A solid disk with a mass of 0.8 kg and a radius of 12cm is rotating at a rate of 2.6 rev/s. Another disk with half the mass and half the radius is dropped on top of the first disk. What is the new angular speed of both disks when they are rotating together?
Answers
Answer:
Law of Conservation of angular momentum states:
The total angular momentum of a system about an axis remains constant, when the net external torque acting on the system about the given axis is zero.
This is applicable in the given question as a stationary second disk is dropped on a rotating disk. It is assumed that second disk is also mounted on the same shaft.
Angular momentum
→
L
=
→
I
×
→
ω
where angular velocity of disk rotating with angular speed
f
is
ω
=
2
π
f
Moment of inertia of disk
I
=
1
2
m
disk
R
2
Initial Angular momentum
=
→
L
1
+
→
L
2
=
∣
∣
∣
→
L
1
∣
∣
∣
+
0
=
(
1
2
m
1
R
2
1
)
(
2
π
f
)
=
(
3.5
×
(
0.15
)
2
)
(
π
×
15
)
=
1.18125
π
If
f
c
is the final common angular velocity of the system,
Final Angular momentum
=
(
→
I
1
+
→
I
2
)
×
→
ω
c
Inserting given values and equating scalar part with (1) we get, (remember to change the given diameter of second disk to its radius)
(
1
2
m
1
R
2
1
+
1
2
m
2
R
2
2
)
2
π
f
c
=
1.18125
π
⇒
(
3.5
×
(
0.15
)
2
+
5.0
×
(
0.18
2
)
2
)
π
f
c
=
1.18125
π
⇒
f
c
=
1.18125
3.5
×
(
0.15
)
2
+
5.0
×
(
0.18
2
)
2
f
c
=
9.9
rev per sec
, rounded to one decimal place.
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