a rod revolving60 times in a mintue about an axis passing through an end at right angle to the length has kinetic energy of 400joule,find moment of inertia of rod
Answers
Answer:
Explanation:
We defined the moment of inertia I of an object to be
I
=
∑
i
m
i
r
2
i
for all the point masses that make up the object. Because r is the distance to the axis of rotation from each piece of mass that makes up the object, the moment of inertia for any object depends on the chosen axis. To see this, let’s take a simple example of two masses at the end of a massless (negligibly small mass) rod ((Figure)) and calculate the moment of inertia about two different axes. In this case, the summation over the masses is simple because the two masses at the end of the barbell can be approximated as point masses, and the sum therefore has only two terms.
In the case with the axis in the center of the barbell, each of the two masses m is a distance R away from the axis, giving a moment of inertia of
I
1
=
m
R
2
+
m
R
2
=
2
m
R
2
.
In the case with the axis at the end of the barbell—passing through one of the masses—the moment of inertia is
I
2
=
m
(
0
)
2
+
m
(
2
R
)
2
=
4
m
R
2
.
From this result, we can conclude that it is twice as hard to rotate the barbell about the end than about its center.
Figure A shows a barbell of the length 2 R with the masses m at the ends. It is rotating through its center. Figure B shows a barbell of the length 2 R with the masses m at the ends. It is rotating through one end.
Figure 10.23 (a) A barbell with an axis of rotation through its center; (b) a barbell with an axis of rotation through one end.
In this example, we had two point masses and the sum was simple to calculate. However, to deal with objects that are not point-like, we need to think carefully about each of the terms in the equation. The equation asks us to sum over each ‘piece of mass’ a certain distance from the axis of rotation. But what exactly does each ‘piece of mass’ mean? Recall that in our derivation of this equation, each piece of mass had the same magnitude of velocity, which means the whole piece had to have a single distance r to the axis of rotation. However, this is not possible unless we take an infinitesimally small piece of mass dm,