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				   two blocks of masses m and m' are connectd by a light spring on a horizantal frictionless table.the spring is compressed and then released.Find the ratio of accelaration and masses
				   

6 years ago

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Dear apraitm,


Let k be the spring constant of the spring.


Here reduced mass concept will be applicable. so reduced mass of the system is M= mm'/(m+m')


or this solution, I will already assume that all motion takes place in the x-direction. So, I can write Newton's second law as:


Screenshot 37


Where Fx is the total force (there is only one in this case anyway). Please forgive me, but I am going to drop the "in the x-direction" notation since EVERYTHING is only in the x-direction. The force from the spring is:


Screenshot 38


Remember that this spring has zero natural length, so the x position IS the "stretch". Putting this together, I get:


Screenshot 39


Here I wrote acceleration as the second time derivative of position. If that is something completely foreign to you, don't worry - you will see this later maybe. Anyway, what I have here is a differential equation. How do you solve a differential equation? There are lots of strategies, but I find the best one is to guess. Yes, just guess a solution and see if it works. First, let me re-write the differential equation:


Screenshot 40


If you look at this equation, it says "take the derivative with respect to time twice and get something times the original function" (really, it says that, you might have to listen closely). Once function that does that is ....cosine. So, let me try the function:


Screenshot 41


Where A and ? are constants. Let me take the first derivative:


Screenshot 42


And now the second derivative:


Screenshot 43


So this means that:


Screenshot 44


ACCELERATION=Kx(m+m')/(mm')


 






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Sagar Singh


B.Tech IIT Delhi


sagarsingh24.iitd@gmail.com





6 years ago

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