Parameters:
L = 2[m],m1 = 5[kg],m2 = 0.01[kg],v = 400[m/s],x = 0.1[m]
a. Conservation on angular momenta around the rod's axis (The gravity is
parallel to the displacement vector):
Lz = (L − x)m2 v sin(90) = Iω
(1)
The moment of inertia of the rod and the bullet is:
I=
m1 L2
L
+ m1 ( )2 + m2 (L − x)2
12
2
(2)
So the angular velocity after the collision:
ω=
m1 L2
12
(L − x)m2 v sin(90)
+ m1 ( L2 )2 + m2 (L − x)2
(3)
b. The distance between the axis and the center of mass:
Lcm =
m1 L2 + m2 (L − x)
m1 + m2
(4)
Using conservation of energy:
Iω 2
= (m1 + m2 )gLcm (1 − cos ϕ)
2
(1 − cos ϕ) =
ϕ = cos−1 (1 −
(5)
2g(m1 L2
Iω 2
+ m2 (L − x))
(6)
2g(m1 L2
Iω 2
)
+ m2 (L − x))
(7)
1
‫א‪ .‬ההתנגשות אלסטית והמוט אינו מקובע בשום ציר‪ .‬בנוסף אין חיכוך או כוחות חיצוניים‬
‫אחרים‪ .‬מכאן‪ -‬הגדלים השמורים במערכת הינם אנרגיה‪ ,‬תנע קווי ותנע זויתי‪.‬‬
‫ב‪ .‬נשתמש בחוקי השימור של האנרגיה‪ ,‬התנע הקווי והתנע הזויתי בהתאמה‪:‬‬
‫‪1 2 1‬‬
‫‪1‬‬
‫‪mv  MU 2  I  2‬‬
‫‪2‬‬
‫‪2‬‬
‫‪2‬‬
‫‪mv  MU‬‬
‫‪L‬‬
‫‪mv  I ‬‬
‫‪2‬‬
‫נפתור עבור יחס המסות‪ ,‬כאשר ידוע כי מומנט ההתמד לסיבוב של מוט סביב מרכז המסה‬
‫‪1‬‬
‫‪m 1‬‬
‫‪ I ‬ונקבל כי‪-‬‬
‫שלו הוא ‪ML2‬‬
‫‪‬‬
‫‪12‬‬
‫‪M 4‬‬
‫‪.‬‬
zlblbe dtilw
zeywan od m`e
.zegek aygl jixv ,dve`z zeywan zel`y m` ,(cinz `l ynn la`) llk jxca
zligza f` ,dwexid `qtewd ly ieeykrd mewina qegid daeb z` rawp m` .dibxp` `id jxcd ,zexidn
ody s` lr) dcear zervan opi` zelblbd ,miff `l zelblbd ixivy oeeikn .qt` `id dibxp`d drepzd
.
0
,dligza dibxp`l ddf zeidl dkixv
h
daeb ixg` dibxp`d jk m` .(gek zelirtn
1
1
1
0 = −m2 gh + m2 v22 + I1 ω12 + I2 ω22
2
2
2
:lagd zexidn oial zizieefd ozexidn oia xyw yi ,zelblba dwlgd oi`y oeeikny `ed oiiprd
ω1 · R = v2
:mbe
ω2 · r2 = v2
:jk dibxp`d xeniy z`eeyn z` meyxl lkep okl
2m2 gh =
m2 v22
+ I1
v 2
2
R
+ I2
v2
r2
2
I2 2
I1 2
v2 +
v
2
m2 R
m2 r22 2
2gh
v22 =
I1
1 + m2 R2 + mI22r2
2gh = v22 +
2
T
T
. 1 a iwte`d lagd z`e , 2 a ikp`d lagd z` onqp
.mihpnene zegek aygl jxhvp ,dve`zd liaya
:lawp wexid sebd lr .mitebd zyelyl oeheip ly ipyd wegd z` meyxp
m2 g − T2 = m2 a2
:lawpe ,mihpnen aygp dlegkd zlblbd lr
T2 r2 − T1 r2 = I2 α2
:dxet`d dtilwd lre
T1 R = I1 α1
oia xyw lawle eze` xefbl ozip ,cinz xnyp mcew ep`vny zeiexidnd oia xywde ,dwlgd oi`y oeeikn
:zeieewl zeizieefd zeve`zd
a2
R
a2
α2 =
r2
α1 =
:`ed ipyd wegdn eplaiwy ze`eeynd hqy jk
m2 g − T2 = m2 a2
a2
T2 r2 − T1 r2 = I2
r2
a2
T1 R = I1
R
1
:dxabl`a xcq zvw
m2 g − T2 = m2 a2
a2
T2 − T1 = I2 2
r2
a2
T1 = I1 2
R
(1)
(2)
(3)
:cgi ze`eeynd zyely z` xagp eiykre
I1
I2
a + 2 a2
2 2
r2
R
g
a2 =
1 + mI22r2 + mI21R2
m2 g − T2 + T2 − T1 + T1 = m2 a2 +
2
la` ,caekd dve`za ltep sebdy milawn epiid ,dqn dziid `l zelblbl m`y al eniy .dve`zd efe
.xzei h`l ltep `ed dqn odl yiy oeeikn
zpzip iwte`d laga zegiznd .epl did xaky ze`eeynd hq mr cearl jixv heyt ,zeiegiznl xywa
(3)
I1
I1
T1 = 2 a2 = 2
R
R 1+
I2
m2 r22
+
I1
m2 R2
(1)
T2 = m2 g − m2 a2 = m2 g −
1+
d`eeyn ici lr
g
m2 g
I2
+ mI21R2
m2 r22
m2 g
=
1+
d`eeynn ikp`d lagde
I2
m2 r22
+
I1
m2 R2
I2
m2 r22
+
I1
m2 R 2
:`ed dtilwd ly cnzdd hpnen ,cnzdd ihpnenl xywa
2
I1 = m1 R2
3
:ze`wqic ly cnzd ihpnen xtqn xagp zlblbd ly cnzdd hpnen z`e
1
1
1
I2 = m3 r32 + m3 r32 + m3 r22
2
2
2
2