"a. When the displacement of a mass on..." was answered by our community

mronjo7n

Answered question

2021-11-06

a. When the displacement of a mass on a spring is $\frac{1}{2}$ A, what fraction of the mechanical energy is kinetic energy and what fraction is potential energy? b. At what displacement, as a fraction of A, is the energy half kinetic and half potential?

Answer & Explanation

Gloria Lusk

Beginner2021-11-07Added 18 answers

The total is given by the equation:

E = \frac{1}{2} K A^{2}

For part a) when the displacement is

x = \frac{1}{2}

A the potential energy is calculated as:

U = \frac{1}{2} k x^{2} = \frac{1}{2} k (\frac{1}{2} A) = \frac{1}{4} (\frac{1}{2} A) = \frac{1}{4} (\frac{1}{2} k A^{2}) = \frac{1}{4} E \Rightarrow K = E - U = \frac{3}{4} E

For part b) To have

U = \frac{1}{2} E

, we need:

U = \frac{1}{2} k x^{2} = \frac{1}{2} E = \frac{1}{2} (\frac{1}{2} k A^{2}) \Rightarrow x = \frac{A}{\sqrt{2}}

Result: a)

U = \frac{E}{4} and K = 3 \frac{E}{4}

b) x = \frac{A}{\sqrt{2}}

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