I did pull-downs at the gym the other day and felt unusually strong. I was lifting 300 lbs like it was nothing, and then, despite my ego’s warning, I decided to check out the equipment because it seemed too easy. It turns out the weights had not one but two pulleys on them! This means I was actually only lifting one quarter of the total weight.

**How many pulleys would a person need to lift the Empire State Building?**Pulleys can provide additional force to help move an object. By attaching a pulley to a weight as shown here and pulling the rope with some force, you get double that force on the weight because there are two ropes pulling up on it. If you add

*N*pulleys to the weight, you will get 2*N*times as much force. The trade-off for adding a pulley is that you now have to pull the rope twice the distance that you want to lift the weight. For example, if you’re using a single pulley to lift an object 1.0 m off the ground, you need to pull the rope 2.0 m. If you want to do the same thing but with two pulleys attached to the object, then you have to pull the rope 4.0 m.The Empire State Building weighs about 7.3

**×**10^{8}lbs (~365,000 tons). A reasonably strong person can lift 100 lbs. To lift the Empire State Building, we’ll need the force applied by the pulleys to balance the weight of the building,(weight of the Empire State Building) = 2

**·**(# of pulleys)**·**(force applied).From this, we can solve for the number of pulleys,

# of pulleys = (weight of the Empire State Building) / [2

**·**(force applied)]= (7.3

**×**10^{8}lbs) / [2**·**(100 lbs)]= 3.7

**×**10^{6}pulleys.
It seems to me that each pulley would give 2^N times as much force, not 2N, or am I mistaken? If that's the case, you'd only need 23 pulleys; 2^23 is 8,388,608, and you only need 7,300,000 times your original 100 lbs of force.

ReplyDeleteI could be wrong here; you're the mathematician, not me.

To lift 730 million pounds, 23 pulleys sounds a bit low, don't you think?

ReplyDeleteIf 2^N were the principle, you could lift planet Earth (1.32*10^25 lbs) with just 84 pulleys...

It's fun to wonder at what point friction would make the lifting unfeasible: all those pulleywheels turning on their hinges, all that rope straining in grooves -- not to mention the added weight of 3,700,000 pulleys and 4,500 miles of rope.