QS(a) A centrifuge in Figure Q5(a) is often used for astronaut training activity. The radius arm of the centrifuge is 15m and initially rotates according to 8 = 0.3t² where t is in seconds and o is in radians. (i1) Determine the astronaut's angular velocity and linear speed when t = 3.5 s. (i) Determine the astronaut's magnitude of tangential and normal acceleration when t = 3.5 s. 15m Cockpit X. 2, 2 0 = 0.3t? Figure Q5(a)
You may use these formulas please try not to use other than these formulas
Fundamental Equation of Dynamics
KINEMATICS
Particle Rectilinear motion:
Equations of Motion:
Variable a
?=???? ?=???? ???=???
Constant a = ac ?=?0+??? ?=?0+?0?+12???2 ?2=?02+2??(?−?0)
Particle: Σ?=??
Rigid Body: Σ??=?(??)?
(Plane Motion) Σ??=?(??)?
Σ??=??? or Σ??=Σ(ℳ?)?
Particle Curvilinear Motion:
Cartesian Coordinates (x,y,z) ??=?̇ ??=?̈ ??=? ̇ ??=?̈ ??=?̇ ??=?̈
Polar Coordinates (r,θ,z) ??=? ̇ ??=?̈−??̇2 ??=??̇ ??=??̈+2?̇?̇ ??=?̇ ??=?̈
Normal-Tangential Coordinates (n,t,b)
?=?̇ ; ??=?̇=????? ; ??=?2?
?ℎ??? ?=[1+(??/??)2]32⁄|?2?/??2|
Principle of Work and Energy: ?1+?1−2=?2
Kinetic Energy
Particle: ?=12??2
Rigid Body: ?=12???2+ 12???2
(Plane Motion) or ?= 12???2
Work
Variable Force: ??=∫?cos???
Constant Force: ??=(?cos?)Δ?
Weight: ??=−?Δ?
Spring: ??= −(12??22−12??12)
A couple of Moment: ??=? Δ?
Power and Efficiency ?=????=??, ∈= ???????=???????
Relative Motion ?B=?A+?B/A ?B=?A+?B/A
Rigid Body Motion About a Fixed Axis
Conservation of Energy Theorem ?1+?1=?2+?2
Potential Energy ?=?g+?e
Where:
?g=± Wy, ?e=+12??2
Variable a ?=?????=???? ???=???
Constant a = ac ?=?0+??? ?=?0+?0?+12???2 ?2=?02+2??(?−?0)
Principle of Linear Impulse and Momentum:
Particle : ??1+Σ∫???= ??2
Rigid Body: ?(??)1+Σ∫???= ?(??)2
For Point P
s=θr , v=ωr , a?=αr , a?=ω2r
Relative General Plane Motion-Translating Axes ?B=?A+?B/A(pin)?B=?A+?B/A(pin)
Relative General Plane Motion-Tran. And Rot. Axis ?B=?A+?×?B/A+(?B/A)??? ?B=?A+?̇×?B/A+?×(?×?B/A)+2?×(?B/A)???+(?B/A)???
Kinetics
Mass Moment of Inertia ?=∫?2??
Parallel-Axis Theory ?=??+? ?2
The radius of Gyration ?=√??
Conservation of Linear Momentum: Σ(??)1=Σ(??)2
Coefficient of Restitution ?=(??)2−(??)2(??)1−(??)1
Principle of Angular Impulse and Momentum:
Particle : (??)1+Σ∫????= (??)2
Where ??=(?)(??)
Rigid Body : (??)1+Σ∫????= (??)2
(Plane Motion)
Where ??=??? (??)1+Σ∫????= (??)2
Where ??=???
Conservation of Angular Momentum Σ(?)1=Σ(?)2
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