Tension and Friction with Acceleration

[jonnyboy1234]

Homework Statement

A building is on fire and the occupant of a room ties together some bed sheets, attaches them to something immovable in the room and throws the bedsheet ‘rope’ out the window. The person weighs 800 N and the bed sheets can only hold a force of 600 N before they break. If the person descends (starting with zero vertical velocity) 12 m (about 4 stories) using this bedsheet rope, applying just the right amount of friction on the rope to provide maximum braking without breaking the rope, how fast is the person going after 12 m of descent? Give you answer in miles per hour so you have a feel for how fast this is. (26.8 m/s = 60 mph)

Homework Equations

I'm sorry, I do not know how to use the LaTex Math Typesetting, but the equations are pretty basic.

Fnet = ma

Fnet = F1 + F2 + F3 + ... + Fn

Vf^2 - Vo^2 = 2ay

The Attempt at a Solution

Okay, so what I did is say that the person must be using a maximum downward force of 600N on the rope so it doesn't break. This makes then friction equal to 200N, since it acts in the opposite direction of the 800N downward of the occupant.
So, (-600N)/(9.8kg*m/s^2) = -7.35 m/s^2. This is person's acceleration downwards.
Plugging this into the final equation yields Vf = √(2*(-7.35 m/s^2)*(-12m)) = - 13.3 m/s or -29.7 mph.

HOWEVER, my classmate says that the rope pulls upward with 600N and since the rope is applying the force of friction, the there're 600N of friction. So that would make the net force -200N. Redoing the calculation gets -17 mph.

So, which is correct? My classmate's makes sense, but it seems off to me...so any input is very, very appreciated! We could also both be wrong haha. Thank you for taking your time!

 

[voko]

If the rope exerts some amount of friction, then by Newton's third law it is itself subject to the same amount of force (acting in the opposite direction).

 

[jonnyboy1234]

So, then the friction must be equal to the tension limit of 600N?

 

[voko]

It is in the interest of the occupant to use the max friction afforded by the rope.

 

[Nickie]

I would like to clarify that both approaches are correct, but they are calculating different speeds. The first approach is calculating the speed at the end of the 12m descent, while the second approach is calculating the speed throughout the descent.

In the first approach, the friction force is not taken into account because it is assumed to be just enough to prevent the rope from breaking. Therefore, the net force acting on the person is only the weight force, which results in a lower speed at the end of the descent.

In the second approach, the friction force is considered and included in the net force calculation. This results in a lower acceleration and a lower speed throughout the descent, as the friction force is acting in the opposite direction of the weight force.

To determine the correct approach, it would be helpful to have more information about the situation, such as the length and thickness of the rope, the surface it is rubbing against, and the angle at which the person is descending. These factors can affect the amount of friction and the speed of descent. Additionally, it is important to note that the calculations assume ideal conditions and do not take into account other factors such as air resistance.