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GENERAL PHYSICS PH 221-3A
(Dr. S. Mirov)
Test 4 (12/03/07)
STUDENT NAME: ________________________ STUDENT id #: ___________________________ -------------------------------------------------------------------------------------------------------------------------------------------
Key
Sample
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:
Motion along a straight line with a constant acceleration
Motion
along a straight line with a constant acceleration
v aver. speed
= [dist. taken]/[time trav.]=
S/t;
v aver.vel.
x
/^ Δ
t;
v ins
=d
x
/^ Δ
t;
a
aver.
v aver. vel
t;
a
= d
v
/^ Δ
t;
v = v
o^
at;
x= 1/2(v
+v)t; x = vo^
t + 1/2 ato
v
2
= v
(^2) o
(if
x
=0 at to
=0)o
Free fall motion (with positive direction
g
= 9.80 m/s
y = v
aver.
t
v aver.
= (v+v
)/2o
v = v
o^
;^
y = v
o^
t - 1/2 g t
v 2
= v
(^2) o
(if
y
=0 at to
=0)o
3
Motion in a plane
3
.^
Motion
in a plane
v x^
= v
o^
cos
v y^
= v
o^
sin
x = v
ox
t+ 1/2 a
x^
(^2) t
;^
y = v
oy
t + 1/2 a
y^
(^2) t
;^
v x^
= v
ox
;^
v y^
= v
oy
4.
Projectile motion (with positive direction
v x^
= v
ox
= v
o^
cos
x = v
ox
t
x
v ox
t
x max
= (2 v
(^2) o sin
cos
)/g = (v
(^2) o
sin
)/g
for
y
in
= y
fin
v y^
= v
oy
o^
sin
y = v
oy
t - 1/2 gt
Uniform circular Motion
a=v
2 /r,
T=
r/v
Relative motion
P^ A
P B
B A
P A
P B
v^
v^
v
a^
a
=^
r^
r^
r
r^
r
7.
Component method of vector addition
A
A
1
A
2
; A
= x
A
x
A
x
and
A
y^
A
y
A
y
;^
A^
A^
A
x^
y
=^
2
2
= tan
A
y^
/A
⏐x
The scalar product
A
=^
c o s
a^
b^
a b
φ
⋅^
r
r
ˆ^
ˆ^
ˆ^
ˆ^
(^
)^
(^
x^
y^
z^
x^
y^
z
a^
b^
a^
i^
a^
j^
a^
k^
b^
i^
b^
j^
b^
k
⋅^
=^
+^
+^
⋅^
+^
r
r
r^
=^
x^
x^
y^
y^
z^
z
a^
b^
a^
b^
a^
b^
a^
b
⋅^
+^
r The vector product
ˆ^
ˆ^
ˆ^
ˆ^
(^
)^
(^
x^
y^
z^
x^
y^
z
a^
b^
a^
i^
a^
j^
a^
k^
b^
i^
b^
j^
b^
k
×^
=^
+^
+^
×^
+^
r
r
ˆ^
ˆ^
y^
z^
x^
y
x^
z
x^
y^
z
y^
z^
x^
y
x^
z
x^
y^
z
i^
j^
k^
a^
a^
a^
a
a^
a
a^
b^
b^
a^
a^
a^
a^
i^
j^
k
b^
b^
b^
b
b^
b
b^
b^
b
×^
=^
−^
×^
=^
=^
−^
+^
r^
r
r^
r
ˆ^
(^
)^
(^
)^
(^
y
y^
z^
y^
z^
z^
x^
z^
x^
x^
y^
x^
y
a^
b^
b^
a^
i^
a^
b^
b^
a^
j^
a^
b^
b^
a^
k
=^
−^
+^
−^
+^
Second Newton’s Law m
a
F
net
;
Kinetic friction
f =k^
μ
Nk
;
St ti
f i ti
f^
N
St
atic friction
f
=s
μ
N;s
Universal Law of Gravitation:
F=GMm/r
2 ;
G=6.67x
Nm
2 /kg
2 ;
Drag coefficient
2
D
C
A v
ρ
6.
