Pulley puzzle

Classical Mechanics Level 3

A special type of chain pulley can be used to manually lift very heavy objects like car engines.

There are two pulleys, red (with radius 5) and blue (with radius 4), that move together and determine the maximum weight that can be lifted. Pulling on the slack part of the chain (black dots) wrapping around the pulleys lifts the box.

What is the mechanical advantage of this pulley? (That is, if I pull with a force of 1 Newton, how many times of upward force is applied on the brown box?)

Assumptions:

Assume that the two lines that pull on the brown box are vertical (that is, there is no loss of mechanical advantage due to a bad angle).
Assume that the pulleys and the chain are weightless and that there is no friction.

The answer is 10.

4 solutions

Lance Kuanwu
Jun 4, 2018

Pulling 5m down causes the weight to move up by 2.5m. At the same time the smaller gear will release 4m of chain and cause the weight to move down by 2m. Net movement of the weight is 0.5m. The work done by you must be equal to the work done on the weight, since the energy have no place to go.

0.5m x F = 5m x 1N, Thus F = 10N

How do you know that pulling 5 meters down moves the weight up by 2.5 meters, and that 4m of chain released by the blue gear drops it back down by 2? I don't see where the 1/2 ratio is coming from.

Ryan Conti - 3 years ago

I do not see either. If both gears have same radius, the answer is 1. So just reducing one radius by 20% leads to 10 times more force? Hard to believe.

carslover14 . - 3 years ago

I found this very confusing until I looked up the Wikipedia article, which shows a pulley hoisting the weight. Once I assumed that the rope could freely move by the box, things became much clearer. See Antoine's explanation below.

Lawrence Quinnett - 2 years, 11 months ago

If you pull up by x meters, and the weight move up by x meters, then 1) the top part of the chain increase by x meters, 2) but the bottom part decrease by x meters on each side Which means the whole chain magically become shorter.

The amount you pull up has to be shared equally by the chain on the left and right side of the weight, thus x/2

Lance Kuanwu - 2 years, 12 months ago

Antoine G
May 24, 2018

I am computing the mechanical advantage through the ratio of the displacement rather than directly computing the ratio of the forces. See clarifications below if you are not familiar with this.

When one pulls 10 elements of chain on the red pulley, 10 chain elements get removed from the part of the chain in gray. But the blue pulley turns along with the red pulley. The ratio of the radii is so that there are 8 elements of chain released by the blue pulley. The part of the chain [in gray] which holds the weight is thus $10 -8 = 2$ chain elements shorter.

So the gray part of the chain gets 2 chain elements shorter. So how much does the weight go up?

Recall that if a 4 meter chain hung in such a U-form the weight would hang 2 meter lower. This is because you need 1 m of the chain to go down and one to go up. If you cut up 2 meter from the chain, the weight just moves one meter up (because you cut 1 meter on the right and 1 meter on the left).

Likewise if the gray chain is shortened by 2 chain elements, then the weight goes up [the height of]1 chain element.

As a recap: we pull down 10 chain elements and the weight is moved up by 1 chain element. The ratio movement of the pulling force to movement of the weight is 1 to 10. Hence the mechanical advantage is 10.

This type of pulley is called a differential pulley (there is a Wikipedia article on them).

Clarifications: The mechanical advantage is the ratio of the output force $F_o$ to the input force $F_i$ : $\dfrac{F_o}{F_i} = \alpha$ .

The crucial property of simple machines and levers is that if the path made by the input force is $s_i$ and by the output force $s_o$ then $\frac{s_i}{s_o} = \alpha$ . [I guess this is how people came to define work... and later realised this as a case of conservation of energy.]

So I computed the ratio of the displacements instead of computing the ratio of the force (because I though it was easier).

[The upcoming paragraph is not related to the pulley puzzle, it's just another phenomenon where the ratio of force is directly felt as a a ratio of displacement] Intuitively, you learned this when you got your first bike with gears: the gear where one needs many turns of the pedal to get one turn of the wheel is the one where you get the most force (good for uphill or to start rolling). Unfortunately, when you go downhill (or after having reached a certain speed) your legs simply cannot move fast enough, so you choose a gear where one turn of the pedals gives many turns of the wheels. In this gear, you can actually use your legs. The force applied by your legs is however much higher than the force with which the bike is pushed forward. This can be felt if you try to start in such a gear: you have to push hard on the pedal to start get some acceleration. In this gear, the mechanical advantage is awful (but at least you can use your legs once the bike has some speed).

Here is also a link to another wikipedia article.

In theory, one could use planetary gears combining geared "inputs" for this same kind of differential mechanical advantage, which can be made extreme, but in practice it doesn't work as well. Regardless of how heavy the load is, the differential pulley doesn't lock up, but the planetary gears version will.

Michael Mendrin - 3 years ago

Interesting! can you explain a bit more what you mean by "the differential pulley doesn't lock up"?

I got the inspiration for the problem as I saw a similar pulley a in an old garage, it seemed to work ok (but the ratio of the wheels was probably different).

Antoine G - 3 years ago

Back in my 20s, I had a machine shop, and I needed a way to deliver a very slow output shaft speed without using a lot of gears, so I tried to make use of a planetary gearbox with some more gearing for inputs. It worked in delivering the extremely slow output shaft speed but it was a mechanical failure in that it did not deliver the increase in torque force as would have been expected from ordinary gearing. Later, I'll try to provide graphics about what I'm talking about.

