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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