P221/S2001/Problem 3A
Consider a body of mass moving in a circular orbit of radius 9 about a massive body of
mass 4 ˆ . The system might be a planet moving around the Sun or a satellite in orbit
around the Earth.
(a) (1 point) Write down the expression for the gravitational potential energy of the
system.
The gravitational potential energy is < ~ c
.4
9
.
(b) (3 points) Using Newton's second law of motion, show that the orbital speed of the
smaller body is # ~ l .4 °9À
Apply Newton's second law to uniform circular motion, for which ~ #2 °9 ¢
- ~ so .4 °9 2 ~ #2 °9 so #2 ~ .4 °9 and # ~ l.4 °9 .
(c) (3 points) Prove that the total mechanical energy of the system is , ~ c .4
29 À
Using the result of part (b), 2 ~ #2 ~ ².4 °9³ ~ .4 °29.
The mechanical energy , ~ 2 b < ~ .4 °29 c .4 °9 ~ c .4 °29
(d) (3 points) Determine an algebraic expression (in terms of some or all of the quantities
Á 4 Á .Á 9Á 9E ) for the energy that must be added to move a satellite of mass from a
circular orbit of radius 29E to one of radius 39E , where 9E is the radius of the Earth.
The energy that must be added is
,f c ,i ~ c .4 °2²39E ³ c ² c .4 °2²29E ³ ~ ².4 °9E ³ ²1/4 c 1/6³
~ ².4 °9E ³ ²2/24³ ~ .4 °129E .
(e) (3 bonus points) Determine an algebraic expression (in terms of .Á Á and 9P ) for the
escape speed from the surface of a planet of uniform density and radius 9P .
To escape, the total mechanical energy , ~ 2 b < € 0, or 2 ‚ c < ~ b
.4
9
The escape speed is the speed for which this inequality is an equality: ²#esc ³2 ~
.4
9
or #esc ~ .4 °9À Using 4 ~
4
3
²9P ³3 and 9 ~ 9P , this becomes
#esc ~ l.4 °9 ~ m2.² 34 ²9P ³3 ³°9P ~ l(8/3).(9P )2 .
.
P221/S2001/Problem 3B
A 2.0-kg object oscillates in one dimension with simple harmonic with the displacement %
from its equilibrium position varying with time according to the graph:
4.0
x ( t ) [m ]
3.0
2.0
1.0
0.0
-1.0 0
1
2
3
4
5
-2.0
-3.0
-4.0
t [s]
(a) (2 points) Determine the amplitude (in m) of the oscillation.
( ~ 4.0 m since the displacement oscillates between b 4.0 m and c 4.0 m.
(b) (2 points) Determine the period (in s) and the frequency (in Hz) of the oscillation.
The period is 3 s, so the frequency is ~ 1°; ~ 1°3 s ~ 0.33 Hz.
(c) (4 points) Determine the displacement %²!³ as a function of !.
Since % ~ 0 at ! ~ 0 we expect that %²!³ is a sin function (because sin 0 = 0). The value of
~ 2 ~ (6.28)(1/3 s) š 2 rad/s.
Finally, then, %²!³ ~ c (4.0 m) sin (2 rad/s) !. Alternatives: b (4.0 m) sin "(2 rad/s) ! f rad#
or c (4.0 m) cos "(2 rad/s) ! c /2 rad# or b (4.0 m) "cos (2 rad/s) ! b /2 rad#
The c sign is necessary because for small ! € 0, %²!³ is negative rather than positive.
(d) (2 points) Determine the velocity component #% (in m/s) of the particle at ! ~ 0.
#% ²!³ ~ %²!³°! ~ c (4.0 m)(2 rad/s) cos (2 rad/s) ! ~ c (8 m/s) cos (2 rad/s) !
so at ! ~ 0, #% ~ c 8 m/s.
(e) (2 bonus points) Determine the mechanical energy of the system.
, ~ (2 ~ 2
(2 ~ ²0.5³²2.0 kg³²2 rad/s³2 ²4.0 m³2
š 1 d (2)2 d 16 J ~ 64 J.
P221/S2001/Problem 3C
An electric dipole consists of a positive
charge 8 on the positive & axis at (0, ) and a
negative charge c 8 on the negative & axis
at (0, c ) , as shown.
hB
l
TTTTTTTTA
h
m
W at a point (%Á 0)
(a) (4 points) Find the direction and magnitude of the electric field ,
[marked A] on the positive % axis.
At a point on the positive % axis, the electric fields from the two charges have canceling %
components and negative & components that add up.
The two charges are each a distance ~ l2 b %2 from the point A, so they each
produce an electric field there of magnitude 8°²2 b %2 ³. The vertical component of each of
these electric fields is ° times this magnitude, so overall (taking into account the fact that each
produces a negative component in the & direction,
,W ~ c ²28°²2 b %2 ³3/2 ³Vj .
