Problem 1

A mass

m

is attached to a light spring of stiffness

k

on a smooth horizontal surface, with the other end fixed. Let equilibrium be

x=0

, neglect spring mass and resistance, and release from rest at

x=A>0

. (a) Write the equation of motion and find the period. (b) Find maximum speed and its location. (c) Find speed as it passes

x=A/2

toward equilibrium. Keep the sign convention consistent and use appropriate SI units throughout. Each part builds on intermediate results from the preceding part; apply a physical relation only where the stated model assumptions hold. Finally compare the answers with the initial conditions, expected trends, and an independent check when available.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{T=2\pi\sqrt{m/k},\quad v_{max}=A\sqrt{k/m},\quad v(A/2)=\frac{\sqrt3}{2}A\sqrt{k/m}}

problems.proof.analysis

Collect answers to every part, including signs, directions, and image or process properties where relevant. Dimensions match the physical quantities.