Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova

Lecture 42

Please return you optional homework

A bullet of mass m is fired with velocity of magnitude into a block of mass M. The block is connected to a spring constant k and rests on a frictionless surface. Find the velocity of the block as a function of time. (Assume the bullet comes to rest infinitely quickly in the block, i.e. during the collision the spring doesn’t get compressed.)

mV

mV

Exercise 4

k1 k2m

x=0

Resonance

)cos()(

cos

2

2222

0

02

2

t

mb

mk

mF

tx

tFkxdt

dxb

dt

xdm

D

DD

D

tF Dcos0

amplitude

D

Frames of reference

This means that in the freely-falling elevator cabin you don’t feel any effects of gravity! You and all objects around you experience the same acceleration.

In outer space you can imitate the effect of gravity by acceleration.

Newton’s First Law

Second Law

Third Law

F

F1N

gm1

1N

1N

gm2

1N

2N

Kinematics

dt

dVa

dt

dVa

dt

Vda y

yx

x ;;

dt

dyV

dt

dxV

dt

rdV yx ;;

dtVydtVx

dtaVdtaV

yx

yyxx

;

;;

is given, you can find If V

and a

r

Work Energy Theorem

22

222

1

initialfinalr

r

total

mVmVrdFW

22

222

1

2

1

initialfinaly

y

totaly

x

x

totalx

mVmVdyFdxFW

veconservatinonveconservati WWW

veconservatiW does NOT depend on path!

)]()([ 12 rUrUW veconservati

y

UF

x

UF yx

;

22)]()([

21

22

12

mVmVWrUrU

WWW

veconservatinon

veconservatinonveconservati

2)(

2)(

21

1

22

2

mVrU

mVrUW veconservatinon

2)(

2)(

21

1

22

2

mVrU

mVrUW veconservatinon

2)(

2)(

,02

11

22

2

mVrU

mVrU

WIf veconservatinon

Mechanical energy is conserved!

Conservation of Momentum

extFdt

pd

)()(

)()(

,0,0

afterpbeforep

afterpbeforep

ConstpConstp

Constpdt

pdFIf

yy

xx

yx

ext

If the collision is perfectly elastic, the kinetic energy is conserved!

Circular Motion

rdt

drar

dt

rda

maFmaF

r

rr

2;

;

22

2

y

x

r

ri

i

rirr

r

dt

drV

dt

drVr ;

Conservation of Angular Momentum

Fr

rhrmriridt

drmrprL

dt

Ldrext

tot

);(][; 2

ConstLIf totext

,0

For symmetrical objects rotating about their axis of symmetry:

2);( ii

irmIrhrIL

22

2

1

2

1 IVmKE ii

i

)(rhrIdt

Ld totext

Second Law: m1

m2

R I

Harmonic Motion

m

k

tBtAtx

kxdt

xdm

sincos)(

02

2

Resonance:

)cos()(

cos

2

2222

0

02

2

t

mb

mk

mF

tx

tFkxdt

dxb

dt

xdm

D

DD

D

Final exam: room 101 Richardson, 10 am- noon,

Friday, December 11.

Help Sessions: Tuesday (by appointment)Wednesday, December 9th, 2pm 203 MPHYS