A rigid body undergoing planar motion is the sum of the translational and rotational kinetic energy. The motion is in reference to the mass center, G.
Translation: rectilinear or curvilinear translation is T = .5 m VG^2
Rotation about a fixed axis: T = .5 m VG^2 + .5 IG ω^2 or T = .5 IO ω^2
General Plane Motion: T = .5 m VG^2 + .5 IG ω^2 or T = .5IIC ω^2
Work
Work done by:
- Variable force: UF = ∫F cos(θ) ds
- Constant Force: UFc = Fc cos(θ)
- Weight: Uw = -WY Where Y is the change in height, work is negative when the displacement is upward.
- Linear Elastic Spring Force: Us = -(.5ks2^2 - .5ks1^2) The work will be negative when a spring stretches or compresses.
- Couple Moment(M = Fr): UM = ∫M dθ work is positive when the moment and dθ have the same sense of direction.
- Forces acting at a point.
- Having a direction perpendicular to displacement (normal force or weight of a body moving horizontally)
Work and Energy
T1 + U1-2 = T2
The initial kinetic energy plus the work done by external forces is equal to the final kinetic energy.
Conservation of Energy
If the work of a force is independent of the path, only depending on the initial and final position then it is a conservative force. Example: weight and spring
∑T1 + ∑V1 = ∑T2 + ∑V2
V = VG + Ve
- VG is the Gravitational Potential Energy: VG = WYG
- Ve is the Elastic Potential Energy: Ve = .5 ks^2
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