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Very hard mechanics question (1 Viewer)

ultra908

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isnt this the grammar q16 lol
which part of the question are you having problem with?
 

Trebla

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For part (i), use differentiation to find the velocity and acceleration equations and do a bunch of substitutions to reach the form of the acceleration equation given
 

CNSie

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For part (ii) solve .
For part (iii) note that the total distance should be

where

Get a general formula for (same process as part 2 but using the general solution for tan) and note that your total distance will be in terms of A (constant), and k/n which is a function of P/Q
 

ultra908

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For part (ii) solve .
For part (iii) note that the total distance should be

where

Get a general formula for (same process as part 2 but using the general solution for tan) and note that your total distance will be in terms of A (constant), and k/n which is a function of P/Q
its should be 2|x_r| (apart from x_0=A), since you need to go down and up, and the distance cant be negative.
 

CNSie

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its should be 2|x_r| (apart from x_0=A), since you need to go down and up, and the distance cant be negative.
Oh yeah true! But the reasoning holds at least haha
 

Nav123

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Part ii/iii written out:
In part i you would find:


To find when particle is rest set which you would do in part ii


so solve:

mucking around with auxiliary angle (or you could just divide by cos) we get:

so solve:
not sure if general solutions is in the syllabus so I'll just list out the solutions

Then sub this into x (keeping in mind the displacement will be negative) you will get the answer for ii.

For part iii like previous posters said take the infinite sum so we get the sum (i'll write it out long assuming general sol isn't in the syllabus):

Notice the pattern for odd multiples of cosine is negative and for even multiples of it's positive. So simplifying the sum:


We can see the sum is convergent quite easily but a formal proof is prob required. Using the GP sum we finally get:







.

Therefore every part of the expression for depends solely on the fraction (notice that A is constant so we don't have to worry about that).

Sidenote: Sorry for the terrible latex:eek:. I also probably included too many steps which wouldn't be required in a typical exam.
 
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