Problem

For a certain oscillator the net force on the bod…

06:46

Problem 83 Easy Difficulty

Two point masses $m$ are held in place a distance $d$ apart. Another point mass $M$ is midway between them. $M$ is then displaced a small distance $x$ perpendicular to the line connecting the two fixed masses and released. (a) Show that the magnitude of the net gravitational force on $M$ due to the fixed masses is given approximately by $F_{\mathrm{na}}=\frac{16 \mathrm{GmMx}}{d^{3}}$ if $x \ll d$ . What is the direction of this force? Is it a restoring force? (b) Show that the mass $M$ will oscillate with an angular frequency of $(4 / d) \sqrt{G m / d}$ and period $\pi d / 2 \sqrt{d / G m} .$ (c) What would the period be if $m=100 \mathrm{kg}$ and $d=25.0 \mathrm{cm} ?$ Does it seem that you could easily measure this period? What things prevent this experiment from easily being performed in an ordinary physics lab? (d) Will $M$ oscillate if it is dis- placed from the center a small distance $x$ toward either of the fixed masses? Why?

Answer

A. This is a restoring force.
B. $k=\frac{16 G m M}{d^{3}} . \omega=\sqrt{\frac{k}{m}}=\frac{4}{d} \sqrt{\frac{G M}{d}} . T=\frac{2 \pi}{\omega}=\frac{\pi d}{2} \sqrt{\frac{d}{G M}}$
C.
D.

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Physics 101 Mechanics

University Physics with Modern Physics

Chapter 13

Periodic Motion

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Video Transcript

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