GENERAL PHYSICS PH 221-3A (Dr. S. Mirov) Test 4 (12/03/07)

STUDENT NAME: ________________________ STUDENT id #: ___________________________ -------------------------------------------------------------------------------------------------------------------------------------------

KeySample

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 acceleration1. Motion along a straight line with a constant accelerationvaver. 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 = vo

2 + 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 = vo

2 – 2gy (if yo=0 at to=0)

3 Motion in a plane3. Motion in a plane 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;

4. Projectile motion (with positive direction ) vx = vox = vo cos; x = vox t; x vox t;xmax = (2 vo

2 sin cos)/g = (vo2 sin2)/g for yin = yfin;

vy = voy - gt = vo sin - gt; y = voy t - 1/2 gt2;

5. Uniform circular Motion a=v2/r, T=2r/v

6. Relative motion P A P B B A

P A P B

v v va a

7. Component method of vector addition 1

A = A1 + A2 ; Ax= Ax1 + Ax2 and Ay = Ay1 + Ay2; A A Ax y 2 2 ; = tan-1 Ay /Ax;

The scalar product A = c o sa b a b

ˆ ˆˆ ˆ ˆ ˆ( ) ( )x y z x y za b a i a j a k b i b j b k

= x x y y z za b a b a b a b

The vector product ˆ ˆˆ ˆ ˆ ˆ( ) ( )x y z x y za b a i a j a k b i b j b k

ˆˆ ˆˆˆ ˆy z x yx z

x y zy z x yx z

x y z

i j ka a a aa a

a b b a a a a i j kb b b bb b

b b b

ˆˆ ˆ( ) ( ) ( )

y

y z y z z x z x x y x ya b b a i a b b a j a b b a k

1. Second Newton’s Law ma=Fnet ;

2. Kinetic friction fk =kN;

3 St ti f i ti f N3. Static friction fs =sN;

4. Universal Law of Gravitation: F=GMm/r2; G=6.67x10-11 Nm2/kg2;

5. Drag coefficient 212

D C A v

6. Terminal speed 2

tm gv

C A

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: c o sW F d F d

10. Kinetic energy: 212

K m v

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: c o sgW m g d 2

1. Work done in Lifting and Lowering the object:

; ; f i a g f i a gK K K W W i f K K W W

2. Spring Force: ( H o o k ' s l a w )xF k x ( )x

3. Work done by a spring force: 2 2 21 1 1; i f 0 a n d ; 2 2 2s i o i f sW k x k x x x x W k x

4. Work done by a variable force: ( )f

i

x

x

W F x d x

5. Power: ; ; c o sa v gW d WP P P F v F v

t d t

6. Potential energy: ; ( )f

i

x

xU W U F x d x

7. Gravitational Potential Energy:

( ) ; 0 a n d 0 ; ( )f i i iU m g y y m g y i f y U U y m g y

8. Elastic potential Energy: 21( )2

U x k x

9. Potential energy curves: ( )( ) ; ( ) ( )m e c

d U xF x K x E U xd x

10. Work done on a system by an external force:

F r i c t i o n i s n o t i n v o l v e d W h e n k i n e t i c f r i c t i o n f o r c e a c t s w i t h i n t h e s y s t e m

m e c

m e c t h

t h k

W E K UW E E

E f d

i n tm e c t hW E E E E 11. Conservation of energy: i n t

i n tf o r i s o l a t e d s y s t e m ( W = 0 ) 0m e c t h

m e c t hE E E

12. Power: ; a v gE d EP Pt d t

;

13. Center of mass: 1

1 n

c o m i ii

r m rM

14. Newtons’ Second Law for a system of particles: n e t c o mF M a

3

1. Linear Momentum and Newton’s Second law for a system of particles: a n d c o m n e td PP M v Fd t

2. Collision and impulse: ( ) ; ;

f

i

t

a v gtJ F t d t J F t

when a stream of bodies with mass m and

speed v, collides with a body whose position is fixed a v gn n mF p m v v

t t t

t t t

Impulse-Linear Momentum Theorem: f ip p J

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 mi fP P

4. Inelastic collision in one dimension: 1 2 1 2i i f fp p p p

5. 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.

