Circular Motion

2D Forces and Motion

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

Merry-Go-Round

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

Animals further out have a greater linear speed.

Rotational Speed

Also calledAngular SpeedCircular Speedω (lower case

Omega Ω)

Linear Speed (v)Also calledTangential Speedv or vT

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

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 Lab QUESTIONS:

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

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

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

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

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

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

Tangential Speed

€

vT = CT

= 2πrT

HOW IS RADIUS RELATED 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 TO REVOLUTION SPEED?

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

smaller rotational speed.

v

r

€

lrθdldt

= drθdt

vT = r dθdt

vT = rω

Rotational Speed

€

vT

r€

vT = 2πrT

€

2πrTr

= 2πT

€

T = 1f

€

2πf

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

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

constant force inward. You reduce the radius.

WHAT IS THE DIRECTION OF ACCELERATION OF AN OBJECT

IN UNIFORM CIRCULAR MOTION?

€

a = ΔvΔt

€

Δvv

= Δrr

€

Δv = Δrr

v

€

ar = vΔrurΔt

= v2

r

WHAT IS THE DIRECTION OF ACCELERATION OF AN OBJECT

IN UNIFORM CIRCULAR MOTION?

€

aT =d vdt

€

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 vdt

ˆ θ − v2

rˆ r

€

a = a r2 + aT

2

€

θtan−1 aT

ar

⎛ ⎝ ⎜

⎞ ⎠ ⎟

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

and the bottom weight

Centripetal Force Fc

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

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

Centripetal means “center-seeking”Always acts inwards

Centripetal Acceleration ac

2T

cvar

Tangential Velocity

Radius

Centripetal Force

2

2

c c

c

Tc

F maF ma

varmvFr

The banked ramp exit

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

The banked ramp exit

Fg

FN FNy

FNx

€

Fy∑FNy − Fg = 0

€

Fx∑FNx = mac

€

FNy = Fg

€

FN cosθ = mg

€

FN sinθ = mac

€

FN = mgcosθ

€

mgcosθ

sinθ = mac

€

gtanθ = ac

€

θtan−1 ac

g ⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

θtan−1 v2

rg

⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

θ ≈20°