A particle executes S.H.M of amplitude a

1. At what distance from mean position is it's K.E is equal to its P.E

2.At what point is its speed half the maximum speed

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Solution

Kinetic energy = 0.5mw²(A² - x²)
Potential energy = 0.5mw²x²

At certain distance from mean position,

Kinetic energy = Potential energy

0.5mw²(A² - x²) = 0.5mw²x²
$0.5 m w squared A squared minus 0.5 m w squared x squared equals 0.5 m w squared x squared 0.5 m w squared A squared equals m w squared x squared x squared equals A squared over 2 x equals square root of A squared over 2 end root equals fraction numerator A over denominator square root of 2 end fraction$

x = $plus-or-minus fraction numerator A over denominator square root of 2 end fraction$

Therefore, at a distance ± $fraction numerator A over denominator square root of 2 end fraction$ from mean position, energy is half Kinetic and potential.

Part 2

Kinetic energy = 0.5mw²(A² - x²)=0.5m
[General formula of kinetic energy is 0.5m]
v = $fraction numerator w A over denominator 2 end fraction$

Where wA is max velocity
Substitute this value of v in kinetic energy equation.
0.5m
$w squared left parenthesis A squared minus x squared right parenthesis equals v squared w squared A squared minus w squared x squared equals left parenthesis fraction numerator w A over denominator 2 end fraction right parenthesis squared w squared A squared minus fraction numerator w squared A squared over denominator 4 end fraction equals w squared x squared fraction numerator 3 w squared A squared over denominator 4 end fraction equals w squared x squared x squared equals fraction numerator 3 A squared over denominator 4 end fraction x equals square root of fraction numerator 3 A squared over denominator 4 end fraction end root equals fraction numerator square root of 3 A over denominator 2 end fraction$

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