Circular Motion
8.01T
Sept 27, 2004
Position and Displacement
Position vector of an object
moving in a circular orbit of
radius R
G
Change in position ∆r
between
time t and time t+∆t . Position vector is changing in
direction not in magnitude.
Magnitude of Displacement
• The magnitude of the
displacement , is the
length of the chord of
the circle
G
∆r
= 2 R
sin(∆θ / 2)
Small Angle Approximation
• When the angle is small, approximate
sin ∆θ ≅ ∆θ
• infinite power series expansion
1
1
3
5
sin ∆θ = ∆θ − (∆θ ) + (∆θ ) − ...
3!
5!
• Using the small angle approximation, the
magnitude of the displacement is
G
∆r ≅ R ∆θ
Magnitude of Velocity and Angular Velocity
Magnitude of the velocity is proportional to the rate of
change of the magnitude of the angle with respect to
time
G
∆r
R ∆θ
∆θ
G
dθ
v ≡ v = lim
∆t →0
∆t
angular velocity
= lim
∆t →0
∆t
= R lim
dθ
ω≡
dt
units: [ rad-sec-1 ]
magnitude of velocity
v = Rω
∆t →0
∆t
=R
dt
Direction of Velocity G
• sequence of chord ∆r
directions as ∆t
approaches zero
• the direction of the
velocity at time t is
perpendicular to
position vector and
tangent to the circular
orbit
Acceleration
When an object moves in a circular orbit,
the acceleration has two components,
tangential and radial.
Tangential Acceleration
the tangential acceleration is just the rate of change of
the magnitude of the velocity
∆vθ
∆ω
dω
d 2θ
aθ = lim
= R lim
=R
=R 2
∆t →0 ∆t
∆t →0 ∆t
dt
dt
Angular acceleration: rate of change of angular velocity
with time
d ω d 2θ
= 2
α≡
dt
dt
−2
⎡
⎤⎦
rad
sec
⋅
units are ⎣
tangential acceleration
atan = Rα
Uniform Circular Motion
• object is constrained to move in a circle
and total tangential force acting on the
object is zero, then by Newton’s Second
Law, the tangential acceleration is zero
atan = 0
• magnitude of the velocity (speed) remains
constant
Period and Frequency
• The amount of time to complete one circular orbit of radius R is
called the period.
• In one period the object travels a distance equal to the
circumference,
s = 2π R = vT • Period:
T = 2π R / v = 2π R /Rω = 2π / ω
• Frequency is the inverse of the period
f = 1/ T = ω / 2π
• Units [sec-1] ≡ [Hz]
Radial Acceleration
• Any object traveling in
a circular orbit with a
constant speed is
always accelerating
towards the center
• Direction of velocity is constantly changing
Change in Magnitude of Velocity
G
• The velocity vector v (t + ∆t) has been
transported to the point P ,
G G
G
• change in velocity ∆v =
v(t + ∆t) − v(t)
Magnitude of Change in Velocity • The magnitude of the change in velocity is
G
∆v = 2v
sin(∆θ / 2)
• small angle approximation sin ∆θ ≅ ∆θ
•
Conclusion
G
∆v ≅ v ∆θ Magnitude of Radial Acceleration
• Magnitude
G
∆v
v ∆θ
∆θ
dθ
= lim
= v lim
=v
= vω
ar = lim
∆t →0 ∆t
∆t →0 ∆t
∆t →0 ∆t
dt
• Recall the magnitude of velocity
• Conclusion:
ar = R ω
2
v = Rω
Alternative Forms for Magnitude of Radial Acceleration
• Radius and speed
• Radius and frequency
• Radius and period
2
v
ar =
R
ar = 4π 2 R f 2
4π 2 R
ar = 2
T
Direction of Radial Acceleration • sequence of Gchord • directions ∆v as
∆t approaches zero
• perpendicular to the
velocity vector
• points radially inward Summary: Circular Motion
• arc length
s = Rθ
• tangential velocity
ds
dθ
=R
= Rω
v=
dt
dt
dvθ
d 2θ
• tangential acceleration aθ =
= R 2 = Rα
dt
dt
v2
2
= Rω
• centripetal acceleration ar = vω =
R
Cylindrical Coordinate System
• coordinates
(r ,θ , z )
• unit vectors
ˆ θˆ , zˆ )
(r,
• Unit vectors at
different points
Circular Motion Vectorial
Description
• Use plane polar coordinates
• Position
• Velocity
• Acceleration
G
r(t ) = Rr̂(t ) G
dθ ˆ
θ(t ) = Rωθ̂(t ) v(t ) = R
dt
G
a = ar r̂ + aθ θ̂
aθ = rα
ar = −rω = −(v / r)
2
2
Class Problem
Two objects of mass m are whirling around a shaft with a constant angular velocity ω. First object is a distance d from central axis, and second object is a distance 2d from the axis. You may ignore
the effect of gravity.
a) Draw separate force diagrams
for each object.
b) What is the tension in the string
between the shaft and the first object?
c) What is the tension in the string
between the first object and the second object?