Quarter-Period Rendezvous

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63.Quarter-Period Rendezvous

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A mass

M

rests in equilibrium on top of a vertical spring of spring constant

k

. Directly above

M

, a pendulum of length

L

hangs from a fixed pivot, with the pivot positioned so that the bob, hanging straight down at rest, would just touch the top of

M

. At

t = 0

the bob is held at a small angle

\theta_0

from vertical and

M

is held a distance

\Delta x

below its equilibrium position; both are released from rest simultaneously. For what value of

L

does the bob first strike

M

at the instant

M

first returns to its equilibrium position?

$A vertical spring stands on the ground with mass $M$ on top at its equilibrium position; label the spring constant $k$. Directly above $M$, at distance $L$ above the top of $M$, mark a fixed pivot point. From the pivot, draw a string of length $L$ angled to the right by a small angle $\theta_0$ from vertical, with a small bob at its end; label the angle $\theta_0$ and the string length $L$. Draw a dashed vertical line from the pivot through $M$'s position to indicate the rest-line of the pendulum. Draw a small downward arrow next to $M$ labeled $\Delta x$ to indicate $M$'s initial displacement below equilibrium (with $M$ shown displaced downward from its equilibrium dashed line).$

L = \dfrac{Mg}{16k}

L = \dfrac{Mg}{4k}

L = \dfrac{Mg}{k}

L = \dfrac{\pi^2 Mg}{8k}

L = \dfrac{4Mg}{k}

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