Physics

Problem

For the same mass–spring system described in Question 2, let the instantaneous displacement of the mass from equilibrium be

x

and its maximum displacement (amplitude) be

A

. a. (i) Write expressions for the instantaneous kinetic energy

K

and potential energy

U

of the mass in terms of

m

k

x

, and

A

. b. (ii) Show that the total mechanical energy

E = K + U

of the system remains constant during the oscillation. c. (iii) Prove that

E = \tfrac12 kA^{2}

. d. (iv) Hence find the kinetic energy of the mass when its displacement is

x=\tfrac{A}{2}

. Express your answer in terms of

k

and

A

only.