Download GENERAL PHYSICS PH 221-1D Test 4 (Sample)

GENERAL PHYSICS PH 221-1D (Dr. S. Mirov)
Test 4 (Sample)
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ALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS.
NOTE: Clearly write out solutions and answers (circle the answers) by section for each part (a., b., c., etc.)
Important Formulas:
1.
Motion along a straight line with a constant acceleration
vaver. speed = [dist. taken]/[time trav.]=S/t;
vaver.vel. = x/t;
vins =dx/t;
aaver.= vaver. vel./t;
a = dv/t;
v = vo + at; x= 1/2(vo+v)t; x = vot + 1/2 at2; v2 = vo2 + 2ax (if xo=0 at to=0)
2.
Free fall motion (with positive direction )
g = 9.80 m/s2;
y = vaver. t
vaver.= (v+vo)/2;
v = vo - gt; y = vo t - 1/2 g t2; v2 = vo2 – 2gy (if yo=0 at to=0)
3.
Motion in a plane
4.
Projectile motion (with positive direction )
5.
Uniform circular Motion
vx = vo cos;
vy = vo sin;
x = vox t+ 1/2 ax t2; y = voy t + 1/2 ay t2; vx = vox + at; vy = voy + at;
vx = vox = vo cos;
x = vox t;
xmax = (2 vo2 sin cos)/g = (vo2 sin2)/g for yin = yfin;
vy = voy - gt = vo sin - gt;
y = voy t - 1/2 gt2;
a=v2/r,
T=2r/v
6. Relative motion



v PA  v PB  v BA


a PA  a PB
7.
Component method of vector addition
1
A 
A x2  A y2 ;  = tan-1 Ay /Ax;


The scalar product A a  b = a b c o s 


a  b  ( a x iˆ  a y ˆj  a z kˆ )  ( b x iˆ  b y ˆj  b z kˆ )
 
a  b = a xb x  a yb y  a zbz


The vector product a  b  ( a x iˆ  a y ˆj  a z kˆ )  ( b x iˆ  b y ˆj  b z kˆ )
A = A1 + A2 ; Ax= Ax1 + Ax2 and Ay = Ay1 + Ay2;
iˆ




a  b  b  a  ax
bx
ˆj
a
b
y
y
kˆ
a
a z  iˆ
b
bz
y
a
y
bz
 ( a y b z  b y a z ) iˆ  ( a z b x  b z a x ) ˆj  ( a x b
y
z
ax
bx
 ˆj
 bxa
y
ax
az
 kˆ
bx
bz
a
y
b
y

) kˆ
1.
Second Newton’s Law
ma=Fnet ;
2.
Kinetic friction fk =kN;
3.
Static friction fs =sN;
4.
Universal Law of Gravitation: F=GMm/r2; G=6.67x10-11 Nm2/kg2;
5.
Drag coefficient
6.
Terminal speed
7.
Centripetal force: Fc=mv2/r
8.
Speed of the satellite in a circular orbit: v2=GME/r
9.
The work done by a constant force acting on an object:
10. Kinetic energy:
D 
vt 
K 
1
C  Av
2
2
2m g
C  A
1
m v
2
W
 
 F d cos  F  d
2
11. Total mechanical energy: E=K+U
12. The work-energy theorem: W=Kf-Ko; Wnc=K+U=Ef-Eo
13. The principle of conservation of mechanical energy: when Wnc=0, Ef=Eo
14. Work done by the gravitational force:
W
g
 m gd cos
2
1.
Work done in Lifting and Lowering the object:
 K

K
f
 K
 W
i
F
a
 W
  k x
; if
g

f
K
Spring Force:
3.
Work done by a spring force:
4.
Work done by a variable force:
W
1
k x
2

s
x


W
x
W
; P
 t
Pavg 
5.
Power:
6.
Potential energy:
7.
d W
d t

 U
i
; W
2
i

 W
a
g
( H o o k 's l a w )
2.
x
K
 W ;  U
2
o
; if x
i

x ; W
s
 
1
k x
2
2
F ( x )d x
i

 
x
x
f


F  v

F ( x )d x
i
 m g ( y
f

y i )  m g  y ; if
y
i
 0 a n d U
1
k x
2
8.
Elastic potential Energy:
U ( x ) 
9.
Potential energy curves:
F ( x )  
10.
Work done on a system by an external force:
F ric tio n is n o t in v o lv e d W
i
 0; U ( y )  m g y
2
d U ( x )
; K ( x ) 
d x
  E
m e c
  K
E
 E
th