W at a point (0, &)
(b) (4 points) Find the direction and magnitude of the electric field ,
[marked B] on the positive & axis.
The positive charge contributes an electric field along the b & direction with
,& ~ b 8°²& c ³2 while the negative charge contributes an electric field along the c &
direction with ,& ~ c 8°²& b ³2 , for a total of
,W ~ ² b 8°²& c ³2 c 8°²& b ³2 ³Vj , which can be simplified to
,W ~ 48&°²& c ³2 ²& b ³2 !Vj
(c) (2 points) Make a rough sketch of ,& versus & below, for points on the & axis. Make
sure you have the correct sign of ,& in each region.
(d) (3 bonus points) Show that in both cases, along the % axis or along the & axis, the
magnitude of the electric field , is proportional to 1/3 for ˆ , where is the distance
from the origin.
Taking the limit for point A on the b % axis as % becomes large,
, ~ lim ²28°²2 b %2 ³3/2 ³ ~ 28°%3 À
%SB
Taking the limit for point B on the b & axis as & becomes large,
, ~ lim ²8&°²& c ³2 ²& b ³2 ³ ~ 8&°& ~ 48°&3 À
&SB
In each case , becomes proportional to 1°3 (where ~ % or & , respectively).
MULTIPLE-CHOICE QUESTIONS
Note that these questions begin with number 21. Start with number 21 on your
machine-gradeable form. These questions are worth 2 points each.
21. A sinusoidal force with a given amplitude is applied to an oscillator. To maintain the
largest amplitude oscillation, the frequency of this applied force should be:
(A) one-fourth the natural frequency of the oscillator.
(B) half the natural frequency of the oscillator.
È(C) the same as the natural frequency of the oscillator.
(D) twice the natural frequency of the oscillator.
(E) as great as possible.
22. For an oscillator subjected to a damping force proportional to its velocity:
(A) The displacement is a sinusoidal function of time.
(B) The velocity is a sinusoidal function of time.
(C) The frequency is a decreasing function of time.
(D) The mechanical energy is constant.
È(E) None of the above is true.
23. A simple pendulum is suspended from the ceiling of an elevator. The elevator is
accelerating upwards with acceleration . The period of the pendulum, in terms of its
length 3, , and , is:
(A) 2l 3°
È(B) 2l 3°² b ³
(D) 2m 3°2
(E) impossible to determine without knowing the elevator's velocity.
24. A meter stick c 1.000 m in length, not
surprisingly c is pivoted at a point ~ 0.3
m above its center and swings as a physical
pendulum. The moment of inertia of a
uniform thin rod about an axis through its
center is 4 32 °12. What is the approximate
angular speed of the stick as it swings back
and forth?
(A) 1.2 rad/s
(B) 2.1 rad/s
(C) 2.8 rad/s
(C) 2l 3°² c ³
ˤ
.̈
RT center
(D) 3.5 rad/s
È(E) 4.2 rad/s
25. A hoop of radius 9 and mass 4 , with
all of its mass concentrated at its rim, rolls
without slipping down an inclined plane. If it
starts from rest, and its center drops a
vertical distance , what will be its speed at
the bottom, assuming nonconservative forces
are not doing work on the hoop?
(A) l °2
È(B) l (C) l 7°5
(D) l 3°2
(E) l 2
26. A positively charge metal sphere A is brought into contact with an uncharged metal
sphere B. As a result:
È(A) both spheres are positively charged.
(B) A is still positively charged and B is uncharged
(C) A is positively charged and B is negatively charged
(D) A is uncharged and B is positively charged
(E) both spheres are negatively charged
W ~ 8in ° . Which one of the followings statements is
W h (
27. Consider Gauss' law ,
true?
W must be the electric field due to the enclosed charge.
(A) ,
W ~ 0 everywhere inside the Gaussian surface.
(B) If 8in ~ 0, then ,
È(C) If the charge inside consists of an electric dipole, then the integral is zero.
W along the closed surface.
W is everywhere parallel to (
(D) ,
W on the surface.
(E) If the charge is placed outside the surface, then it cannot affect ,
28. A meter stick of mass is suspended by two ropes, one at the 20-cm mark and one at
the 10-cm end. Determine the tension in the rope at the 100-cm end.
;20
§
(A) °8
(B) °4
È(C) 3 °8
;100
§
(D) °2
(E) 5 °8
29. A spherical conducting shell has a charge 8. A point charge is placed at the center
of the cavity. The charge on the inner surface of the shell and the charge on the outer
surface of the shell are, respectively:
(A) 0, 8
(B) Á 8 c (C) 8Á 0
È(D) c , 8 b (E) c Á 0
30. Charge is distributed uniformly on the surface of a very large flat plate. The electric
field 2.0 cm from the plate is 40 N/C. The electric field 4.0 cm from the plate is:
(A) 160 N/C
(B) 80 N/C
È(C) 40 N/C
(D) 20 N/C
(E) 10 N/C