circular motion 2d forces and motion. which is faster? the horse on the outside or the horse on the...

Circular Circular MotionMotion

2D Forces and Motion2D Forces and Motion

Which is faster? The horse on Which is faster? The horse on the outside or the horse on the outside or the horse on

the inside?the inside?

Merry-Go-RoundMerry-Go-Round

Same Same rotational speedrotational speed for all animals for all animals on the Merry-Go-Round because they on the Merry-Go-Round because they are attached rigidly. are attached rigidly.

Animals further out have a greater Animals further out have a greater linear speed.linear speed.

Rotational SpeedRotational Speed

Also calledAlso calledAngular SpeedAngular SpeedCircular SpeedCircular Speedω (lower case ω (lower case

Omega Omega ΩΩ))

Linear Speed (v)Linear Speed (v)

Also calledAlso calledTangential Tangential

SpeedSpeedv or vv or vTT

Describe Earth’s motion using Describe Earth’s motion using the words rotate, revolve, and the words rotate, revolve, and

axis.axis.

Rotation: Spin. Axis is located within the object.

Days on Earth.Revolution: Object turns about an external axis.

Earth years.

Axis: Straight line around which rotation takes place.

Days on Earth.

Frequency vs. Period

Period (T)- The time it takes for one full rotation or revolution of an object in seconds.

Frequency (f)- The number or rotations or revolutions per unit time, measured in Hertz (Hz)

1

1

Tf

fT

Revolution LabRevolution Lab

QUESTIONS:QUESTIONS: 1) HOW IS RADIUS RELATED TO REVOLUTION SPEED?1) HOW IS RADIUS RELATED TO REVOLUTION SPEED? 2) WHAT HAPPENS WHEN YOU ARE SPINNING THE 2) WHAT HAPPENS WHEN YOU ARE SPINNING THE

STOPPER AT A CONSTANT RATE AND THEN SUDDENLY STOPPER AT A CONSTANT RATE AND THEN SUDDENLY PULL DOWN ON THE STRING? WHY DOES THIS PULL DOWN ON THE STRING? WHY DOES THIS HAPPEN?HAPPEN?

3) DOES A SPINNING OBJECT ACCELERATE? IF SO, 3) DOES A SPINNING OBJECT ACCELERATE? IF SO, WHAT IS THE DIRECTION OF ACCELERATION?WHAT IS THE DIRECTION OF ACCELERATION?

CHALLENGES:CHALLENGES: 1) Spin a rubber stopper above your head at several 1) Spin a rubber stopper above your head at several

different lengths to answer the question above. Do different lengths to answer the question above. Do multiple trials. multiple trials.

2) Graph your data to help find the mathematical 2) Graph your data to help find the mathematical relationship between radius and revolution speed. relationship between radius and revolution speed.

Tangential Speed

vT C

T

2r

T

HOW IS RADIUS RELATED HOW IS RADIUS RELATED TO REVOLUTION SPEED?TO REVOLUTION SPEED?

In Uniform Circular Motion (fixed tangential speed), a larger radius will result in a

smaller rotational speed.

v

r

vT

r

HOW IS RADIUS RELATED HOW IS RADIUS RELATED TO REVOLUTION SPEED?TO REVOLUTION SPEED?

In Uniform Circular Motion (fixed tangential speed), a larger radius will result in a

smaller rotational speed.

v

r

rddt

drdt

vT rddt

vT r

Rotational SpeedRotational Speed

vT

r

vT 2r

T

2r

Tr

2T

T 1

f

2f

WHAT HAPPENS WHEN YOU ARE WHAT HAPPENS WHEN YOU ARE SPINNING THE STOPPER AT A SPINNING THE STOPPER AT A CONSTANT RATE AND THEN CONSTANT RATE AND THEN

SUDDENLY PULL DOWN ON THE SUDDENLY PULL DOWN ON THE STRING? WHY DOES THIS HAPPEN?STRING? WHY DOES THIS HAPPEN?It spirals in because you apply a

constant force inward. You reduce the radius.

