CHAPTER 9 : ROTATIONAL MOTION
9-10 Why Does a Rolling Sphere Slow Down?
9-8 Rotational Kinetic Energy
9-1 Angular Quantities
9-9 Rotational Plus Translational Motion; Rolling
9-2 Vector Nature of Angular Quantities
9-4 Torque
9-6 Solving Problems in Rotational Dynamics
9-3 Constant Angular Acceleration
9-5 Rotational Dynamics; Torque and Rotational Inertia
9-7 Determining Moments of Inertia
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
- 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.
Torque = R┴ F = RF┴.
τ = mR^2α
I = mi Ri^2
- A large-diameter cylinder has greater rotational inertia than one of equal mass but smaller diameter.
- 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.
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
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
v = R ω
No real sphere is perfectly rigid.
The bottom will deform, and the normal force will create a torque that slows the sphere.