A block of mass on a frictionless horizontal spring of stiffness has displacement…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Calculus of change and accumulation

Mathematical Methods for Physics, Ch The derivative as a physical instrument

Problem

A block of mass

m

on a frictionless horizontal spring of stiffness

k

has displacement

x(t)

from equilibrium and velocity

v(t) = \mathrm{d}x/\mathrm{d}t

, positive or negative with the direction of motion. Mechanical energy is conserved:

\tfrac12 k x^2 + \tfrac12 m v^2 = E,

with

E

constant. Differentiate this relation implicitly with respect to

x

, treating

v

as an unknown function of

x

, to find

\mathrm{d}v/\mathrm{d}x

without first solving for

v

explicitly, on a stretch of the motion where the block is moving (

v\neq0

). Express the result in terms of

k

m

x

and

v

, and check that it has the units of

\mathrm{s^{-1}}

. What happens to the result at a turning point, where

v=0

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