Pulling with and without a pulley

[brotherbobby]

Homework Statement

In the diagram shown below, a block of mass 25 kg is raised by a man weighing 50 kg in two different ways. Answer the following questions :

(1) What is the action on the floor by the man in the two cases?

(2) Will the man find it easier to pull the block in case (b) than in case (a)? Explain.

Relevant Equations

1. Newton's first law : $\Sigma \vec F = m \vec a$, where $\Sigma F$ is the net force acting on the body and $\vec a$ is the acceleration of the body.

2. Newton's 3rd law involving a pair of bodies A and B applying forces on one another : $\vec F_{AB} = -\vec F_{BA}$

Let me draw a free body diagram for the man. I assume that for both cases (a) and (b), the man pulls the block up with uniform velocity.

(1) In case (a), realising that the man is in equlibrium ($\Sigma F = 0$) the normal reaction on the man $N_a = w_M + T$ is given by $N_a = (50+25)\times 9.8 = 735$ N. Hence the normal reaction in case (a) : $\boxed{N_a = 735 \text{N}}$.
In case (b),

we have $T+N_b = w_M \Rightarrow N_b = w_M - T = (50-25)\times 9.8 = 245$N. Hence the normal reaction in case (b) : $\boxed{N_b = 245 \text{N}}$.
(2) I do not know which of the two cases would the man find "easier". All I know is that there is less normal reaction on the man in case (b) than in case (a). I understand, intuitively, that it is easier to use a pulley to lift a bucket from a well (as is done still in my third world country) than it is to pull the bucket up without a pulley. But why? Does the smaller normal reaction have anything to make things easy?

A hint or help would be welcome.

 

[haruspex]

(2) I do not know which of the two cases would the man find "easier". All I know is that there is less normal reaction on the man in case (b) than in case (a). I understand, intuitively, that it is easier to use a pulley to lift a bucket from a well (as is done still in my third world country) than it is to pull the bucket up without a pulley. But why? Does the smaller normal reaction have anything to make things easy?

The leg muscles are much stronger than the arm muscles.
With the pulley, the technique is to hold the rope at some high point and relax the legs. Your own weight pulls up the bucket, then you do the real work by straightening the legs again (having secured the rope). The arm muscles are only used in gripping the rope with the hands.
Without the pulley, you can in principle do much the same. Squat, hold the rope at a low point, then stand up. (One difficulty is avoiding having to bend forwards, straining the back. In diagram a the man would topple over the edge.)

So, think about the work the man does at each stage in the two procedures.

 

[brotherbobby]

So, think about the work the man does at each stage in the two procedures.

In either case, the man pulls the block up by the same amount, hence the work done by the man in both cases is the same : $W_{\text{man}} = \text{Tension} \times {\text{lift}}$

 

[haruspex]

In either case, the man pulls the block up by the same amount, hence the work done by the man in both cases is the same : $W_{\text{man}} = \text{Tension} \times {\text{lift}}$

What about the work done in raising and lowering the man's body?

 

[brotherbobby]

What about the work done in raising and lowering the man's body?

Say the man (a) held somewhere in a long rope, with much of it on the ground. Can the man not lift the block without lowering/raising his body?

Likewise, say the man (b) held somewhere in a long rope coming off the pulley, with much of the rope on the ground. Can the man not lift the block without raising/lowering his body?

 

[jbriggs444]

I do not see that part b has anything much to do with physics. It's all down to ergonomics.

Personally, I would find it easier to raise a 25 kg/50 pound weight pulling downward hand over hand than pulling upward hand over hand. Closing my eyes and imagining the motions...

In the rope-at-the-feet case, I would be bent slightly forward, trying to keep the rope mostly clear of the lip of the floor. I would reach down and grasp the rope at waist height, pulling upward hand over hand and releasing at about belly-button height. [For a lighter load it would be tempting to instead turn it into a curl, grasping at waist height with forearms extended and lifting to neck height. The rope grip is reversed from underhand to overhand in the two approaches]

In the rope-from-the-pulley case, I would be fully upright, reaching upward to grasp the rope at arms length, pulling downward hand over hand and releasing again at breastbone level. [For a lighter load one would lengthen the stroke and shift it downward] . The rope grip is the same regardless.

For a very heavy load, I might switch to the @haruspex approach, using body weight to assist with the pulley. I might ultimately shift to calf muscles for lift and the thighs for support in the rope-at-the-floor situation. You can get a lot of force out of the calves. Much more than one's own body weight.

For the 25 kg mass situation and a fit adult male, the pulley has the advantage because the weight of the arms is helping on the downstroke. You are expending effort both raising and lowering arms, but with a different set of muscles each way. With the rope-at-the-feet, the muscle force is unbalanced, wearing out one set of muscles faster and not stressing the other at all.

The bend at the waist which is present in the lift from floor case but absent in the pull from pulley case gets tiring and is an additional advantage for the pulley. You need that bend -- it shifts the body's center of gravity rearward and improves one's margin of safety for falling. And it gives you space for your arms.

Edit: The diagram in the original post pretty clearly illustrates the light load ergonomics -- the curling motion for lifting from the floor and the "mashed potato" approach for pulling from the pulley. [The arm extension in the light load case reduces the need for the bent waist]

 

[haruspex]

Say the man (a) held somewhere in a long rope, with much of it on the ground. Can the man not lift the block without lowering/raising his body?

Likewise, say the man (b) held somewhere in a long rope coming off the pulley, with much of the rope on the ground. Can the man not lift the block without raising/lowering his body?

It is feasible, but I wouldn't do it, and I'm 60kg. Too much risk of back injury, and unnecessarily difficult.

Anyway, I was answering your original point that intuitively the pulley should make life easier. I believe it does for the reasons I have given.

For a very heavy load

Half the weight of the man is not heavy?

The diagram in the original post pretty clearly illustrates the light load

The diagrams don't match the data. In figure a, the man would topple over the edge.

 

[jbriggs444]

Half the weight of the man is not heavy?

I'd probably do a 50 pound bag hand over hand. But I'm well over 100 pounds, so your point is well taken.