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(a) At what displacement from equilibrium is the energy of a SHO one-sixth KE and five-sixths PE? (Answer in terms of the amplitude A.)

(b) What fraction of the total energy of a SHO is kinetic and what fraction potential when the displacement is four-fifths of the amplitude?
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a)
We know
PE = 1/2 k x^2 ; where x is the distance form the equilibrium;
And KE = 1/2 k (A^2 -x^2)

Given,
PE = 5/6 * TE
=>1/2 k(x^2) = 5/6 *1/2k A^2
=>x^2 =5/6A^2
=>x= sqrt(5/6)A;

b)x= 4/5 A
So, PE = 1/2 K 16/25 A^2 = 16/25 TE
So,
fraction = 16/25
Hence
KE = 1 - 16/25 = 9/25
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