Rotational Motion and the Law of Gravity
7.1 Measuring Rotational Motion
 When something spins around a certain point its called –
circular motion
 A circle of radius “r” has a circumference 2πr and has an
angular measure of 360°
 Circular motion is describes in terms of the angle that the
object moves.
when the CD (compact disc) in Figure 1
rotates about its center—each point in the
object follows a circular arc. Consider a line
from the center of the CD to its edge. Each pit
used to record sound along this line moves
through the same angle in the same amount
of time. The rotation angle is the amount of
rotation and is analogous to linear distance.
We define the rotation angle Δθ to be the
ratio of the arc length to the radius of
curvature:Δθ=Δsr.
 Angles in physics are measured in radians (rad)
 The angular displacement in radians (∆ϴ) is the ratio
of the change in arc length (s) to the radius (r) of a circle
Formula:
∆ϴ = ∆s
r
Units cancel out
and rad is used
Conversions:
2π radians
And
=
360°
=
1 rev
1 rad = 1800/ π = 57.30
Angular Speed describes the rate of rotation
Formula:
ω = ∆ϴ / ∆ t
Units: rad / sec
 Greek letter omega (ω) = angular speed
Average angular speed = angular displacement / time interval
 When an object spins, we can describe how fast its going in
terms of either:
degrees/sec or revolutions/sec or radians/sec
Example Problem:
Earth makes one rotation (3600) about its axis
in one day (24 hours)
If 3600 = 2 π rad and 24 hours = 86400 sec
And 2 x π = 6.28
Then,
2 π rad / 86400 sec =
Earth’s angular speed = 7.27 x 10 -5 rad/sec
In a certain game show,
contestants spin a wheel
when it is their turn. One
contestant gives the wheel
an initial angular speed of
3.20 rad/s. It then rotates
through one-and-onequarter revolutions and
comes to rest on the
BANKRUPT space. Through
what angle has the wheel
turned when its angular
speed is 1.90 rad/s?
Angular acceleration occurs when angular speed changes
Formula α = ∆ω
∆t
Greek letter α = alpha
Average angular acceleration = change in angular
speed / time interval
Comparing angular and linear quantities
Linear
Angular
x
ϴ
v
ω
a
α
7.2 Tangential and Centripetal Acceleration
 Objects in circular motion have a tangential speed
 Any two objects have the same angular speed and
angular acceleration regardless of distance from
center but….
 If the two objects are different distances from the
axis of rotation, they have different tangential speeds
(instantaneous linear speed at that point)
Tangential Speed
Formula: distance from the axis (r) x instantaneous angular speed (ω)
Vtan = r x ω
tangential speed and angular speed
Tangential Acceleration
•instantaneous linear acceleration is tangent to the circular path
atan = r x α
tangential acceleration = distance from the axis x angular
acceleration
Centripetal Acceleration
• Centripetal - “towards the center”
• Since Velocity is a vector there are 2 ways an acceleration
can be produced:
• change in magnitude
• and/or change in direction
• For a car moving in a circular path with constant speed the
object is accelerating due to a change in direction.
• Experienced by any object that travels in a curved path
ac = vtan2
r
2
centripetal acceleration = (tangential speed)
distance from the axis
ac = r ● ω2
Since Vtan = r ωavg
And ac = Vtan2
r
Sooo,
Or
ac = (r2 ● ω2)
r
ac = r ● ω2
7-3 Causes of Circular Motion
 Force that maintains circular motion
Centripetal Force - any force towards center
Examples: Earth’s gravitational pull on moon or the electric
force that pulls electrons around atomic nuclei
According to Newton’s 2nd Law:
Fc = m x ac
or
Fc = m ● vtan2 or
Centripetal force
r
Fc = m x r x ω2
Units are in Newtons (N)
3 Types of Acceleration
1)Linear (tangential) – rate of change of speed… due
to change in speed
units: m/s2
2) Angular – rate of change of angular speed – due
to change in speed
units: rad/s2
3) Centripetal – a center-seeking acceleration – due
to change in direction
units: m/s2
http://www.pbs.org/opb/circus/cla
ssroom/circusphysics/centripetal-acceleration/
Newton’s Law of Universal Gravitation
Why do our planets stay in the sun’s orbit?
Why does the moon stay in orbit around the Earth?
 Mutual force of attraction between 2 objects
 According to Newton’s 2nd Law of Motion
Fg = G m1m2
r2
G = constant of universal gravitation = 6.673 x 10 -11 N m2
kg2
Launch Speed less
than 8000 m/s
Projectile falls to Earth
Launch Speed equal to 8000 m/s
Projectile orbits Earth - Circular
Path
Launch Speed less
than 8000 m/s
Projectile falls to Earth
Launch Speed greater than 8000 m/s
Projectile orbits Earth - Elliptical Path