CSULA
SOLUTION MIDTERM EXAM I
Summer 2006
PHYSICS 202
I. CONCEPTUAL QUESTIONS
1. Between a molecule of gas (mass ~10-6 kg) and the Discovery shuttle (mass ~105 kg) which one
has the biggest escape speed (vesc)?
They have the same escape speed!!! Since escape speed is given by:
vesc 
2GM E
we
R
can see that vesc is independent of the mass of the projectile going out.
2. If the length of a simple pendulum is quadrupled, then its period will be change by the amount of:
(a) ½, (b) 4, (c) 2, (d) 2½
(c) Since:
T  2 L
g
if L becomes 4L, then:
T '  2 4L
g
 2 2 L   2T , therefore
g

the period doubled!!
3. Suppose a grandfather clock is calibrated correctly at a very tall mountain and is taken to the sea
level. Does the grandfather clock run: (a) Slow, (b) Fast, (c) Correctly?
(b) Fast. Since:
T  2 L
g
T 
1
. Then; at sea level g is greater than at top of the
g
mountain. As a result, the period of the pendulum will decrease and the clock will run faster.
4. The pressure at the bottom of a filled glass of water ( = 103 kg/m3) is P. The water is poured out
and the glass is filled with ethyl alcohol ( = 800 kg/m3). The pressure at the bottom of the glass
will be: (a) Smaller than P, (b) Greater than P, (c) Equal to P, (d) Indeterminate
(a) Smaller than P. Given: P   gh  P   , with g and h constant.
Since ethyl alcohol has less density than water, its pressure will be smaller that the
pressure of the water at the bottom of the glass.
II. Solve the following problems
1. Derive an expression for the work (W = E) required to move an Earth satellite of mass m from a
circular orbit of radius 2RE to one of radius 4RE.
E  E f  Ei  G
E  G
M Em 
M m
M m
M m
  G E   G E  G E
2rf
2ri 
2rf
2ri

M Em
M Em
M m1 1 1 M m
G
G E    G E
2  4 RE 
2  2 RE 
RE  4 8  8
RE
2. An object of mass M is hung from the bottom end of a vertical spring fastened to an overhead
beam. The object is set into vertical oscillation having a period of π seconds. Show that the force
constant (k) of the spring has an expression of: k = 4M
T  2
M
M
4 2 M
 T 2  4 2
k 
 4M
k
k
2
3. The horizontal constructed pipe in the figure is known as Venturi Tube can be used to measure
the flow speed of an incompressible fluid. Using Bernoulli’s Equation and Equation of Continuity
2  P1  P2 
show that v2 is given by: v  A
2
1
Bernoulli’s Equation:
P1 
P1 
Since: y1 = y2 , then
Equation of continuity:
1
2
  A12  A22 
 v12   gy1  P2 
1
2
 v12  P2 
A1v1  A2 v2  v1 
1
2
 v22
A2 v2
A1
1
2
 v22   gy2
(1)
(2)
Substitution of (2) into (1) gives:
 A2 v 2
Av 
P1    2 2   P2  12  v22  P1  12   2 2 2
 A
 A1 
 1
2
1
2
 A12  A22
2  P1  P2    

A12

 2
 v2  v2 


2
  P2  12  v2  P1  P2 

2  P1  P2 
 A  A2
 A2

1

2
1
2



 A1
1
2



 1 
A22  2
 v2 
A12 
2  P1  P2 
  A12  A22 
4. Derive an expression for the Average Density of the Earth (E) in function of g, G and RE. (DO
NOT plug in values)
E 
ME
VE
But : g 
GM E
gRE2

M

E
G
RE2
4
And : VE  RE3 
3
gRE2
3gRE2
3 g
E  G 3 

3
4RE 4GRE 4 GRE
3
III. Solve the following problems
1. A piece of metal with mass M and density ρ is suspended from a spring and then completely
immersed in a container with water (ρw). Show that the tension T2 in the spring has an
expression of:
F
y
   w 
T2  Mg 

  
 T2  B  Mg  0  T2  Mg  B  Mg  (  wVmetal )g
Vmetal 
M 

w 
   w 
 T2  Mg   w 
 g  Mg  1 
  Mg 

 
 
  


M
2. In the figure, by Pascal’s Principle, the absolute pressure at B is the same as A. Show that:
  Hg  w 
h1  
 h2
 w 
PA  PB  P0   Hg gh2   w gh  P0   w gh2   w gh   w gh1 
 Hg gh2   w gh2   w gh1   Hg h2   w h2   w h1 

Hg
   w 
  w  h2   w h1  h1   Hg
 h2
 w 
3. Io is a moon of Jupiter, with orbital period of 2 days and an orbital radius of 4 x105 km. From this
data find the mass of Jupiter (MJ)
4 2
T (
)r 3  GM J T 2  4 2 r 3 
GM J
4 2 r 3
4(3.14) 2 (4 x108 m)3
MJ 

 M J  3.60 x1018 kg
2
11
2
2
2
GT
(6.67 x10 N  m / kg )(2 x86400s )
2
4. A uniform rod of mass M and length L is pivoted about one end and oscillates in a vertical plane.
If the moment of inertia about an axis through one end is I  13 ML , show that the period of
2
oscillation (for small amplitude) is: T  2
T  2
I
 2
Mgd
2L
3g
1
ML2
3
 2
L
Mg  
2
1
L
3
 2
1
g
2
2L
3g