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EK Question 35 - Springs
Kipling_3604
#1 Posted : Saturday, May 29, 2021 6:18:55 AM
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Hello, I am a bit confused with EK Lecture 2 Question 35.

If objects A and B are placed on the spring together and object A has twice as much mass as object B. If the spring is depressed and released, propelling the objects into the air, object A will:

The book says that Object A will rise to the same height as object B. When I try to use equations to support this conclusion, I've tried using the idea of energy conservation, so if we use PE = mgh and we would expect that the heavier object A would only reach half as high as object B. Please direct!

Thanks!
INSTR_Katrina_128
#2 Posted : Sunday, May 30, 2021 4:19:37 PM
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Oooh this question is SO MEAN. It goes against our intuition completely. ๐Ÿคจ๐Ÿค”

Let's do the math on it. First, let's choose simple numbers. Let's set Object A to have a mass of 200 kg, so Object B will have a mass of 100 kg.

Then, the total mass on the spring is 300 kg. This is a total FORCE of 3000 N.

Assuming k = 1 for the spring (to make it easier), and that the spring is depressed by 10 cm (or 0.1 m), then the energy that the spring "gives" to the two objects is 300 J.

So, we have a total of 300 J to partition amongst these two objects. IN TOTAL, Object A + Object B get 300 J of energy to jump up into the air and fly baby fly.

How much does Object A get and how much does Object B get? In total, the two get 300 J. Well, based on the relative size of the objects, I would expect that Object A should get more than Object B. In fact, since A is 2x bigger than B, A will get 2x more of the energy than B!

Note: A common mistake here is to assume that Object A gets 300 J AND Object B gets 300 J. That makes no sense!! That would mean energy is being created out of nothing. Be careful!!

So if Object A gets 2x more energy than Object B, and we have 300 J in total, then Object A gets 200 J and Object B gets 100 J.

Using our gravitational potential energy equation, we can now determine the final heights that the two objects achieve: Ep = mgh.

For Object A: 200 = (200)(10)h ---> h = 0.1 m
For Object B: 100 = (100)(10)h ---> h = 0.1 m

WHAT THE?!

Yup. Same height.

The key here is that each object will "grab" an amount of the total energy in proportion to their size. ๐Ÿค๐Ÿค๐Ÿค
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