Problem 1

A mass

m

attached to a light spring of stiffness

k

moves horizontally on a frictionless surface. It is displaced by

A

from equilibrium and released from rest. Find the period, maximum speed, and first time it passes equilibrium. Model each stage with the appropriate conservation law or equation of motion; quantities stated as constant remain fixed during that stage. (a) Establish the governing physical relation for each stage. (b) Determine the requested observables from the stated initial conditions and constraints. (c) Check signs, dimensions, and the physical limiting behavior of the results.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

x(t)=A\cos(\omega t),\qquad v(t)=-A\omega\sin(\omega t)

problems.proof.analysis

Choose positive toward the initial displacement; $x(0)=A$ and $v(0)=0$ give the cosine solution. This follows from the chosen physical model and the stated initial conditions; the sign convention must remain consistent throughout. A dimensional check catches algebraic errors before the result is interpreted.