Chapter 8: Rotational Motion

Parts of Rotational Motion

Pure Translational Motion

  • Pure Rotational Motion (around the axis of rotation)
  • The general motion is the combination of these 2 parts
  • Given that the rotational motion = 0 for linear motion

Angular Motion

  • Angular Displacement is the angle the object rotates through a rotational motion
  • Δθ = θ2 – θ1 (SI unit is given as radian)
  • 1 rad = 360° / 2π ≈ 57.3°
  • x rad = arc length / radius => (θ rad = s/r)
  • counter clockwise angular displacement is positive
  • vice versa for the negative direction
  • Angular Velocity
  • ϖ = Δθ / Δt (SI Unit is given as rad/s)
  • ω = lim Δt \rightarrow 0 Δθ / Δt
  • Angular Acceleration
  • αavg=Δw/Δt\alpha _{avg}=\Delta w/\Delta t (SI Unit is given as rad/s2s_{2} )
  • α = lim Δt-­‐> 0 Δω / Δt

The similar kinematic equations can be applied to the rotational motion given that the acceleration is constant.

Relationship between linear and angular quantities

Vtangent=s/t=rΘ/t=r(Θ/t)V_{tangent}=s/t=r\Theta /t=r(\Theta /t) w=Θ/t\Rightarrow w=\Theta /t \Rightarrow Vtan=wrV_{tan}=wr

At=(vTvTo/t)=[(rw)(rwo)]/t=r((wwo)/t)A_{t}=(v_{T}-v_{T_{o}}/t)=\left [ (rw)-(rw_{o}) \right ]/t=r((w-w_{o})/t) aT=rα\Rightarrow a_{T}=r\alpha 

Centripetal Acceleration – Tangential Acceleration

  • arad=(vT)2/r=(wr)Λ2/r=rw2a_{rad}=(v_{T})_{2}/r=(wr)^{\Lambda }2/r=rw_{2}
  • aT=rαa_{T}=r\alpha also if the net tangential force FT is known, aT=FT/ma_{T}=F_{T}/m
  • a=γ(arad2+aT2)a=\gamma (a_{rad2}+a_{T2}) and φ =tan1=(aT/arad)=tan_{-1} = (a_{T}/a_{rad}) 

Kinetic Energy of Rotation and Moment of Inertia

  • KE=12mvT2=12mr2w2KE=\frac{1}{2}mv_{T2}=\frac{1}{2}mr_{2}w_{2}
  • KE=(12mr2w2)=12(mr2)w2=12w2KE=\sum (\frac{1}{2}mr_{2}w_{2})=\frac{1}{2}\sum (mr_{2})w_{2}= \frac{1}{2}|w_{2}
  • I yields the moment of inertia given:
  • =mArA2+mBrB2+...| = m_{A}r_{A2}+m_{B}r_{B2}+ ... (SI Unit is kg \cdot m2 m_{2} )
  • Rotational Kinetic Energy:
  • kR=12Iw2k_{R}=\frac{1}{2}Iw_{2}
  • Potential Energy of a Rigid Body
  • U=MgycmU=M g y_{cm}
  • I = ½ mL for L is the length of the rod

Note that I is always changing depending on the axis of rotation and for mass constant

The following equation only applies for a particle \rightarrow I=mr2I = mr_{2} I=mr2I = mr \sub 2

Parallel Axis Theorem

I=I= Icm+Mh2I_{cm}+Mh_{2} for that MM is the total mass and hh is the perpendicular distance between the 2 axis

Rotation about a Moving Axis

Total kinetic energy is the sum of its transitional and rotational kinetic energies


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