Michael Mendrin - 3 years ago

@Michael Mendrin –

Okay, one way of getting an extremely slow output shaft speed is to consider this figure, letting green being the output. (One can use either blue, green, or red as the output). If the blue and red input speeds are "balanced", then there is no resulting blue output shaft speed. By very slighly "unbalancing" those speeds by use of suitable input gearing (using 5 gears), one can deliver a very slow green output shaft speed.

For example, let's say that the ratio of the blue and red gears are $2:1$ . Then if we used a pair of gears for each input, the ratios being $\dfrac{26}{55}$ and $\dfrac{87}{92}$ , then the output green rotation will be $\dfrac{1}{10120}$ of the input rotation driving the $55$ and $92$ teeth gears together. The 5th gear (of arbitrary number of teeth) is for providing counter-rotation necessary for this to work and simplifying gear meshing geometry.

Unfortunately, there is no multiplication of torque force, so one wouldn't be able to easily lift a car engine with such a device. In practice, it seems to simply lock up with any significant output shaft resistance.

Let me revisit this problem again to see if there's a way to get around this. I haven't thought about this since my 20s.

Michael Mendrin - 3 years ago

@Michael Mendrin – Very interesting! Just a random thought (as I don't have time at the moment): Could it be that the output torque is small because the torques on the red and blue gears "cancel out"?

Antoine G - 3 years ago

Just a side question: "Hence the mechanical advantage is 10." What is behind of this conclusion of yours? I chose the mechanical work. Did you, too, or is there any other way?

Laszlo Kocsis - 3 years ago

well, it's a general principle (which is actually behind the definition of work) that there are two effects to "simple mechanical machine" (by this I mean a construct which is made of ideal mechanical parts, has no motor, only one input force and one output force). One of them is that it gives a "mechanical advantage" that is: there is an input force $F_i$ and an output force $F_o$ which are related by $F_i = \alpha F_o$ where $\alpha$ is a positive real number which depends only on the machine. This $\alpha$ is the mechanical advantage.

Now it turns out that however you build these machines, if the path made by the input force is $s_i$ and by the output force $s_o$ then $s_i = s_o / \alpha$ . I guess this is how people came to define work...

So to sum up: if the ratio of the displacements is $\frac{s_i}{s_0} = \alpha$ , then the mechanical advantage is $\dfrac{F_o}{F_i} = \alpha$ .

You can also do this directly in terms of work. Look at the work done by the force pulling down on the pulley (here $1 \cdot 5$ ; 1 unit of force times 5 units of displacement). Look at the work done by the weight: $F_o \cdot \tfrac{1}{2}$ . By conservation of energy, $F_o \cdot \tfrac{1}{2} = 1 \cdot 5$ . So the output force is 10 units (if the input force was 1 unit).

Here is also a link to wikipedia: https://en.wikipedia.org/wiki/Mechanical_advantage

Antoine G - 3 years ago

I was thinking the same way, I was just not aware (I forgot, I guess, it was a long time ago) that this is a general principle so it's not necessary to mention it explicitly in a solution.

But thank you very much for the detailed explanation!

Laszlo Kocsis - 3 years ago

@Laszlo Kocsis – Here it's a very common high school topic. But I guess it's a "cultural" emphasis which need not be done, so thanks for asking!

Antoine G - 3 years ago

Please further explain the half?????

Caleb Opoku Oppong - 3 years ago

the part of the chain which holds the weight gets 1 chain element shorter. The chain makes a "U" form holding the weight, so only half of the length of a chain element gets added to the right part of the U and half to the left part. The weight is thus only raised by half a chain element.

Antoine G - 3 years ago

well, some illustration in your solution would be great! or at least more clear explanation of why the load lifts on length 1/2 units when one pulls off 5 units of chain from the red pulley! i figured it out myself, but i reckon that the most of the people who read your explanation don't understand it... maybe even it would be nice to place here some note about how this fancy pulley is different from another common one: where the load is put on the little-radius pulley and the lifting force is applied to a big-radius pulley (i bet many people confuse one with the other)

Nik Gibson - 2 years, 9 months ago

you are of course perfectly right. I just don't have the graphical skills for the task :( nor the time really... I'll add in some details to the text. Tell me if you find this satisfying...

Also there is the wikipedia page https://en.wikipedia.org/wiki/Differential_pulley which provides some additional info and details. But yes

Antoine G - 2 years, 9 months ago

i've read your bicycle analogy and i would substitute the word "speed" with the word "gear", coz that's what it is - the gear... and there are high gears (with bad mech. adv.) and low gears (good mech. adv.) ....like gears in a car or every other mechanism with several pulleys of different radii... And before placing this analogy in the text of your solution i would recommend to put smth like "please do not confuse this junction of the pulleys with the example below"

Nik Gibson - 2 years, 9 months ago

Hagay Kelman
Jun 5, 2018

The sum of moments on the axis should be zero. If the tension on 'segment A' (the chain segment that is carrying the box) is X :

The chain is being pulled with 1N force CW (clockwise) at a radius of 5.
One side of 'segment A' pulles with force X CW at a radius of 4.
The other side of 'segment A' pulles with force X CCW (counter clockwise) at a radius of 5.

5x1N+4xX-5xX=0 So, X=5N The force on the box is 2xX=10N

Marius Kirkeeide
Jun 9, 2018

Pulley puzzle

The answer is 10.

4 solutions

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