Sect. 10.4: Rotational Kinetic Energy

1. Sect. 10.4: Rotational Kinetic Energy

2. Translation-Rotation Analogues & Connections Translation Rotation Displacement x θ Velocity v ω Acceleration a α Mass m ? Kinetic Energy (K) (½)mv2 ? CONNECTIONS s = rθ, v = rω, at= rα, ac = (v2/r) = ω2r

3. Sect. 10.4: Rotational Kinetic Energy • Translational motion (Chs. 7 & 8): K = (½)mv2 • Rigid body rotation, angular velocity ω.Rigid means Every point has the same ω.Object is made of particles, masses mi. • For each mi at a distance rifrom the rotation axis: vi = riω. So, each mass mi has kinetic energy Ki = (½)mi(vi)2. So, The Rotational Kinetic Energy is: KR = ∑[(½)mi(vi)2] = (½)∑mi(ri)2ω2 = (½)∑mi(ri)2ω2 ω2 goes outside the sum, since it’s the same everywhere in the body • Define the moment of inertia of the object, I  ∑mi(ri)2  KR = (½)Iω2(Analogous to (½)mv2)

4. Translation-Rotation Analogues & Connections Translation Rotation Displacement x θ Velocity v ω Acceleration a α Force (Torque) F τ Mass (moment of inertia) m I Kinetic Energy (K) (½)mv2(½)Iω2 CONNECTIONS s = rθ, v = rω, at= rα, ac = (v2/r) = ω2r

5. Example 10.3: Four Rotating Objects 4 tiny spheres on ends of 2 massless rods. Arranged in the x-y plane as shown. Sphere radii are very small.  Masses are treated as point masses. Calculate moment of inertia I when (A) The system is rotated about the y axis, as in Fig. a. (B) The system is rotated in the x-y plane as in Fig. b.

6. Bottom line for rotational kinetic energy • Analogy between the kinetic energies associated with linear motion: K = (½)mv2, & the kinetic energy associated with rotational motion: KR = (½)Iω2. • NOTE: Rotational kinetic energy is not a new type of energy. But the form of this kinetic energy is different because it applies to a rotating object. • Of course, the units of rotational kinetic energy are Joules (J)

7. Sect. 10.5: Moments of Inertia • Definition of moment of inertia:I  ∑mi(ri)2 • Dimensions of I are ML2 & its SI units are kg.m2 • Use calculus to calculate the moment of inertia of an object: Assume it is divided into many small volume elements, each of mass Δmi. Rewrite expression for I in terms of Δmi. & take the limit as Δmi 0. So Make a small volume segment assumption & the sum changes to an integral: ρ≡ density of the object. If ρ is constant, the integral can be evaluated with known geometry, otherwise its variation with position must be known. With this assumption, I can be calculated for simple geometries.

8. Notes on Densities ρ • Volume Mass Density Solid mass m in volume V: r = mass per unit volume: r = (m/V) • Surface Mass Density Mass m on a thin surface of thickness t: σ = mass per unit thickness: σ =rt • Linear Mass Density Mass m in a rod of length L & cross sectional area A: λ= mass per unit length: λ=(m/L)= rA

9. Moment of Inertia of a Uniform Thin Hoop • This is a thin hoop, so all mass elements are the same distance from the center

10. Moment of Inertia of a Uniform Thin Rod • The hatched area has mass dm = λdx = (M/L)dx • Linear Mass Density: Mass per unit length of a rod of uniform cross-sectional area λ = M/L = ρA So, the moment of inertia is

11. Moment of Inertia of a Uniform Solid Cylinder • Divide cylinder into concentric shells of radius r, thickness dr, length L. Volume Mass Density: ρ= (m/V) Then, Ibecomes:

12. Moments of Inertia of Some Rigid Objects

13. The Parallel Axis Theorem NOTE!Idepends on the rotation axis! Suppose we need Ifor rotation about the off-center axis in the figure:  d  If the axis of interest is a distance d from a parallel axis through the center of mass, the moment of inertia has the form:

14. Moment of Inertia of a Rod Rotating Around an End • We’ve seen: Moment of inertia ICM of a rod of length Labout the center of mass is: • What is the moment of inertia I about one end of the rod? Shift rotation axis by D = (½)L & use Parallel Axis Theorem: