Rotational Motion and the Law of Gravity
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