Terminal speed
t
m g
v^
C^
A
ρ
Centripetal force:
F
=mvc
2 /r
Speed of the satellite in a circular orbit:
v
2 =GM
/rE
r
The work done by a constant force acting on an object:
c o s
W
F d
F^
d
φ
=^
=^
⋅^
r
r
Kinetic energy:
2
K
m v
Total mechanical energy:
E=K+U
The work-energy theorem:
W=K
-Kf
;o W
nc
=
Δ
K+
Δ
U=E
-Ef
o
The principle of conservation of mechanical energy:
when
W
nc
=0, E
=Ef^
o
Work done by the gravitational force:
c o s
g
W
m g d
φ
=^
Linear Momentum and Newton’s Second law for a system of particles:
a n d c o m
n e t
d P
P^
M v
F^
d t
=^
r
r^
r
r
Collision and impulse:
(^
)^
;^
t^ f i
a v g
t
J^
F^
t^
d t
J^
F^
t
=^
=^
∫
r^
r^
when a stream of bodies with mass m and
speed v, collides with a body whose position is fixed
a v g
n^
n^
m
F^
p^
m
v^
v
t^
t^
t Δ
=^
−^
Δ^
=^
−^
Δ^
=^
−^
Δ^
Δ^
t^
t^
t
Δ^
Δ^
Impulse-Linear Momentum Theorem:
f^
i
p^
p^
J
−^
=^
r
r^
r
3.
Law of Conservation of Linear momentum:
f o r
c l o s e d , i s o l a t e d
s y s t e m
i^
f
P^
P
r^
r
4.
Inelastic collision in one dimension:
1
2
1
2
i^
i^
f^
f
p^
p^
p^
p
+^
=^
r^
r^
r^
r
Motion of the Center of Mass:
The center of mass of a closed, isolated system of two colliding bodies is
not affected by a collision.
Elastic Collision in One Dimension:
1
2
1
1
1
2
1
1
2
1
2
f^
i^
f^
i
m
m
m
v^
v^
v^
v
m
m^
m
m
=^
+^
Collision in Two Dimensions:
1
2
1
2
1
2
1
2
i x^
i x^
f x^
f x^
i y^
i y^
f y^
f y
p^
p^
p^
p^
p^
p^
p^
p
+^
=^
+^
+^
=^
Variable-mass system:
( f i r s t r o c k e t
e q u a t i o n )
l n
( s e c o n d
r o c k e t e q u a t i o n )
r e l
i
f^
i^
r e l
f
R v
M a
M
v^
v^
v^
M
−^
S
Angular Position:
( r a d i a n
m e a s u r e )
S^ r
θ^
Angular Displacement:
2
1
( p o s i t i v e
f o r
c o u n t e r c l o c k w i s e
r o t a t i o n )
θ^
θ^
Δ^
=^
Angular velocity and speed:
;^
( p o s i t i v e
f o r
c o u n t e r c l o c k w i s e
r o t a t i o n )
a v g
d
t^
d t
=^
Angular acceleration:
a v g
d
t^
d t
=^
angular acceleration:
2
(^
o o
o
o^
t o
t
t^
t
ω^
ω^
θ^
θ^
θ^
=^
−^
=^
−^
=^
2
2
2
(^
o^
o
o^
t^
t
ω^
α^
θ^
θ^
θ^
=^
+^
−^
=^
Linear and angular variables related:
2
2
v^
r
T
π
π
θ^
2
;^
;^
;^
t^
r
s^
r^
v^
r^
a^
r^
a^
r^
T
r^
v
θ
ω
α
ω
ω
=^
=^
=^
=^
=^
=^
3.
Rotational Kinetic Energy and Rotational Inertia:
2
2
2 1
;^
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 i t h
c o n t i n u o u s l y
d i s t r i b u t e d
m a s s.
i^
i
K^
I^
I^
m
r
I^
r^
ω d m
=^
∑
∫^
y^
y
∫
4.
The parallel axes theorem:
2
c o m
I^
I^
M
h
=^
5.
Torque:
s i n
t r F
r^
F^
r F
⊥
=^
=^
6.
Newton’s second law in angular form:
n e t
I
τ
α
7.
Work and Rotational Kinetic Energy:
;^
(^
)^
f o r
f i
f^
i
d^
W
c o n s t
W
θ θ
τ^
τ^
θ^
=^
−^
=^
∫
2
2
;^
w o r k
e n e r g y
t h e o r e m
f o r
r o t a t i n g
b o d i e s
f^
i^
f^
i
d W
P^
K^
K^
K^
I^
I^
W
d t
=^
Δ^
=^
−^
=^
−^
d t
8.
Rolling bodies:
2
2
s i n
f o r
r o l l i n g
s m o o t h l y
d o w n
t h e
r a m p
c o m
c o m
c o m
c o m v^
R
K^
I^
m v
a^
R
g
a
α^
=^
= =^
2
f o r
r o l l i n g
s m o o t h l y
d o w n
t h e
r a m p
c o m
c o m
a^
I^
M
R
=^
9.
