physics 201: lecture 6, pg 1 lecture 6 today’s goals (ch 4.4-6) l discuss uniform and...

Physics 201: Lecture 6, Pg 1

Lecture 6Today’s Goals (Ch 4.4-6)Today’s Goals (Ch 4.4-6)

Discuss uniform and non-uniform circular motion Circular Motion Centripetal (or radial) acceleration (direction of v changes) Tangential acceleration (magnitude of v changes)

Relative motion and reference frames

1st Exam (Chapters 1-4, ~10 Multiple choice, 4 short answer)Where: 2103, 2223 & 2241 Chamberlin Hall (& quiet room)When: Monday, February 20 7:15-8:45 PMFormat: Closed book, one 8 x11” sheet, hand writtenElectronics: Any calculator is okay but no web/cell access Quiet room: Test anxiety, special accommodations, etc.Conflicts: E-mail for approval (on or before Monday, Feb 12th) , academic/UW athletic reasons only


Circular Motion is common so specialized terms

Angular position (CCW + CW -) Radius is r Arc distance s = r & ds = r d Tangential “velocity” vt = ds /dt Angular velocity, ≡ d/dt (CCW + CW -) vt = ds/dt = r d/dt = r

r

vt

s


Reformulating changes with vector notationCartesian Coordinates Polar Coordinates

ji

ji

dydxrd

yxr

rdrr if

r r

r

drdrrd

rr

rdrr if

For a very small change dr

r

θ r drdrrd

r drθ dr

rd rθ


Circular Motion (with constant |r|)

r and

r

vt

s

θ r drdrrd

θ

θ θ r

rv

dtdθr

dtdθr

dtdr

dtrd

rv ||


Uniform Circular Motion (with constant |r| and |v|)

Time to go once around is the “period” T Distance once around is 2 r Tangential “velocity” is vt = 2r/T

r

vt

s

222 rr

vrT

rv ||


Uniform Circular Motion (UCM) has only radial acceleration

UCM changes only the direction of v1. Particle doesn’t speed up or slow down!

2. Velocity is always tangential; acceleration perpendicular !

a

v v

|v||v |

vΔ

vv

tva

vv

ΔΔ

Δ

avg

vΔ


Uniform Circular Motion (UCM) has only radial acceleration

UCM changes only in the direction of v1. Particle doesn’t speed up or slow down!

2. Velocity is always tangential, acceleration perpendicular !

path radial aa

a

v v

dtdv

tva

t

Δ

Δlim0

c2

r

2r

/ arva

rrva


Again

Centripetal/radial Acceleration

-ac = ar = -v2/r

Circular motion involves continuous radial acceleration

v

r

ac

Uniform circular motion involves only changes in the direction of the velocity vector

Acceleration is perpendicular to the trajectory at any point, acceleration is only in the radial direction.


Mass-based separation with a centrifuge

How many g’s (1 g is ~10 m/s2)?|ar |= vt

2 / r = 2 r f = 6000 rpm = 100 rev. per second is typical with r = 0.10 m

ar = (2 102)2 x 0.10 m/s2

Before After

bb5

ar = 4 x 104 m/s2 or ca. 4000 g’s !!!

but a neutron star surface is at 1012 m/s2


Consequence of no radial acceleration…a demo

In this demonstration we have a ball tied to a string undergoing horizontal UCM (i.e. the ball has only radial acceleration)

1 Assuming you are looking from above, draw the orbit with the tangential velocity and the radial acceleration vectors sketched out.

2 Suddenly the string brakes.

3 Now sketch the trajectory with the velocity and acceleration vectors drawn again.


Concept test What does the path look like once the string is cut?

A:

B:

C: D:

E:

v

r

ac


Non UCM (with constant |r| and changing |v|)

The speed of the particle increases or decreases d|v|/dt ≠ 0

Always tangent to the path! r

vt

stangential

|| adtvd

rv ||


Acceleration with both speed and direction change

1. Particle speeds up or slows down!

2. Acceleration has tangential and radial components !

tva

rvtva

vv

t

t

ΔΔlim

/ΔΔlim

|'|||

t

0t

2r

0r

v

v

ra a

ta

tvrv

θr tr aaa


Non-uniform Circular MotionFor an object moving along a curved trajectory, with varying

speed

Vector addition: a = ar + at (radial and tangential)

ar

at

2tangential

2radial|a| aa

rv2

tradiala

dtvd ||a t

tangential

a


Total acceleration

rvT

A stunt plane is performing a loop-the-loop of radius 100 m while accelerating (see figure). When its nose is pointed directly down, the speed of the plane is 50 m/s and the acceleration, tangent to the path, is 2g (i.e., 20 m/s2).

What is magnitude of the total acceleration? In x,y vector notation, what is the total acceleration?

ji

j m/s 20 2t a

i m/s 100/50i m/s / 2222r rva

222/122 m/s 32m/s )2025( a

j m/s 20i m/s 25 22 a


Concept Check: Which answer is best

E1. You drop a ball from rest, how much of the acceleration from gravity goes to changing its speed?A. All of itB. Most of itC. Half of itD. None of it

E2. A hockey puck slides off the edge of a horizontal table, just at the instant it leaves the table, how much of the acceleration from gravity goes to changing its speed?A. All of itB. Most of itC. Half of itD. None of it

j ga

onaccelerati radial

av j ga v

v


Relative Motion and reference frames?

If you are moving relative to another person do you see the same physics?

Two observers moving relative to each other generally do not agree on the outcome of an experiment (path)

For example, observers A and B below see different paths for the ball


Reading Assignment

Chapter 5.1-6

physics 201: lecture 6, pg 1 lecture 6 today’s goals (ch 4.4-6) l discuss uniform and...

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