Equal-Time Dead Zone

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71.Equal-Time Dead Zone

+10

An imperfect spring exerts no force on a mass within a dead zone

|x| \leq d

, and exerts a linear restoring force with stiffness

k

outside the dead zone, so that

F(x) = -k(x-d)

for

x > d

and

F(x) = -k(x+d)

for

x < -d

. A mass

m

is attached and released from rest at

x = A

with

A > d

. For what value of the ratio

d/A

does the mass spend equal total time inside the dead zone and inside the spring regions over one full period?

$Plot of $F$ versus $x$ on horizontal $x$-axis and vertical $F$-axis. The graph is zero (lies on the $x$-axis) for $-d \leq x \leq d$, with $d$ and $-d$ marked on the $x$-axis. For $x > d$ the graph is a straight line of slope $-k$ passing through $(d, 0)$ and descending. For $x < -d$ the graph is a straight line of slope $-k$ passing through $(-d, 0)$ and continuing upward to the left. Mark the release point $x = A$ on the $x$-axis with $A > d$, and indicate the slope label $-k$ on one of the linear segments.$

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