A block of weight hangs from a vertical spring of stiffness

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Scientific programming for physicists · Scientific programming for physicists

Scientific Computing for Physicists, Ch Minimisation with guarantees, with constraints and with randomness

Problem

A block of weight

W=4.9\ \mathrm{N}

hangs from a vertical spring of stiffness

k=20\ \mathrm{N\,m^{-1}}

. Let

z

be the block's height above the position where the spring has its natural length, so that its potential energy is

U(z)=\tfrac12kz^2+Wz

. A rigid floor lies

h=0.15\ \mathrm{m}

below that position, so the block must satisfy

g(z)=z+h\ge0

. ((a)) Find the unconstrained minimum of

U

and show that it violates the constraint. ((b)) Solve the KKT conditions in this chapter's convention,

U'(z)=\lambda g'(z)

with

\lambda\ge0

and

\lambda g=0

, for

z

and

\lambda

. Read

\lambda

as a force and check it with a force balance on the block. ((c)) The floor is lowered to

h=0.30\ \mathrm{m}

. Find

z

and

\lambda

now. ((d)) Someone imposes the floor with the equality penalty

\tfrac12\mu g^2

. What does that do to the block of part (c), and why?

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