficient way to solve this problem using conservation laws.

One possible approach is to use the conservation of angular momentum. Initially, the system has no angular momentum since both the monkey and the object are at rest. As the monkey starts to climb up the rope, the rope will start to unwind from the disc, resulting in a decrease in the disc's angular velocity. However, the angular momentum of the system must remain constant. This means that the decrease in the disc's angular velocity must be compensated by an increase in the angular velocity of the object bound to the rope.

Using the equation for conservation of angular momentum, we can write:

I_initial*ω_initial = I_final*ω_final

Where I is the moment of inertia and ω is the angular velocity. Since the disc is weightless, its moment of inertia can be considered negligible. This means that we can write:

0 = I_final*ω_final

Since the monkey and the object have the same mass and are rotating at the same distance from the center of rotation, their moments of inertia (I) will also be the same. Therefore, we can write:

0 = I*ω_final

Rearranging this equation, we get:

ω_final = 0

This means that the object bound to the rope will have no angular velocity and will remain at rest. This also means that the rope will not unwind from the disc.

So, to summarize, using the conservation of angular momentum, we can conclude that the object bound to the rope will remain at rest while the monkey climbs up the rope. This is a simpler and more efficient approach compared to using Newton's laws and considering the motion of the rope and the monkey separately.

I hope this helps. Keep up the good work in solving problems and using conservation laws.

Sincerely,