# 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°
• 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
• ﻿$\alpha _{avg}=\Delta w/\Delta t$﻿ (SI Unit is given as rad/﻿$s_{2}$﻿ )
• 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

﻿$V_{tangent}=s/t=r\Theta /t=r(\Theta /t)$﻿ ﻿$\Rightarrow w=\Theta /t$﻿ ﻿$\Rightarrow$﻿ ﻿$V_{tan}=wr$﻿

﻿$A_{t}=(v_{T}-v_{T_{o}}/t)=\left [ (rw)-(rw_{o}) \right ]/t=r((w-w_{o})/t)$﻿ ﻿$\Rightarrow a_{T}=r\alpha$﻿

### Centripetal Acceleration – Tangential Acceleration

• ﻿$a_{rad}=(v_{T})_{2}/r=(wr)^{\Lambda }2/r=rw_{2}$﻿
• ﻿$a_{T}=r\alpha$﻿ also if the net tangential force FT is known, ﻿$a_{T}=F_{T}/m$﻿
• ﻿$a=\gamma (a_{rad2}+a_{T2})$﻿ and φ ﻿$=tan_{-1} = (a_{T}/a_{rad})$﻿

### Kinetic Energy of Rotation and Moment of Inertia

• ﻿$KE=\frac{1}{2}mv_{T2}=\frac{1}{2}mr_{2}w_{2}$﻿
• ﻿$KE=\sum (\frac{1}{2}mr_{2}w_{2})=\frac{1}{2}\sum (mr_{2})w_{2}= \frac{1}{2}|w_{2}$﻿
• I yields the moment of inertia given:
• ﻿$| = m_{A}r_{A2}+m_{B}r_{B2}+ ...$﻿ (SI Unit is kg ﻿$\cdot$﻿ ﻿$m_{2}$﻿ )
• Rotational Kinetic Energy:
• ﻿$k_{R}=\frac{1}{2}Iw_{2}$﻿
• Potential Energy of a Rigid Body
• ﻿$U=M g y_{cm}$﻿
• 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 ﻿$\rightarrow$﻿ ﻿$I = mr_{2}$﻿ ﻿$I = mr \sub 2$﻿

### Parallel Axis Theorem

﻿$I=$﻿ ﻿$I_{cm}+Mh_{2}$﻿ for that ﻿$M$﻿ is the total mass and ﻿$h$﻿ 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

﻿$K_{total}=\frac{1}{2}Mv_{cm2}+\frac{1}{2}I_{cm}w2$﻿