 U ( x )
m e c
  U
W h e n k in e tic fric tio n fo rc e a c ts w ith in th e s y s te m
  E
W
m e c
  E
th
  E
in t
fkd
Conservation of energy:
Pavg 
 E
; P
 t
12.
Power:
13.
Center of mass:
14.
f
Gravitational Potential Energy:
 U
11.
 0 a n d x
f
F v c o s 

; P
1
k x
2

rc o m

W
  E
  E
m e c
  E
fo r is o la te d s y s te m

1
M
d E
d t
i 1
  E
in t
m e c
  E
th
 0
;
n

th
(W = 0 )  E
m
i

ri
Newtons’ Second Law for a system of particles:

F net 

M a
c o m
3
1.
2.




dP
Linear Momentum and Newton’s Second law for a system of particles: P  M v c o m a n d F n e t 
dt

J 
Collision and impulse:

t

F ( t ) d t ; J  F a v g  t ; when a stream of bodies with mass m and
f
ti
F avg  
speed v, collides with a body whose position is fixed

p
Impulse-Linear Momentum Theorem:

 p
f
n
n
 m
 p  
 v
m  v  
 t
 t
 t

 J
i


Pi  P f f o r c l o s e d , i s o l a t e d s y s t e m
3.
Law of Conservation of Linear momentum:
4.
Inelastic collision in one dimension:
5.
Motion of the Center of Mass: The center of mass of a closed, isolated system of two colliding bodies is


p 1i  p

 p1
2 i
f

 p
2 f
not affected by a collision.
6.
Elastic Collision in One Dimension:
7.
Collision in Two Dimensions:
8.
Variable-mass system:
9.
Angular Position:
v
 vi  v
10. Angular Displacement:

m
m

 m
 m
1
1
 p1
2 ix
fx
2
v1i ;
v
2 f
2
 p
2 fx
;

m
p 1 iy  p
2m1
1  m
2 iy
v1i
2
 p1
fy
 p
2 fy
ln
rel
M
M
i
( s e c o n d ro c k e t e q u a tio n )
f
(ra d ia n m e a s u re )
   
11. Angular velocity and speed:
12. Angular acceleration:
f
 M a (firs t ro c k e t e q u a tio n )
rel
S
r
 
p 1 ix  p
R v
f
v1
avg


2
 
1
(p o s itiv e fo r c o u n te r c lo c k w is e ro ta tio n )

 
;
 t
 
;
 t
 
avg
 
d
dt
(p o s itiv e fo r c o u n te rc lo c k w is e ro ta tio n )
d 
dt
4
  
1.
angular acceleration:
o

  
o
  ot 
2
 
  
o
  )t
o
1
 t2
2
 2  (   o )
2
o
  t 
1
 t
2
v   r;
a
  r;
t
a
r
1
I
2

I 

2
r 2d m
; I 

2
r;
T

2 r
2


v
m i ri 2 f o r b o d y a s a s y s t e m o f d i s c r e t e p a r t i c l e s ;
f o r a b o d y w ith c o n tin u o u s ly d is trib u te d m a s s .
 I
 M h
4.
The parallel axes theorem: I
5.
Torque:
6.
Newton’s second law in angular form:

7.
Work and Rotational Kinetic Energy:
W
P 
dW
dt
com
2
  r F t  r F  r F s i n 
;  K
 K
v
Rolling bodies:
f
 K
i
2
f

net
 I

1
I
2


2
i
f
 d ; W
  (
f
  i ) fo r   c o n st;
i
 W
w o rk e n e rg y th e o re m fo r ro ta tin g b o d ie s
  R
com
a
com
a
com


Torque as a vector:
1
I
2

1
I com 
2
  R
K
9.
v 2
 
r

Rotational Kinetic Energy and Rotational Inertia:
K
8.
2
Linear and angular variables related:
s   r;
3.
1
(
2
  

2.
  t
o


2

1
m v
2
g s in 
1  I com / M R


 r  F ;
2
2
com
fo r ro llin g s m o o th ly d o w n th e ra m p
  r F s in   r F

 r F
5
1.
2.





l  r  p  m (r  v );
Angular Momentum of a particle:
l  r m v s in   r p

Newton’s Second law in Angular Form: 
n et


L 
3.
5.
6.
n

i1
Angular momentum of a system of particles:


4.
 rm v


 r p  r m v

d l
d t
L  I


Conservation of Angular Momentum: L i  L
n et ext


li

d L
d t
Angular Momentum of a Rigid Body:
Static equilibrium:
if a ll th e fo rc e s lie in x y p la n e
F
n et,x
 0;
F
n et, y
 0;

 0
net,z
s tre s s = m o d u lu s  s tra in
Elastic Moduli:
8.
Tension and Compression:
9.
Shearing:
 G
10. Hydraulic Stress:
 L
, G
L
p  B
11. Simple harmonic motion:
12. The Linear Oscillator:
13. Pendulums:
(is o la te d s y s te m )


F net  0 ;  net  0
7.
F
A
f
F
A
 E
 L
, E i s t h e Y o u n g 's m o d u l u s
L
is th e s h e a r m o d u lu s
 V
V
, B is th e b u lk m o d u lu s
x  x
 
m
c o s( t   ); v    x
k
, T
m
 2
m
s in ( t   ); a   
2
x
m
c o s( t   )
m
k
T
 2
I
, to rs io n p e n d u lu m
k
T
 2
L
, s im p le p e n d u lu m
g
T
 2
I
, p h y s ic a l p e n d u lu m
m g h
6
1.
Damped Harmonic Motion:
2.
Sinusoidal waves:
x (t) 
x
m
 b t
e
2 m
k
m
c o s ( ' t   ),  ' 
2


b 2
4 m
1
T
, v 
3.
Wave speed on stretched string:
4.
Average power transmitted by a sinusoidal wave on a stretched string:
Pavg 
5.
Interference of waves:
6.
Standing waves:
7.
Resonance:
8.
Sound waves:
f

y
v

B

Interference:
 L
 

2
 (2 m
Sound level in decibels:
12.
Standing wave patterns in pipes:
13.
f

v
v

Beats:

 1 )
1
 v
2
11.


f


k


1
 v
2
2
2
fo r m
f
2
m
s
2
m
 (1 0 d B ) lo g
, I 
I
, I
Io
 0 ,1 , 2 , 3 ..., d e s tr u c tiv e in te r f e r e n c e
Ps
4 r
o
2
 1 0
 1 2
W
/ m
2
n v
, n  1 , 3 , 5 , ..., f o r p ip e c lo s e d a t o n e e n d a n d o p e n e d a t th e o th e r
4 L
f1 
y
 0 ,1 , 2 , 3 ..., c o n s tr u c tiv e in te r f e r e n c e


m
1
1
 ] s in ( k x   t 
 )
2
2
n v
, n  1 , 2 , 3 , ..., f o r p ip e o p e n e d f r o m
2 L
b ea t
 b t
  f
T

f
e
s in k x ] c o s  t
 m (2 ) fo r m
Sound Intensity:

co s
m
2
10.
f
2
2
m
v
, f o r n  1 , 2 , 3 , ...
2 L
P
, I 
A
I 
,
1
k x
2
,

 L
m

, E (t) 


v 
y '( x , t )  [ 2 y
 n
v 
s in ( k x   t ), k 
m
y '( x , t )  [ 2 y
 
9.
y ( x , t) 
2
b o th e n d s
2
7
1.
The Doppler effect:
f '  f (1 
vR
),
vs
vR the speed of the receiver; vs the speed of the
sound;(vs=331m/s);
+ for receiver approaching stationary emitter,
- for receiver moving away from the stationary emitter;



1 
f'  f 
,
v
E
1

vs 

vE the speed of the emitter, vs the speed of the sound,
- for emitter approaching stationary receiver,
+ for emitter moving away from the stationary receiver;
f ' f
vs  vR
general Doppler Effect
vs  vE
8
9
2.
Your grandfather clock’s pendulum has a length of 0.9930 m. if the clock
loses half a minute per day, how should you adjust the length of the
pendulum?
For a simple pendulum T  2
L
g
Suppose clock's pendulum oscillates "n" times in a day.
 nT1  (24  3600  30)  86370s
after the adjustment of the pendulum's length
nT2  (24  3600)  86400 s
L2
g
T2


Take ratio
T1
L
2 1
g
2
L2 86400

L1 86370
864002  0.9930
 L2 
 0.9937
2
86370
 L2  L1  0.9937  0.9930  7  104 m
10
11
4.
12
5.
13
6.
14