[Donald A. McQuarrie, John D. Simon] Physical Chem(BookZZ.org)
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[Donald A. McQuarrie, John D. Simon] Physical Chem(BookZZ.org)


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, a n d ( b ) a d i f f e r e n t c o o r d i n a t e 
s y s t e m f o r a b o d y f a l l i n g f r o m a h e i g h t x
0
\u2022 
5 9 
60 Chapter 2 I The Classical Wave Equation 
2-17. Derive an equation for the maximum height a body will reach if it is shot straight upward 
with a velocity v0 . Refer to Figure 2.9b but realize that in this case v0 points upward. How 
long will it take for the body to return to its initial position, x = 0? 
2-18. Consider a simple pendulum as shown in Figure 2.1 0. We let the length of the pendulum 
be l and assume that all the mass of the pendulum is concentrated at its end as shown in 
Figure 2.10. A physical example of this case might be a mass suspended by a string. We 
assume that the motion of the pendulum is set up such that it oscillates within a plane so 
that we have a problem in plane polar coordinates. Let the distance along the arc in the 
figure describe the motion of the pendulum, so that its momentum is mdsjdt = mld() jdt 
and its rate of change of momentum is mld2() / d t2 \u2022 Show that the component of force in 
the direction of motion is -mg sin(), where the minus sign occurs because the direction of 
this force is opposite to that of the angle (). Show that the equation of motion is 
d2() 
ml-2 = -mg sin() dt 
Now assume that the motion takes place only through very small angles and show that the 
motion becomes that of a simple harmonic oscillator. What is the natural frequency of this 
harmonic oscillator? Hint: Use the fact that sin() ~ () for small values of(). 
2-19. Consider the motion of a pendulum like that in Problem 2-18 but swinging in a viscous 
medium. Suppose that the viscous force is proportional to but oppositely directed to its 
velocity; that is, 
ds d() 
fviscous = -A dt = -At dt 
where A is a viscous drag coefficient. Show that for small angles, Newton's equation is 
d 2() d() 
ml-2 +Al- +mg() = 0 dt dt 
Show that there is no harmonic motion if 
Does it make physical sense that the medium can be so viscous that the pendulum undergoes 
no harmonic motion? 
s FIGURE 2.10 
The coordinate system describing an oscillating pendulum. 
P r o b l e m s 
2 - 2 0 . C o n s i d e r t w o p e n d u l u m s o f e q u a l l e n g t h s a n d m a s s e s t h a t a r e c o n n e c t e d b y a s p r i n g t h a t 
o b e y s H o o k e ' s l a w ( P r o b l e m 2 - 7 ) . T h i s s y s t e m i s s h o w n i n F i g u r e 2 . 1 1 . A s s u m i n g t h a t t h e 
m o t i o n t a k e s p l a c e i n a p l a n e a n d t h a t t h e a n g u l a r d i s p l a c e m e n t o f e a c h p e n d u l u m f r o m t h e 
h o r i z o n t a l i s s m a l l , s h o w t h a t t h e e q u a t i o n s o f m o t i o n f o r t h i s s y s t e m a r e 
d
2
x 
m - = - m o i x - k ( x - y ) 
d t 2 0 
d 2 y 2 
m - = - m w y - k ( y - x ) 
d t 2 0 
w h e r e w
0 
i s t h e n a t u r a l v i b r a t i o n a l f r e q u e n c y o f e a c h i s o l a t e d p e n d u l u m , [ i . e . , w
0 
= ( g j l ) 
1
1
2
] 
a n d k i s t h e f o r c e c o n s t a n t o f t h e c o n n e c t i n g s p r i n g . I n o r d e r t o s o l v e t h e s e t w o s i m u l t a n e o u s 
d i f f e r e n t i a l e q u a t i o n s , a s s u m e t h a t t h e t w o p e n d u l u m s s w i n g h a r m o n i c a l l y a n d s o t r y 
x ( t ) = A e i w t y ( t ) = B e i w t 
S u b s t i t u t e t h e s e e x p r e s s i o n s i n t o t h e t w o d i f f e r e n t i a l e q u a t i o n s a n d o b t a i n 
( w
