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2m
gl (
1−
tot
s
ω
=
I
tot
•
However, we are not seeking ω, but the initial ve-
locity of the projectile v
tween the two values is given be the equation for
the conservation of angular momentum directly
before and after the collision:
L
= L
K
tot
with the angular momentum of the projectile
L
= m
I
v
K
K
K
0
before the collision and the total angular momen-
tum
ω
L
= I
tot
tot
after the collision. Inserting Eqs. 7 and 8 into Eq.
6 gives:
ω
m
I
v
= I
K
K
0
tot
•
Resolving this for ω and equating with Eq. 5 leads
to the following relationship
1
=
v
2
I
m gl
0
tot
tot
m l
K K
•
The moment of inertia is in principle determined
from the integral
=
2
∫
I
l dm
tot
m
where l is the distance of each mass element from
the axis of rotation. Since in this case it is not the
moment of inertia that we seek to derive I
also be calculated from the period T of the pen-
dulum (with projectile and any extra weights). For
a physical pendulum the following is valid for
small deflections
1
:
T
=
I
m gl
tot
tot
s
π
2
•
This means that all the variables are now known
or calculable. For the above example, the follow-
ing table emerges:
No m
/ kg
m
/ kg
K
tot
1
0.00695
0.06295
2
0.00695
0.06295
3
0.00695
0.06295
4
0.00695
0.09795
5
0.00695
0.09795
6
0.00695
0.09795
1
Recknagel, A.: Physik Mechanik, 3te Auflage, VEB Verlag Technik Berlin, 1958.
ϕ
cos
)
(5)
. The relationship be-
0
(6)
(7)
(8)
(9)
−
ϕ
(
)
1 cos
(10)
s
(11)
can
tot
2
(12)
I
/ m
T / s v
in m/s
s
0
0.218
1.01
3.39
0.218
1.01
4.82
0.218
1.01
6.88
0.252
1.07
3.51
0.252
1.07
4.98
0.252
1.07
6.99
•
The numeric values should be determined sepa-
rately for every pendulum, since material and
manufacturing tolerances mean that values may
differ from one to another.
4.1.3.2 Elastic collision
•
For a swinging pendulum Eq. 5 is still valid for
the motion after a collision, the only difference
being that the moment of inertia I
without the projectile but with any extra weights
(pendulum mass m
2
m gl
ω
=
P
•
To determine the relationship between ω and the
initial velocity v
lar momentum and the conservation of energy
before and after the collision must now be used.
The additional equation is required since the sys-
tem has an additional degree of freedom in the
projectile velocity v
9, the following is true for the angular momen-
tum:
m
I
v
= m
K
K
0
⇔
I
=
−
P
v
v
2
0
m I
K K
•
If this velocity v
the conservation of energy
1
1
=
2
m v
K
0
2
2
by rearranging in various steps the following ex-
pression is obtained for v
1
=
ω
v
l
0
K
2
•
If Eq. 13 is plugged in here and I
in Eq. 12, then v
inelastic collision:
No m
/ kg
m
/ kg
K
P
7
0.00695
0.0560
8
0.00695
0.0560
9
0.00695
0.0560
•
These values for v
those obtained for inelastic collisions. This can
be explained by the fact that the elastic collisions
are not entirely ideal.
7
is determined
P
):
P
−
ϕ
(
)
1 cos
s
(13)
I
P
both the conservation of angu-
0
after the collision. As for Eq.
2
ω
I
v
+ I
K
K
2
P
ω
(14)
is inserted into the equation for
2
1
+
ω
2
2
m v
I
(15)
K
2
P
2
0
I
+
P
1
s
(16)
2
m I
K K
determined as
P
can be calculated for an ideal
0
I
/ m
T / s v
in m/s
s
0
0.211
1.008
2.88
0.211
1.008
4.05
0.211
1.008
5.65
are about 18% smaller than
0
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