PHYS-222 Worksheet 3
Problem 3-1 (See also P22.61 )
Step 1 , Let first suppose the density of planet is ρ, the radius R, the little green men’s masses m.
1
4π 3
3 r ,
recall formula: the volume of a sphere is V =
A = 4πr2 .
3
mass m = ρ 4π
3 r . The surface area of a sphere is
Step 2 , We know the distance, in both cases are
? In order to get time t, we need to know the
acceleration a. By Newton’s second law, F = ma, we need to know force F .
Step 3 , The gravity F~ = GMr12M2 r̂12 has the same form as Coulomb’s law. So you can guess, the Gauss law
is also hold for gravity.
gravity
electrostatic
GM m
r̂12
F~ =
2
r12
kQq
F~ = 2 r̂12
r12
mass M, m
charge Q, q
density
ρ
ρ
constant
G
k
force
G≡
constant
Field
Gauss law
1
4πe0
F~
=
m
I
=
1
4π0
k=
~
F
q
~
· dA
Q
=
0
~
=E
I
~ · dA
~
E
Step 4 , In the first case, neglect the mine because it’s very small comparing with a planet.
. What does it look like?
Use Gauss law to find the gravity force at any point of the mine. F =
What is the force of a spring (Hooke’s law F = −kr) ? Is it a simple harmonic oscillator? [Yes]/[No].
If yes, the spring constant ks =
?
q
Recall that the simple harmonic oscillator period T = 2π kms . How many period during the fall of the
little green man
? So the time of the first little green man fell down is t1 =
.
Step 5 , In the second case, we have a hollowed sphere. It equals a whole sphere plus
density of the first sphere is ρ, and the density of the second sphere is
?
? The
Step 6 , The total field equals the superposition of the field from first sphere and that second sphere. Use
Gauss law what is the gravity fields respectively? We have obtain the magnitude in Step 4. Now
,
.
combined with the direction, the gravity fields are:
Step 7 , The total force is
is in the cavity?
~ is a constant vector. So what kind of force
? Notice that ~r − r~0 = 12 R
Step 8 , What is the acceleration a and the time t2 that the second little green drops down? t2 =
Recall for uniform accelerated motion, the displacement s = v0 t + 21 at2 .
Step 9 , Take the ratio:
. Which little green men is more miserable?
Conguratulations! You have solved the problem for EXINCA! They might offer you a vacation to Naboo
(See Star War).
Key: t1 : t2 = π : 4
2