Rotational Motion and the

Law of Gravity

= angle from 0r = radius of circles = arc length (in radians)

= (in radians) r

s

If you go all of the way around the circle,

so for s = 360° = 2r, so =r

r2

(rad) = (°) 180

= 2 rad

Angular Displacement

r

s

angular displacement (rad) = change in arc length radius

If an ant is crawling around the lip of a coffee cup that is 15 cm wide, and walks for 3 cm, what is its angular displacement in radians and degrees?

r = 15 cm/2 = 7.5 cm s = 3 cm

cm

cm

r

s

5.7

3

= .4 rad

)(180

(deg) rad

= 22.9°

Angular Speed

Just like linear speed is displacement over time v = x/t

Average angular speed is angular displacement over time avg = t

avg is “omega average”

My Mustang has 19” rims. If the tire spins with an angular speed of 45 rad/s, and you time it for 7 s, How many times did it spin?

avg = t avg = 45rad/s t = 7 s

= avgt = 45 rad/s (7s) = 315 rad

One spin = 2 rad 315 rad/2 = 50 spins

Angular Acceleration

Just like linear acceleration is velocity / timea = v/t

Angular acceleration is angular speed / time= t

= “alpha”

Tangential Speed

= t, , a b

Which point is moving faster, a or b?

vt = r

at = r

If time is kept small, tangential speed and acceleration are tangent to the radius

vt

at

Centripetal Acceleration

An object moving in a circle is accelerated toward the center

2

2

ra

r

va

c

tc

Forces that maintain circular motion

Something moving in a circular path has a tendency to want to continue in a tangent path. A force must hold it in its circular path.

22

mrFr

mvF c

tc

Universal Gravitation

All matter creates a force of gravity with other matter

2

21

r

mmGFg

2

21110673.6

kg

NmxG

2

21

r

mmGFg

Fg = mg

2

21

r

mmGmg

2E

E

r

mGg

Find your mass (kg) (weight in lbs ÷ 2.2 kg/lb)

Find “g” on Mars and your weight there, if Mars has a mass of 6.424 x 1023Kg and average radius of 3.398 x 106 m.

2Mars

Mars

r

mGg 2

26

23

2

211 /71.3

)10398.3(

10424.6)10673.6( sm

mx

kgx

kg

Nmx

Fg = myou(3.71m/s2) x .455 lb/kg