# Person pulling themselves up a double pulley

• SadDan
In summary, the man is hanging on a rope attached to the inner disk and is pulling on the rope attached to the outer disk. He is trying to lift himself up using only the tension in the ropes. He is experiencing an acceleration of 5.3 meters per second squared.
SadDan

## Homework Statement

A man is pulling himself up to a pulley that consists of two disks welded together as shown(same center). The man is currently pulling straight down on the rope in his hands with a force of magnitude 447.2 N (on the bigger disk). The other rope is also vertical and is attached to the man's waist at his center of mass(this is on the smaller disk). The man's mass is 76.3 kg, the pulley's total moment of inertia is 2.74 kg⋅m2, the radius of the small disk is 0.33 m, and the radius of the big disk is 0.62 m.

The man is hanging on a rope attached to the inner disk and is pulling on the rope attached to the outer disk. if you need a better description of the picture let me know.

## Homework Equations

a(t) = angular acceleration*R
n2l for rotation and translation

## The Attempt at a Solution

R=outer radii r=inner radii
sys: man Tpm+Tpm2-Mg=Ma
a=(Tpm+Tpm2-mg)/M
sys: pulley Tmp*R - Tmp2*r=angular acceleration*I
Tmp*R - Tmp2*r= a*I/r
(r*Tmp*R - Tmp2*r*r)/I= a
so far i have 2 equations and 3 unknowns, how would i solve for a from here?

Can you list the three unknowns? I see only 2.

the 2 different tension forces and the acceleration

Aren't you essentially given one of the tension forces?

SadDan
yes, yes i am. Thank you I didn't really make that connection

The statement of the problem does not really state what you are supposed to find.
Is the following picture representative of the setup?

SadDan
sorry i guess i forgot to include that, i am supposed to find the magnitude of acceleration at the mans waist or CM. And yes that picture is the right setup

SadDan said:
sorry i guess i forgot to include that, i am supposed to find the magnitude of acceleration at the mans waist or CM. And yes that picture is the right setup
OK. I think you have set up the equations correctly.

SadDan
Hold on. Check your signs in your equations to make sure your positive direction for ##a## corresponds to your positive direction for ##\alpha##.
[EDIT: Never mind, I think everything is OK!]

So i solved for a=(F*R*r+F-M*g)/(M*r^2+I) and got the wrong answer which is probably a sign problem. If I made a positive clockwise torque and defined positive y to be up would the second tension force be negative in the first equation?

SadDan said:
So i solved for a=(F*R*r+F-M*g)/(M*r^2+I) and got the wrong answer which is probably a sign problem. If I made a positive clockwise torque and defined positive y to be up would the second tension force be negative in the first equation?
No, both tensions act upward on the person. So, I think you are OK there.

What did you get for the ##a##?

-18.99 m/s^2

That's now what I get. I get about 5.3 m/s2.

SadDan said:
So i solved for a=(F*R*r+F-M*g)/(M*r^2+I) a
You can see something's wrong with this equation. The terms in the parentheses do not have the same dimensions.

SadDan
5.3m/s^2 worked but how did you get that? Did you just solve for Tmp in terms of a and plug that into the other equation?

I plugged your two equations (from your first post) into Mathematica and let it do the grunt work!

ok i tried again and got the right answer, just an algebra mistake. Thank you so much

OK, good work. Did you notice that this person is going to go head-over-heels heels-over-head?

SadDan
haha yes

On second thought, I think it is head-over-heels. (Counterclockwise rotation of the person's body). Oh well.

## What is a double pulley?

A double pulley is a type of pulley system consisting of two pulleys attached to a rope or cable. The rope or cable is threaded through both pulleys, with one pulley attached to a fixed point and the other attached to the load being lifted. This system allows for a mechanical advantage, making it easier to lift heavy objects.

## How does a double pulley work?

A double pulley works by distributing the weight of the load being lifted between the two pulleys. This reduces the amount of force needed to lift the load, making it easier for a person to pull themselves up.

## What are the benefits of using a double pulley?

Using a double pulley can provide a mechanical advantage, making it easier to lift heavy objects. It can also help reduce the strain on a person's muscles and joints, as the weight of the load is distributed between the two pulleys.

## What are the potential risks of using a double pulley?

One potential risk of using a double pulley is the potential for the rope or cable to break under the weight of the load. It is important to use a strong and sturdy rope or cable and to regularly check for any signs of wear and tear. Another risk is improper use of the pulley system, which can lead to accidents or injuries.

## Are there any safety precautions that should be taken when using a double pulley?

Yes, there are several safety precautions that should be taken when using a double pulley. These include using a strong and sturdy rope or cable, regularly checking for any signs of wear and tear, and properly securing the pulley system. It is also important to follow proper lifting techniques and to avoid overloading the pulley system.

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