2 
- w~ - ~) A = - ~ B 
( w
2 
- w~ - ~) B = -~A 
N o w w e h a v e t w o s i m u l t a n e o u s l i n e a r h o m o g e n e o u s a l g e b r a i c e q u a t i o n s f o r t h e t w o a m -
p l i t u d e s A a n d B . W e s h a l l l e a r n i n M a t h C h a p t e r E t h a t t h e d e t e r m i n a n t o f t h e c o e f f i c i e n t s 
m u s t v a n i s h i n o r d e r f o r t h e r e t o b e a n o n t r i v i a l s o l u t i o n . S h o w t h a t t h i s c o n d i t i o n g i v e s 
( w 2 -w~- ~Y = (~Y 
N o w s h o w t h a t t h e r e a r e t w o n a t u r a l f r e q u e n c i e s f o r t h i s s y s t e m , n a m e l y , 
2 2 
W I = W o 
a n d 
w~ = w 2 + 2 k 
0 -
m 
I n t e r p r e t t h e m o t i o n a s s o c i a t e d w i t h t h e s e f r e q u e n c i e s b y s u b s t i t u t i n g w i a n d w~ b a c k i n t o 
t h e t w o e q u a t i o n s f o r A a n d B . T h e m o t i o n a s s o c i a t e d w i t h t h e s e v a l u e s o f A a n d B a r e 
c a l l e d n o r m a l m o d e s a n d a n y c o m p l i c a t e d , g e n e r a l m o t i o n o f t h i s s y s t e m c a n b e w r i t t e n a s 
a l i n e a r c o m b i n a t i o n o f t h e s e n o r m a l m o d e s . N o t i c e t h a t t h e r e a r e t w o c o o r d i n a t e s ( x a n d y ) 
i n t h i s p r o b l e m a n d t w o n o r m a l m o d e s . W e s h a l l s e e i n C h a p t e r 1 3 t h a t t h e c o m p l i c a t e d 
F I G U R E 2 . 1 1 
x T w o p e n d u l u m s c o u p l e d b y a s p r i n g t h a t o b e y s H o o k e ' s l a w . 
6 1 
62 Chapter 2 I The Classical Wave Equation 
vibrational motion of molecules can be resolved into a linear combination of natural, or 
normal, modes. 
2-21. Problem 2-20 can be solved by introducing center-of-mass and relative coordinates 
(cf. Section 5-2). Add and subtract the differential equations for x(t) and y(t) and then 
introduce the new variables 
17 = x + y and ~ = x - y 
Show that the differential equations for 17 and ~ are independent. Solve each one and 
compare your results to those of Problems 2-20. 
M A T H C H A P T E R 
B 
P R O B A B I L I T Y A N D S T A T I S T I C S 
I n m a n y o f t h e f o l l o w i n g c h a p t e r s , w e w i l l d e a l w i t h p r o b a b i l i t y d i s t r i b u t i o n s , a v e r a g e 
v a l u e s , a n d s t a n d a r d d e v i a t i o n s . C o n s e q u e n t l y , w e t a k e a f e w p a g e s h e r e t o d i s c u s s s o m e 
b a s i c i d e a s o f p r o b a b i l i t y a n d s h o w h o w t o c a l c u l a t e a v e r a g e q u a n t i t i e s i n g e n e r a l . 
C o n s i d e r s o m e e x p e r i m e n t , s u c h a s t h e t o s s i n g o f a c o i n o r t h e r o l l i n g o f a d i e , 
t h a t h a s n p o s s i b l e o u t c o m e s , e a c h w i t h p r o b a b i l i t y p j , w h e r e j = 1 , 2 , . . . , n . I f t h e 
e x p e r i m e n t i s r e p e a t e d i n d e f i n i t e l y , w e i n t u i t i v e l y e x p e c t t h a t 
N . 
1
. J 
p = l i D -
j N - + o o N 
j = 1 , 2 , . . . , n 
( B . l ) 
w h e r e N . i s t h e n u m b e r o f t i m e s t h a t t h e e v e n t j o c c u r s a n d N i s t h e t o t a l n u m b e r o f 
J 
r e p e t i t i o n s o f t h e e x p e r i m e n t . B e c a u s e 0 : = : N j : = : N , p j m u s t s a t i s f y t h e c o n d i t i o n 
0 : S p j : S 1 
( B . 2 ) 
W h e n p . = 1 , w e s a y t h e e v e n t j i s a c e r t a i n t y a n d w h e n p . = 0 , w e s a y i t i s i m p o s s i b l e . 
J J 
I n a d d i t i o n , b e c a u s e 
n 
L N j = N 
j = i 
w e h a v e t h e n o r m a l i z a t i o n c o n d i t i o n , 
n 
L P j = 1 
( B . 3 ) 
j = i 
6 3 
64 Math Chapter B I P R 0 B A B I Ll T Y AN D STAT I S T I C S