Torque as a vector:
;^
s i n
r^
F^
r F
r F
r^
F
φ^
⊥^
⊥
=^
×^
=^
=^
r
r^
r
1.
Damped Harmonic Motion:
2
2
2
2
1
(^
)^
c o s (
'^
) ,
'^
,^
(^
)
4
2
b t^
b t
m^
m
m^
m
k^
b
x^
t^
x^
e^
t^
E^
t^
k x
e
m^
m
ω
φ^
ω
−^
−
=^
+^
=^
−^
≈
2
1
(^
)^
i^
(^
)
k^
k^
f^
f
π^
ω
ω^
λ^
λ
2.
Sinusoidal waves:
(^
,^
)^
s i n (
) ,
,^
,
2
m
y^
x^
t^
y^
k x
t^
k^
f^
v^
f
T^
k^
T
ω
λ
λ
π
=^
−^
=^
=^
=^
=^
=^
=
3.
Wave speed on stretched string:
v
τ μ
=
4.
Average power transmitted by a sinusoidal wave on a stretched string:
2
2
1 2
a v g
m
P^
v^
y
μ^
ω
=
5.
Interference of waves:
1
1
' (^
,^
)^
[ 2
c o s
] s i n (
)
2
2
m
y^
x^
t^
y^
k x
t
φ
ω
φ
=^
−^
6.
Standing waves:
' (^
,^
)^
[ 2
s i n
] c o s
m
y^
x^
t^
y^
k x
t ω
=
7.
Resonance:
,^
f o r
1 , 2 , 3 ,...
2
v^
v
f^
n^
n
L
λ =^
=^
=
8.
Sound waves:
, B
v
ρ
=
9.
Interference:
2
( 2
)^
f o r
0 , 1 , 2 , 3... ,
c o n s t r u c t i v e
i n t e r f e r e n c e
2
( 2
1 )
f o r
0
1
2
3
d e s t r u c t i v e
i n t e r f e r e n c e
L^
m^
m
L^
m^
m
φ
π
π
λ
φ
π
π
Δ =^
=^
=
Δ =^
=^
+^
=
2
( 2
1 )
f o r
0 , 1 , 2 , 3... ,
d e s t r u c t i v e
i n t e r f e r e n c e
m^
m
φ
π
π
λ =^
=^
+^
=
10.
Sound Intensity:
2
2
2
1
,^
,
2
4
s
m
P^
P
I^
I^
v^
s^
I
A^
r
ρ^
ω
π
=^
=^
=
11.
Sound level in decibels:
1 2
2
( 1 0
) l o g
,^
1 0
/
o
I o
d B
I^
W^
m
I
β^
−
=^
=
12.
Standing wave patterns in pipes:
,^
1 , 2 , 3 ,... ,
f o r
p i p e
o p e n e d
f r o m
b o t h
e n d s
2
,^
1 , 3 , 5 ,... ,
f o r
p i p e
c l o s e d
a t
o n e
e n d
a n d
o p e n e d
a t
t h e
o t h e r
4
v^
n v
f^
n L
v^
n v
f^
n L
λ λ =^
=^
=
=^
=^
=
13.
Beats:
1
2
f^ b e a t
f^
f
=^
−
The Doppler effect:
f^
f^
v v
R s
'^
(^
)
=
± 1
,^
v^ R
the speed of the receiver; v
s^ the speed of the
sound;(v
=331m/s);s
i^
hi
i^
i
- for receiver approaching stationary emitter,- for receiver moving away from the stationary emitter;
f^
f^
v
E
'^ =
⎛ ⎜ ⎜⎜
⎞ ⎟ ⎟⎟
1
1
m
,^
v^ E
the speed of the emitter, v
s^ the speed of the sound,
v E
s
⎜^ ⎝
⎟⎠
1
m
- for emitter approaching stationary receiver,+ for emitter moving away from the stationary receiver;
'^
general Doppler Effect
s^
R
v
v
f^
f^
±
=
general Doppler Effect
s^
E
f^
f
v
v m
Your grandfather clock’s pendulum has a length of 0.9930 m. if the clockloses half a minute per day, how should you adjust the length of thependulum?
L
For a simple pendulum
Suppose clock's pendulum oscillates "n" times in a day.
L
T
g
1
pp
p
y
after the adjustment of the pendulum's length
nT
s
⋅^
2
2
nT
s
L g
T
L
⋅^
2 1
1
Take ratio
g
T
L
T
L g
2 1
L
2
2
2
4
L L
L
m
−
2
1
L
L
m
