1. Section 10.8: Energy in Rotational Motion

2. Translation-Rotation Analogues & Connections
Displacement x θ
Velocity v ω
Acceleration a α
Force (Torque) F τ
Mass (moment of inertia) m I
Newton’s 2nd Law ∑F = ma ∑τ = Iα
Kinetic Energy (KE) (½)mv (½)Iω2
CONNECTIONS: v = rω, atan= rα
aR = (v2/r) = ω2r , τ = rF , I = ∑(mr2)

3. The radial component of F
Work done by force F on an object as it rotates through an infinitesimal distance ds = rdθ
dW = Fds = (Fsinφ)rdθ
dW = τdθ
The radial component of F
does no work because it is
perpendicular to the
displacement.

4. Power The rate at which work is being done in a time interval Δt is
This is analogous to P = Fv for translations.

5. Work-Kinetic Energy Theorem
The work-kinetic energy theorem in rotational language states that the net work done by external forces in rotating a symmetrical rigid object about a fixed axis equals the change in the object’s rotational kinetic energy

6. Ex : Rod Again

7. Sect Rolling Objects
The curve shows the path moved by a point on the rim of the object. This path is called a cycloid
The line shows the path of the center of mass of the object
In pure rolling motion, an object rolls without slipping
In such a case, there is a simple relationship between its rotational and translational motions

8. Rolling Object The velocity of the center of mass is
The acceleration of the

9. A point on the rim, P, rotates
to various positions such as
Q and P. At any instant, a
point P on the rim is at rest
relative to the surface since
no slipping occurs
Rolling motion is thus a combination of pure translational motion and pure rotational motion

10. Total Kinetic Energy K = (½)Mv2 + (½)Iω2
The total kinetic energy of a rolling object is the sum of the translational energy of its center of mass and the rotational kinetic energy about its center of mass
K = (½)Mv2 + (½)Iω2
Accelerated rolling motion is possible only if friction is present between the sphere and the incline

11. Example:   y = 0 Sphere rolls down incline
(no slipping or sliding).
KE+PE conservation:
(½)Mv2 + (½)Iω2 +MgH
= constant, or
(KE)1 +(PE)1 =
(KE)2 + (PE)2
where KE has 2 parts:
(KE)trans = (½)Mv2
(KE)rot = (½)Iω2
v = 0
ω = 0 
v = ? 
 y = 0

12. Summary of Useful Relations

13. Translation-Rotation Analogues & Connections
Displacement x θ
Velocity v ω
Acceleration a α
Force (Torque) F τ
Mass (moment of inertia) m I
Newton’s 2nd Law ∑F = ma ∑τ = Iα
Kinetic Energy (KE) (½)mv (½)Iω2
CONNECTIONS: v = rω, atan= rα
aR = (v2/r) = ω2r , τ = rF , I = ∑(mr2)

14. Ex. 10.12: Energy & Atwood Machine