Class 40: Moment of Inertia and Torque Course Evaluation: 1. Started - - PowerPoint PPT Presentation

Class 40: Moment of Inertia and Torque

Course Evaluation:

• 1. Started yesterday Apr 13th, ends Dec 29th (Wednesday).
• 2. Go to http://pa.as.uky.edu/
• 3. Click at “UNDERGRADUATES” in the top menu and then choose the

first item: Physics & Astronomy Course Evaluations

• 4. Follow instructions from there.
• 5. Make sure remember or write down any given key or password. You

need this to re-enter the system if you cannot finish the evaluation in

• ne time.

Velocity of Center of mass

total iz i i i i iz i i N 2 1 Nz N 2z 2 1z 1 CM CMz total iy i i i i iy i i N 2 1 Ny N 2y 2 1y 1 CM CMy total ix i i i i ix i i N 2 1 Nx N 2x 2 1x 1 CM CMx

M v m m v m m m m v m v m v m dt dy v M v m m v m m m m v m v m v m dt dy v M v m m v m m m m v m v m v m dt dx v

        

                             

Example Before collision: 2kg 4kg 4m/s 5m/s After collision: 2kg 4kg 1m/s #1 #2 #2 #1

• 1. v1f=?
• 2. What is the velocity of the CM before the collision?
• 3. What is the velocity of the CM after the collision?

v1f=?

Newton’s Second Law

for Translational Motion of a Rigid Body

t d r d M a M F

CM CM ext

    

We can “condense” the rigid body into a particle at the CM. This fictitious particle will follow Newton’s second law of motion with the external forces acting on the rigid body.

Examples

CM performing projectile motion

Newton’s Second Law

for Translational Motion of a Rigid Body

t d r d M a M F

CM CM ext

     We can “condense” the rigid body into a particle at the CM. This fictitious particle will follow Newton’s second law of motion with the external forces acting on the rigid body. F1 F2 F3 CM M F1 F3 M F2 This will give you 3 equations of motion (translational motion).

Example Write down the translational equations of motion for the CM in the following two cases: 1. 2.

Rolling without slipping Yo-yo

Rotational Motion In this class we will study only “simple” rotational motion: the axis of rotation is perpendicular to the page. In this simple case, angular velocity  and angular acceleration  can be considered as vectors along the axis

• f rotation, perpendicular to the page.

Constant angular acceleration

Translational Motion Rotational Motion x  v  a 

2 i i f

t a 2 1 t v x x    t

i f

     t a v v

i f

 

2 i i f

t 2 1 t        ) x (x 2a v v

i f 2 i 2 f

   ) ( 2

i f 2 i 2 f

       

Moment of Inertia Moment of inertia to rotational motion is like mass to translational motion. It measures the difficulty or easiness (inertia) in changing the angular velocity. The moment of inertia not only depends on mass (M), but it also depends on the mass distribution about the axis of rotation. So it depends on the shape of the object and the location and orientation of the axis of rotation.

dm r I

2





Unit of moment of inertia: kgm2

Moment of Inertia of Some Common Shapes