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CHAPTER 9 : ROTATIONAL MOTION - Coggle Diagram
CHAPTER 9 : ROTATIONAL MOTION
9-10 Why Does a Rolling Sphere Slow Down?
No real sphere is perfectly rigid.
The bottom will deform, and the normal force will create a torque that slows the sphere.
9-8 Rotational Kinetic Energy
kinetic energy of a rotating object :
K = Σ (1/2) mv^2 = 1/2 (I ω^2)
object that both translational and rotational kinetic energy:
K = (1/2) m vcm^2 + 1/2 (Icm ω^2)
W = T ∆θ
9-1 Angular Quantities
Angular displacement, Δθ = θ2 – θ1.
angular velocity, ω = Δθ / Δt
instantaneous angular
velocity, ω = dθ / dt
angular acceleration, α= Δω / Δt
instantaneous acceleration, α = dω / dt
v = Rω
*Objects farther from the axis of rotation will move faster.
tangential
acceleration ( If the angular velocity of a
rotating object changes ), atan = dv/dt = R (dω /dt) =Rα
centripetal
acceleration (Even if the angular velocity is constant), αR = v^2 / R = (Rω)^2 / R = ω^2 R
frequency, f = ω / 2π
1Hz= 1 s^-1
T = 1 / f
9-9 Rotational Plus Translational Motion; Rolling
v = R ω
9-2 Vector Nature of Angular Quantities
The angular velocity vector points along the axis
of rotation ( use right-hand rule for the direction )
If the direction he direction of the rotation axis
does not change, angular acceleration vector
points along it as well.
9-4 Torque
Torque = R┴ F = RF┴.
9-6 Solving Problems in Rotational Dynamics
Draw a diagram.
Decide what the system comprises.
Draw a free-body diagram, , including all the forces
acting on it and where they act.
Find the axis of rotation; calculate the torques
around it.
Apply Newton’s second law for
rotation
. If
the
rotational inertia is not provided
, need to find it before proceeding with this step.
Apply Newton’s second law for translation
and other laws and principles as needed.
Solve it and check the answer for units and correct
order of magnitude.
9-3 Constant Angular Acceleration
9-5 Rotational Dynamics; Torque and Rotational Inertia
τ = mR^2α
I = mi Ri^2
A large-diameter cylinder has greater rotational inertia than one of equal mass but smaller diameter.
9-7 Determining Moments of Inertia
I = ∫ R^2 dm
The parallel-axis theorem gives the
moment of inertia about any axis parallel to an axis that goes through the
center of mass of an object:
I = Icm + Mh^2
