Two point masses m are held in place a distance d apart. Another point mass M is midway between...

Question

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 apprmumate1 y by F, 16GmMx/d3 if x « d. What
(b) Show that the mass M will oscillate with an angular frequency of (4/d) Y Gm/d and period wd/2, Id/Gm.
(c) What would the period be if m = 100 kg and d = 25.0 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 displaced from the center a small distance x toward earlier of the fixed masses? Why?

Robert · Accepted Answer

GRAVITATIONAL FORCES
Dr. Michele Laino
March 14, 2016
1 Solution
Parts a) b) and c) The situation of the exercise, is like below:
Figure 1: figure 1
After a litle displacement z, there are two gravitational forces which are
acting on the object of mass M . By symmetry, the resultant of such gravita-
tional forces, has no horizontal component, whereas the vertical component,
is:
Fz =
−2GMmz((
d
2
)2
+ z2
)3/2
1
which can be rewritten as below:
Fz =
−2GMmz((
d
2
)2
+ z2
)3/2 = −16GMmz
d3
(
1 +
(
2z
d
)2)3/2
so, the motion of the object of mass, can be described by this differential
equation M :
Mz̈ =
−16GMmz
d3
(
1 +
(
2z
d
)2)3/2
or, if we divide by the mass M to both sides:
z̈ =
−16Gmz
d3
(
1 +
(
2z
d
)2)3/2
As we can see,

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...

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