95.141 Apr 3 , 2013 PHYSICS I Lecture 17faculty.uml.edu/pchowdhury/95.141/Lectures/LECTURE17.pdf ·...

CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 17

Course website: faculty.uml.edu/pchowdhury/95.141/

www.masteringphysics.com

Course: UML95141SPRING2013

Lecture Captureh"p://echo360.uml.edu/chowdhury2013/physics1Spring.html

95.141 Apr 3 , 2013 PHYSICS I Lecture 17

Last Lecture Today

Chapter 9 2-D collisions Systems of particles (extended objects) Center of mass

Chapter 10 Rotational Motion Rotational kinematics Rotational dynamics Torque


Center of Mass: Wrap-up

A 50 kg person stands on the right most edge of a uniform board of mass 25 kg and length 6 m, lying on a frictionless surface. She then walks to the other end of the board. How far does the board move?

50 kg 25 kg

25 kg 50 kg

c.m.


(m1 +m2 )xCM =m1x1 +m2x2

6m

3m 2m

1m

?

Rotational Motion In addition to translation, extended objects can rotate

Need to develop a vocabulary for

describing rotational motion

There is rotation everywhere you look in the universe, from the nuclei of atoms to spiral galaxies


Angular Position Easiest to describe rotation in polar coordinates

y

x

R θ

! = arc length

! =!R

! = R!

R,!

R

Axis of rotation

θ in radians!


12 θθθ −=Δ

R

Axis of rotation Axis of rotation

θΔ

Angular Displacement


12

12

ttt −

−=

Δ

Δ=

θθθω

! = lim!t" 0

!"!t

=d"dt

dtd

ttωω

α =Δ

Δ→Δ

=0

lim12

12

ttt −

−=

Δ

Δ=

ωωωα

Angular Velocity & Acceleration Average angular velocity

Instantaneous angular velocity

Average angular acceleration

Instantaneous angular acceleration


Clicker Quiz

Two children are on a merry-go-round which is turning with a period of rotation of 20 s. If child A is 3 m and child B is 5 m from the axis of rotation, what is the difference in their angular velocities?

A)  0 rad/s B) π/10 rad/s C) π/20 rad/s D) 2π rad/s


An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle θ in the time t, through what angle did it rotate in the time t/2 ?

A) θ B) θ/2

C) θ/4

D) 2θ

E) 4θ

12


Clicker Quiz

Angular vs Linear •  Angular variables need an axis to be defined •  Each point on a rotating rigid body has the same

angular displacement, velocity, and acceleration! •  The corresponding linear (or tangential) variables

depend on the radius

vtan =d!dt d! = Rd!! = R! vtan = R

d!dt

atan =dvtandt

= R!

= R d!dt

= R!

!atotal =!atan +

!aRaR =vtan2

R=! 2R

!aR

!atan


Frequency and Period We can relate the angular velocity of rotation to the frequency of rotation:

1 rev/s =2! rad/s

f = 1T=!2"

1 hertz (Hz) = 1 rev/s


Vector Nature of Angular Quantities Both ω and α can be treated as vectors

– Choose vector in direction of axis of rotation – But which direction?

Right Hand Rule -Curl fingers on right hand to trace rotation of object -Direction of thumb is vector direction for angular velocity, acceleration -z

+z


Rotation vs. Translation

The equations of motion for translational and rotational motion (for constant acceleration) are identical

! =!o +"ot +12#t2

2oωω

ω+

=2

)(221

22

2

o

oo

oo

o

vvv

xxavv

attvxx

atvv

+=

−+=

++=

+=

( )oo θθαωω −+= 222

! =!o +"t

!! v

!! a

!! x

t! t


F

F

F F

F

F

No net force

No motion

rotation

F

F

F translation

translation

translation and rotation

Net force

Motion of extended objects


The diagram shows the top view of a door, hinge to the left and door-knob to the right. The same force F is applied differently to the door. In which case is the turning ability provided by the applied force about the rotation axis greatest?

Clicker Quiz

A B C

D

E


Torque : turning force

Force Lever arm Angle

F

axis θ r

r

F θ

What causes rotation? F

F

Rotational Dynamics

F

θ

Fsinθ

Fcosθ ! = rF sin"

F


Torque : turning force

axis r

Torque causes angular acceleration In analogy with F = ma

Rotational Dynamics

F

θ

Fsinθ

Fcosθ

! = I"

! = rF sin"

I is the rotational equivalent of mass Moment of Inertia or Rotational Inertia

! = r(F sin" )

! = F(rsin" )

θ

r sinθ


Summary

Rotational Motion Rotational kinematics Rotational dynamics Torque


95.141 Apr 3 , 2013 PHYSICS I Lecture 17faculty.uml.edu/pchowdhury/95.141/Lectures/LECTURE17.pdf ·...

Documents

Transcript of 95.141 Apr 3 , 2013 PHYSICS I Lecture 17faculty.uml.edu/pchowdhury/95.141/Lectures/LECTURE17.pdf ·...