WHAT IS THE DIRECTION OF WHAT IS THE DIRECTION OF ACCELERATION OF AN OBJECT ACCELERATION OF AN OBJECT

IN UNIFORM CIRCULAR IN UNIFORM CIRCULAR MOTION?MOTION?

a v

t

v

v

r

r

v r

rv

ar vru

rt

v2

r

WHAT IS THE DIRECTION OF WHAT IS THE DIRECTION OF ACCELERATION OF AN OBJECT ACCELERATION OF AN OBJECT

IN UNIFORM CIRCULAR IN UNIFORM CIRCULAR MOTION?MOTION?

aT d v

dt

ar v2

r

There is a radial component of

acceleration responsible for the constant

direction change, and a tangential component of acceleration which

results in an increase or decrease in tangential

speed.

a d v

dtˆ

v2

rˆ r

a ar2 aT

2

tan 1 aT

ar

Describe the path of the stopper IF you Describe the path of the stopper IF you were to cut the string between the tube were to cut the string between the tube

and the bottom weightand the bottom weight

CentriCentripetalpetal Force F Force Fcc

A force of some kind is required to A force of some kind is required to maintain circular motion. Why?maintain circular motion. Why?

Any force that causes an object to Any force that causes an object to follow a circular path is called a follow a circular path is called a centripetal forcecentripetal force..

Centripetal means “center-seeking”Centripetal means “center-seeking”Always acts inwardsAlways acts inwards

Centripetal Centripetal Acceleration aAcceleration acc

2T

c

va

r

Tangential Velocity

Radius

Centripetal ForceCentripetal Force

2

2

c c

c

Tc

F ma

F ma

va

r

mvF

r

The banked ramp exitThe banked ramp exit

The goal is to design a banked ramp exit The goal is to design a banked ramp exit that drivers can round safely even on ice.that drivers can round safely even on ice. radius of curve is 50mradius of curve is 50m speed of cars- 13.4m/sspeed of cars- 13.4m/s What should the angle of the bank be?What should the angle of the bank be?

The banked ramp exitThe banked ramp exit

Fg

FN FNy

FNx

FyFNy Fg 0

FxFNx mac

FNy Fg

FN cos mg

FN sin mac

FN mg

cos

mg

cossin mac

gtan ac

tan 1 ac

g

tan 1 v2

rg

20

Review ProblemsReview Problems

Derive the expression (fully simplified) that will determine the

time it will take for a projectile launched on flat ground to reach its

maximum height.How long will it take to land?

t v0 sin

g

t 2v0 sin

g


What is the range

R?a)750mb)375mc)105md)210me)150m

v0x=15m/s

h=250m

R

c) 105m


What is the speed of the object when

it hits the ground?a)72m/sb)15m/sc)150m/sd)70m/se)21m/s

v0x=15m/s

h=250m

R

a) 72m/s


v0

A (peak)

What is the direction of the acceleration vector and velocity vector

at point A?a)0m/s2 and 0m/s d) a vb) a v e) a 0m/sc) a v

c)


A very agile physics student is standing on one of those spinny things in a

playground without slipping. Which force provides the student’s centripetal

acceleration?a)Normal Force d) Centrifugal Forceb) Weight e) Nonec) Friction on shoes f) Abnormal force

c)

Review ProblemsReview ProblemsTwo quarters are on a spinning turntable. One head side up and one tail side up.

Heads is at a distance R/2 from the center.Tails is a distance R from the center.

What is the ratio of accelerations (ah/at)?

a) 2/1b)1/2c) 1d)2^(1/2)e)2^-(1/2)

b)

ah

a t

vh2

rh

vt2

rt

vh

2

rh

rt

vt2

2rh2

rh

rt

2rt2

rh

rt

R

2R

1

2


AB

C

A B (along rope)

C

D (tangent to curve)

E

For the pendulum on the left, which vector on the right possibly shows the direction

of acceleration at point A?

C


AB

C

AB

C

DE

For the pendulum on the left, which vector on the right shows the direction of

acceleration at point B?

A


For the conical pendulum above, find a fully simplified expression for the period in

terms of theta, L, g, and other constants.

Lθ


L θ

v 2r

T

We need an equation with T in it!

What is r?r

r L sinOk, so what is v?

FTx mac

FTy Fg 0Take it easy, make an FBD.

T 2r

v


L θ

T 2r

vr

r L sin

FTx mac

FTy Fg 0

FT sin mv2

r

FT cos mg

mg

cossin m

v2

r

v rgtan


L θ

T 2r

vr

r L sin

v rgtan

T 2L sin

L singtan

T 2L cos

g

sintan

sinsincos

cos

L sinL sin

L sin

circular motion 2d forces and motion. which is faster? the horse on the outside or the horse on the...

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