6. Elastic Collision in One Dimension: 1 2 11 1 2 1

1 2 1 2

2; f i f im m mv v v vm m m m

7. 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 yp p p p p p p p

8. 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

if i r e l

f

R v M aMv v vM

S9. Angular Position: ( r a d i a n m e a s u r e )S

r

10. 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 )

11. 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 gd

t d t

12. Angular acceleration: ; a v gd

t d t

4

1. angular acceleration: 2

1 ( )2

12

o

o o

o o

t

t

t t

2 2

2

2 ( )12

o o

o t t

2. Linear and angular variables related:

22 2 2v rT 2; ; ; ; t rs r v r a r a r T

r v

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 ;2

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 iK I I m r

I r d m

y y

4. The parallel axes theorem: 2c o mI I M h

5. Torque: s i ntr F r F r F

6. Newton’s second law in angular form: n e t I

7. Work and Rotational Kinetic Energy: ; ( ) f o r ;f

if id W c o n s tW

2 21 1; 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 s2 2f i f i

d WP K K K I I Wd t

2 2d t

8. Rolling bodies:

2 21 12 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 Rga

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 p1 /c o m

c o m

aI M R

9. Torque as a vector: ; s i nr F r F r F r F

5

1. Angular Momentum of a particle: ( ) ;

s i nl r p m r vl r m v r p r m v r p r m v

d l

2. Newton’s Second law in Angular Form: n e td ld t

3. Angular momentum of a system of particles: 1

n

ii

n e t e x t

L l

d Ld t

4. Angular Momentum of a Rigid Body: L I

5. Conservation of Angular Momentum: ( i s o l a t e d s y s t e m )i fL L

6. Static equilibrium: n e t

, , ,

0 ; 0i f a l l t h e f o r c e s l i e i n x y p l a n e 0 ; 0 ; 0

n e t

n e t x n e t y n e t z

FF F

7. Elastic Moduli: s t r e s s = m o d u l u s s t r a i n

8. Tension and Compression: , E i s t h e Y o u n g ' s m o d u l u sF LEA L

9. Shearing: , G i s t h e s h e a r m o d u l u sF LGA L

10. Hydraulic Stress: , B i s t h e b u l k m o d u l u sVp BV

11. Simple harmonic motion: 2c o s ( ) ; s i n ( ) ; c o s ( )m m mx x t v x t a x t

12. The Linear Oscillator: , 2k mTk

m k

13. Pendulums:

2 , t o r s i o n p e n d u l u m

L2 , s i m p l e p e n d u l u m

ITk

Tg

I

2 , p h y s i c a l p e n d u l u mITm g h

6

1. Damped Harmonic Motion: 2

222

1( ) c o s ( ' ) , ' , ( )4 2

b t b tm m

m mk bx t x e t E t k x em m

2 1( ) i ( )k k f f 2. Sinusoidal waves: ( , ) s i n ( ) , , , 2 my x t y k x t k f v f

T k T

3. Wave speed on stretched string: v

4. Average power transmitted by a sinusoidal wave on a stretched string: 2 212a v g mP v y

5. Interference of waves: 1 1' ( , ) [ 2 c o s ] s i n ( )2 2my x t y k x t

6. Standing waves: ' ( , ) [ 2 s i n ] c o smy x t y k x t

7. Resonance: , f o r 1 , 2 , 3 , . . .2

v vf n nL

8. Sound waves: ,Bv

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

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 em m

10. Sound Intensity: 2 22

1, , 2 4

sm

P PI I v s IA r

11. Sound level in decibels: 1 2 2( 1 0 ) l o g , 1 0 /oo

Id B I W mI

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 s2

, 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 r4

v n vf nL

v n vf nL

13. Beats: 1 2b e a tf f f

7

1. The Doppler effect: f f vv

R

s

' ( ) 1 , vR the speed of the receiver; vs the speed of the

sound;(vs=331m/s); + f i hi i i + for receiver approaching stationary emitter,

- for receiver moving away from the stationary emitter;

f f vE'

1

1, vE the speed of the emitter, vs the speed of the sound,

vE

s 1

- for emitter approaching stationary receiver, + for emitter moving away from the stationary receiver;

' general Doppler Effects Rv vf f general Doppler Effect

s E

f fv v

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?

LFor a simple pendulum 2

Suppose clock's pendulum oscillates "n" times in a day.

LTg

1

pp p y(24 3600 30) 86370

after the adjustment of the pendulum's lengthnT s

2

2

(24 3600) 86400

2

nT s

LgT L

864002

1 1

Take ratio 2

gT LT L

g

2

1

8640086370L

2

2 2

4

86400 0.9930 0.9937863700 9937 0 9930 7 10

L

L L m

10

2 1 0.9937 0.9930 7 10L L m

11

4.

12

5.

13

6.

14