1. Aim: How do we explain rotational kinetic energy?

2. Rotational Kinetic Energy
A rotating rigid body consists of many particles. We can express the kinetic energy of a single particle Ki = 1/2mivi2
Ki = 1/2mivi2

3. Rotational Kinetic Energy
If we add up the kinetic energy of every particle in the rigid body, we get KR = ∑1/2mivi2 How can we rewrite this using angular speed?

4. Rotational Kinetic Energy
KR=1/2(∑miri2 ω2) We say that I = ∑miri2 and that I is a quantity called the moment of inertia of a rigid body. KR =1/2I ω2 and is the rotational kinetic energy of a rigid body.

5. No single moment of Inertia
Mass is analogous to moment of inertia but there is a BIG difference Mass is an inherent property of an object. Moment of inertia depends on your choice of rotation axis. The minimum moment of inertia occurs at a point that passes through the axis that goes through the center of mass.

6. Total Kinetic Energy
We now know that the total kinetic energy of a rigid body is equal to its translational kinetic energy plus its rotational kinetic energy. Write K = KT + KR

7. Finding the Moment of Inertia using calculus
I = ∫r2dm Moment of Inertia for an extended continuous object (rigid body)

8. Moment of Inertia of Homogeneous Rigid Bodies with Different Geometries

9. Thought Question 1
Two spheres, one hollow and one solid, are rotating with the same angular speed about their centers. Both spheres have the same mass and radius. Which one, if either, has the higher rotational kinetic energy? The Hollow Sphere because it has a greater moment of inertia
Hollow Sphere

10. Question 1-The oxygen molecule
Consider the diatomic oxygen molecule O2 which is rotating in the xy plane about the z axis passing through its center, perpendicular to its length. The mass of each oxygen atom is 2.66 x kg, and at room temperature , the average separation between the two oxygen atoms is d = 1.21 x m.
Calculate the moment of inertia of the molecules about the z axis.
A typical angular speed of a molecule is 4.6 x 1012 rad/s. If the oxygen molecule is rotating with this angular speed about the z axis, what is the rotational kinetic energy?
a) 1.95 x kg m2 b) 2.06 x J

12. Question 2-Four rotating masses
Four small spheres are fastened to the corners of a frame of negligible mass lying in the xy plane.
If the rotation of the system occurs about the y axis with an angular speed ω, find the moment of Inertia Iy, about the y axis and the rotational kinetic energy about the axis.
Suppose the system rotates in the xy plane about an axis through O (the z axis). Calculate the moment of inertia about the z axis and the rotational kinetic energy.
a) 2Ma^ KR = Ma^2w^2 b)2Ma^2 +2mb^ KR=(Ma^2 + mb^2)w^2

13. Moment of Inertia

15. Thought Question 2
There exists an assembly of three identical point masses attached by a massless rod as shown below. If we were to calculate the rotational inertia of the system about an axis going through each mass, about which axis will the moment of inertia be the greatest? Least?

17. Question 3-
Rigid rods of negligible mass lying along the y axis connect 3 small particles. If the system rotates about the x axis with an angular speed of 2 rad/s find,
The moment of inertia about the x axis and the rotational kinetic energy
The tangential speed of each particle and the total kinetic energy evaluated by adding the kinetic energy of each particle.
,4,8

18. Picture